Chaos-enhanced manganese electrolysis: nodule suppression and improved efficiency using controllable chaotic electrical signals

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# Chaos-enhanced manganese electrolysis: nodule suppression and improved efficiency using controllable chaotic electrical signals

Jie Yang 1,5 , Chunbiao Li 3,5 , Qian Zhang 1 , Zhihao Wu 1 , Xin Zhang 3 , Peiqiao Liu 1 ,
Zuohua Liu 1,2  , Changyuan Tao 1,2 , Guocan Zheng 1 , Yong Yang 1,4 & Hanke Wei 1,4

The control and industrial application of chaotic systems is a major obstacle limiting the diffusion
of chaos theory. In this study, we proposed a novel, universally applicable methodology for
constructing an offset boosting function for chaotic systems. By integrating this approach with
traditional techniques, a four-dimensional chaotic system with two-dimensional offset boosting was
developed and successfully implemented by a real chaotic circuit for manganese metal electrolysis,
replacing conventional DC. It has been shown that the use of time-varying electricity facilitates the
suppression of electrochemical oscillations, and inhibits the growth of spherical manganese nodules.
An examination of current efficiency for different oscillations (period 1, period 2, chaos-a and chaos-b)
and various current amplitudes has established that chaos-a electrical signals are most suitable for
inhibiting the growth of manganese nodules. The Mn nodule area ratios can be reduced by 38% with
a 5.83% increase in current efficiency, representing an energy consumption saving of 0.58 kWh/kg.
This interdisciplinary approach holds promise for advancing the study of nonlinear dynamic behavior
in electrochemical reaction processes and addressing critical challenges in various fields, such as
energy dissipation, quality improvement of electrodeposited products, and regulation of by-product
properties.

Keywords Chaotic system, System control, Chaotic-circuit application, Electrodeposition, Manganese,
Current efficiency

Background and significance
Electrodeposition, as a widely applied technique for metal fabrication, plays a crucial role in industrial production
and fundamental scientific research 1,2 . However, conventional direct current (DC) electroplating methods can
suffer from low efficiency, high energy consumption, and difficulty in terms of precision in controlling the
morphology of the deposited products 3,4 . Improving the current efficiency in high-energy-consuming industries
is a pressing issue as the cost of electrical energy is a major component of product costs, representing a critical
requirement for sustainable development 5 .

This study considers the design and implementation of a novel chaotic circuit with controllability features
that include amplitude control, multi-waveform output, and two-dimensional offset boosting, to meet complex
engineering requirements. The developed chaotic circuit is applied to a manganese metal electrolysis system,
replacing traditional DC sources, in order to explore the possibility of suppressing electrochemical oscillations
and controlling the growth of manganese nodules. The work undertaken has provided the first experimental
evidence of the benefit of chaotic currents relative to conventional DC and other forms of currents (such as

1School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 400044, China. 2State
Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing 400044, China. 3 School of Artificial
Intelligence, Nanjing University of Information Science and Technology, Nanjing 210044, China. 4 South manganese
group limited, Chongzuo 532315, China. 5 Jie Yang and Chunbiao Li have contributed equally to this study.  email:
liuzuohua@cqu.edu.cn

periodic currents) in enhancing current efficiency, reducing energy consumption, and improving product
quality.

This study establishes the unique benefits of chaotic currents in electroplating, providing empirical support
for application in complex electrochemical systems 6–8 . An improvement in electroplating efficiency with lower
energy consumption is critical on an industrial scale, opening new avenues for sustainable development in
high-energy-consuming industries, enabling energy conservation with associated economic benefits. This work
advances the development of natural sciences and supports technological progress with energy conservation in
high-energy-consuming industries.

Literature review
Systems change over time, notably those in engineering, physics, chemistry, biology, electromagnetism, and
those associated with atmospheric and geological phenomena, which has led to the concept of dynamic systems 9 .
There is a significant number of nonlinear motions in nature, which are described by nonlinear equations
including ordinary differential, partial differential, and algebraic equations 10,11 . The chaotic state represents a
special case in nonlinear systems because of its complexity and infinite periodic characteristics. Chaos theory
may be considered an evolutionary theory between orderly and disorderly state systems. An exploration of
chaos requires an analysis of irregular and unpredictable phenomena and processes 12,13 . Chaos theory, relativity
and quantum mechanics are recognized as the greatest discoveries in the 20th century and masterpieces of the
scientific community 14,15 .

Chaotic systems represent a distinctive class of nonlinear systems with initial value-sensitive dynamics, where
a minor alteration may have a considerable impact on the dynamic behavior 16,17 . Consequently, the capacity to
exert stable control over chaotic systems represents a pivotal advancement in expanding the possible applications
of chaotic circuits. The most prevalent control methods include amplitude control and offset boosting, which are
employed in the context of discrete 18–20 and continuous chaotic systems 21,22 . A system with two simultaneous
bias directions is termed a two-dimensional offset boosting 23, and a chaotic system with both amplitude and
offset controllability is designated a completely controllable system 24 . Leutcho G. D. et al. 25–27 have conducted
extensive research on amplitude control, offset boosting, and the multistable evolution of chaotic systems.

Manganese is regarded as a strategic metal element, commonly employed as a fundamental ingredient for the
fabrication of diverse alloys and novel energy materials 28–31 . The process of metal preparation by electrodeposition
necessitates a significant consumption of electricity which is an important cost factor for electrolysis processes 32 .
An improvement in current efficiency (CE) and reduction of energy consumption (EC) are essential commercial
considerations 33 . Current efficiency is an important factor in energy consumption that is influenced by a range
of factors. Chen 34 employed a slurry electrolysis system to treat reclaimed copper smelting fly ash with a CE of
91.22%, enabling an effective recovery of copper. At an extended electrolysis time, the pH in both the anode and
cathode regions decreases, with a more significant drop at the anode. Xu 35 has reported an aluminum extraction
from AlCl3-NaCl molten salts resulting in a deposited aluminum coating with a purity of 99.7%, and a CE of
44-64%. The decrease in CE was mainly related to the formation of dendritic or powdery aluminum deposits
that readily entered the electrolyte.

Product quality and production capacity are essential factors in commercial manganese metal
electrodeposition processes. Optimizing production capacity while guaranteeing product quality is challenging.
Manganese electrodeposition is a non-equilibrium kinetic process, where temporal, spatial and energetic
variations are complex 36,37 . Electric energy serves as the primary source of energy input for the electrodeposition
process, and the primary driving force for the spatially oriented deposition of manganese nodules. Therefore,
it is theoretically feasible to modulate the kinetic behavior of manganese metal through variations in the input
characteristics of the current waveform, with the potential to extend this technique to a broader range of
electrochemical applications.

We have recently completed three phases of research on the chaotic current (CC) electrodeposition of
manganese metal. In the first stage, the design and associated problems with CC electrodeposition equipment
were overcome, and the electrodeposition of manganese metal was successfully realized with self-designed
chaotic circuits and power amplifiers. Three issues were identified in the initial iteration of chaotic electrolysis.
The first issue relates to the narrow CC fluctuation range, which complicates a differentiation from DC
electrolysis. The second issue pertains to the self-designed current amplifier module, which exhibits a heating
problem with a consequent loss of output stability. The third issue is associated with the difficulty in effectively
suppressing electrochemical oscillations 38 .

In the second phase of the research, a novel chaotic circuit with a broader voltage range was designed to
address the limitation of the narrow fluctuation CC range. To overcome the problem of unstable operation, a
commercial power amplifier was utilized to achieve a stable and continuous output of chaotic electrical signals. The
results have demonstrated a complete suppression of electrochemical oscillation for the first time. To overcome
the difficulty of quantifying Mn nodules, a semi-quantitative calculation of the nodule distribution area was
achieved using a self-designed image processing technique. It has been demonstrated that a significant number
of manganese nodules are formed when the current density is excessively high 39 . The growth of manganese
nodules results in the dissipation of energy, a reduction in product quality, a decline in current efficiency, and
an increase in energy consumption. The inhibition of Mn nodule growth at higher current represents a pivotal
factor in enhancing production capacity. However, the extent of the CC remains unregulated.

The third stage focused on the effect of current fluctuation range on Mn electrodeposition. A chaotic circuit
with complete control (amplitude control and offset boosting) was designed and systematically investigated. It
was demonstrated that a CC with a larger fluctuation range contributes to improved electrodeposition CE 40 .
Nevertheless, the drawback associated with effect of the current waveform and electrodeposition time has not
been investigated.

Innovation and contributions
Building on the findings of the preceding research, four new studies are presented in this paper. A comparison
with reported literature and previous work is shown in Table 1:

The principal contributions of these studies are as follows:

1. To satisfy complex engineering requirements, a new chaotic circuit with two-dimensional offset boosting

has been devised to enhance the adaptability of chaotic power. The current can be regulated by the offset
boosting voltage dimension, which streamlines the configuration of chaotic power. In addition, this study
reports for the first time the use of offset boosting technology to achieve control over the current magnitude,
replacing the conventional series resistor voltage divider module and optimizing the chaotic power supply
electrolysis apparatus.
2. For the first time, this study investigates the impact of different waveform currents on the electrodeposition

behavior of metallic manganese. The significant influence of the electrical signal waveform on the current
fluctuation range of the current is established, demonstrating the superiority of CC over DC and periodic-state currents.
3. The effect of CC electrolysis time on the growth of manganese nodules is investigated for the first time. The

gradual growth of spherical nodules was monitored using microscopy. It was found that Mn nodule growth
can be significantly inhibited by CC at varying electrolysis times. In addition, the appreciable impact of electrolysis time on the electrolysis CE has been established. It is proposed that the effective inhibition of manganese nodule growth and improved CE can be achieved by altering the initial interfacial electrochemical
reactions through CC.
4. It is proposed that the inhibition of manganese nodules can improve product quality, which is supported by

X-ray photoelectron spectroscopy (XPS) analysis showing differences in the valence and elemental distributions of metallic manganese products with varying nodule content. This suggests that CC has the potential to
improve the quality of manganese products by effectively inhibiting nodule growth.

Methods
Reagents and materials
The experimental setup involved one cathode and one anode. The cathode consisted of a single rectangular plate
of stainless-steel type 304 having an area (S = 2 cm × 2 cm × 1 sides) of 4 cm 2 and a thickness of 0.5 mm. The
anode consisted of plates of a lead-based quaternary alloy (PbSn0.4Ag0.014Sb0.014) with a total area (S = 2 cm ×2 cm
× 2 sides) of 8 cm 2, and an approximate thickness of 3 mm. The electrolytic cell, formed from an acrylic sheet
had an internal length of 11.5 cm, width of 3.6 cm, and height of 6.2 cm. The distance between the cathode and
anode was 5 cm. A nonwoven fabric layer separated the cathode and anode to ensure a stable pH. Purified water
was prepared using a water purification system (HMC − WS10, Human Company, South Korea). The manganese
sulfate electrolyte was composed of 92 g/L MnSO4, 120 g/L (NH4)2SO4 and 30 mg/L SeO2 (analytical pure
reagents). Aqueous ammonia was used to adjust the pH to 7, and simulate electrolysis in industrial manganese
electrodeposition applications. As temperature influences ion hydrolysis, and affects pH, the solution was heated
to 40 °C before adjusting the pH to 7.

Electrochemical device
Electrochemical experiments were performed using a two − electrode electrochemical workstation (CHI660E,
Shanghai ChenHua Co., China). The distance between the anode, cathode and reference electrode was
2 ± 0.02 cm. A square cathode was used in the electrochemical oscillation tests, with a thickness of 1 mm and an
area of 1 cm 2 ; the anode was cone − shaped and immersed in the solution.

Amplitude
control Ofset boostingf Hardware Application domain

Source

Coexisting
attractors

Three-dimensional offset
Zhang et al. 41 No Yes Yes No
boosting

Chen et al. 42 Yes No Initial-offset boosting Yes Neuron model

Li et al. 43 Yes Yes No Yes Review article

Jiri Petrzela 44 Yes No No Yes Lumped chaotic systems

Qiu et al. 45 Yes No No Yes Secure communication

Xie et al. 46 No No No Yes

Reduced amplitude of electrochemical oscillations.
Yang et al. 38 Yes No No Yes

Yang et al. 39 No No No Yes First complete suppression of electrochemical oscillations.

A systematic study of the effect of current fluctuation
Yang et al. 40 Yes Yes No Yes
amplitude on Mn metal electrodeposition.

The effect of waveform and time on the current efficiency
This work Yes Yes Yes Yes
of Mn metal was studied for the first time.

Table 1 . Chaotic system reported in previous literature.

Characterization methods
The phase composition was analyzed by the X-ray diffraction (XRD, PANalytical X’Pert Powder, Panalytical
B.V., Netherlands). The microstructure was determined using scanning electron microscopy (SEM, JSM-7800 F,
JEOL, Japan). Analysis of the chemical composition and structure of the electrodeposition products employed
X-ray photoelectron spectroscopy (XPS, Thermo Scientific K − Alpha, Thermo Fisher Scientific, USA), Raman
spectroscopy (RS, LabRAM HR Evolution, HORIBA Jobin Yvon S.A.S., France), and transmission electron
microscopy (TEM, Talos F200S, Thermo Scientific, USA).

Data processing
Equations (1) and (2) were used to calculate current efficiency and energy consumption, respectively 47 .

∆ _M_ c
_ηc_ = _EMnI_ c∆ _t_ (1)

_WMn_ = _E_ _V_ Mn _×_ 1000 _× η_ c (2)

where _η_ c represents the cathodic current efficiency expressed as a percentage, ∆ _Mc_ /∆ _t_ (g/s) is the weight
gain of the cathode at the time interval∆ _t_, _Ic_ denotes current (A), and _E_ Mn is the electrochemical equivalent,
which is equal to 1.025 g/A·h for Mn. The _W_ Mn parameter represents the unit energy consumption of Mn
(kW·h/t), and _V_ is the cell voltage (V). To minimize experimental error, the average current and voltage during
the entire electrodeposition process are reported in this paper, based on data collected by the current and voltage
analyzer.

The path from theory to application
Method for constructing offset boosting function for any-chaotic system
A variety of tunable multi-channel chaotic signal sources are required to accommodate increasingly complex
systems and meet engineering requirements. The variable-boostable (VB) chaotic system, designated as VB11
(Eq. 3) 48, featured only one-dimensional offset boosting. Therefore, the incorporation of a two-dimensional
offset boosting can enable the output of more chaotic signals, thereby meeting diverse industrial requirements.

A new one-dimensional variable _w_ is introduced into the original chaotic system to achieve offset boosting.
Te format is as follows:

_dw_

_dt_ = _cz −_ _dw_ + _d_ 2 (3)

where _c_, _d_, and _d_ 2 are arbitrary constants.

To further validate the universal applicability of this method, the Lorenz system and the Chen system were
chosen as examples to undergo coupling computations with offset functionality (Table 2). It was observed that
the attractor projection structures of the original chaotic systems on the _x-z_ plane remained virtually unaltered
after the coupling, confirming that the dynamics of the original chaotic systems were not perturbed by this
method (Fig. 1a-f). Therefore, the signal characteristics of the original systems can be well-preserved.

Adjustments to parameters _c_ and _d_ enabled the scaling of signals in the _w_ -dimension of the chaotic system
(Fig. 1g-l) and resulted in attractor shifts in both the Lorenz and Chen systems. Moreover, the parameter _d_ 2
has been successfully employed to achieve arbitrary offsets in the _w_ -dimension for all three systems, which
demonstrated the feasibility and universality of this method (Fig. 1m-o). After transformation, the system VB 11
can achieve simultaneous offset boosting in the _x_ and _w_ dimensions.

Dynamic properties of the chaotic system
The Lyapunov exponent is a key indicator of whether a dynamical system is chaotic or not. A positive Lyapunov
exponent indicates that the system is in a chaotic state. The Lyapunov exponent of system (5) is LE1 = 0.06950,
LE2 = 0.0000, LE3 = -0.4674, LE4 = -10.0000 when _a_ = 1, _b_ = 2.75, _c_ = 12, _d_ = 10, and _d_ 2 = 0 the system is chaotic.

According to the chaos theory, a high value of Kaplan–Yorke dimension directly corresponds to high
complexity of system dynamics 51 . The corresponding Kaplan-Yorke dimension for the suggested system (5) is
computed as follows:

D _L_ = 3 + _LE_ 1 + _LE_ 2 + _LE_ 3 = 3 _._ 05369 (10)

_|LE_ 4 _|_

The fractional nature of the Kaplan-Yorke dimension is seen from Eq. (5). Thus, the attractor is chaotic because
of the fractional Lyapunov dimension 52 . For clear plots of attractors and Lyapunov exponents, the transient
process is ignored, where the initial 100,000 points are neglected.

To gain further insight into the dynamic behavior of the system with parameter variation, bifurcation
diagrams and Lyapunov exponents (Fig. 2a) were calculated. These demonstrate that system (5) exhibits clear
dynamic behavior, which includes the state of period 1 (Fig. 2b), period 2 (Fig. 2c), chaos-b (Fig. 2d), and chaos-a
(Fig. 2e). Moreover, the system also exhibits coexisting attractors that illustrate the initial value sensitivity of a

Chaotic system Primitive 3D-chaotic system Reconstructed 4D-chaotic system

_dzdt_ = _x_ + _by_

_dxdt_ = _y_ + _z_

_dy_

(4)

_dzdt_ = _x_ + _by_

~~~~




_dydt_ = _y_ 2 _−_ _az_

VB11 chaotic system 48

Lorenz chaotic system 49

Chen chaotic system 50

~~~~








~~~~








_dz_

_dxdt_ = _δ_ ( _y −_ _x_ )

_dz_

(5)

(7)

where _a_ = 1, _b_ = 2.7.

_dwdt_ = _cz_ _−_ _dw_ + _d_ 2

~~~~




_dzdt_ = _xy −_ _ηz_

_dxdt_ = _δ_ ( _y −_ _x_ )





~~~~

_dy_

(6)

_dzdt_ = _xy −_ _ηz_



~~~~




_dydt_ = _γx −_ _y −_ _xz_





_dydt_ = _γx −_ _y −_ _xz_

_dxdt_ = _A_ ( _y −_ _x_ )

_dz_

where _δ_ = 10, _η_ = 8/3, _γ_ = 28.





~~~~

_dwdt_ = _cz_ _−_ _dw_ + _d_ 2

_dzdt_ = _xy −_ _Bz_

_dxdt_ = _A_ ( _y −_ _x_ )

(9)





_dzdt_ = _xy −_ _Bz_



~~~~




_dydt_ = ( _C_ _−_ _A_ ) _x −_ _xz_ + _cy_

(8)

where _A_ = 35, _B_ = 3, _C_ = 28.

_dwdt_ = _cz_ _−_ _dw_ + _d_ 2

where c, d and _d_ 2 are constant terms for the system parameters.

Table 2 . Chaos system equations and their corresponding parameters.

chaotic system. The coexisting attractors and corresponding attraction basin are presented in Fig. 2f-k. The
waveform characteristics of different dimensional signals and corresponding frequency spectrum characteristics
(Fig. S1a, b), and chaotic attractor structures in various planes when the system is in a chaotic state are illustrated
in Fig. S1c-f.

System (5) has a special structure where the introduction of two new constants does not disturb the other
dimensions, providing the condition for offset boosting. To clearly show this mechanism, a constant _d_ 1 was
added to the variable _x_ and a constant _d_ 2 was added to the variable _w_ :






_dxdt_ = _y_ + _z_

_dx_

_dydt_ = _y_ 2 _−_ _az_

_dy_





_dzdt_ = _x_ + _d_ 1 + _by_

_dz_

(11)

_dwdt_ = _cz −_ _d_ ( _w_ + _d_ 2)

_dw_

where _a_ = 0.9, _b_ = 2.9, _c_ = 12, _d_ = 10, and the initial value was (0.4 0 0.1 0).

The chaotic attractor of system (11) can be controlled by the constant _d_ 1 located at various positions in the
direction of _x_, where the values in the other dimensions are not changed. The positions of the attractor in the
direction of _w_ can also be controlled by the constant _d_ 2 according to the same principle. The chaotic attractor of
system (12) can be controlled by the constant _d_ 1 locating at various positions in the direction of _x_ (Fig. S2a), and
the values in the other dimensions are not changed (Fig. S2b). The positions of the attractor in the direction of
_w_ can also be controlled by the constant _d_ 2 (Fig. S2e), and the values in the other dimensions are not changed
(Fig. S2f). The transformation does not result in a variation of the dynamic behavior of the chaotic system, which
has been proven by results of the Lyapunov exponents (Fig. S2d, h) and the bifurcation diagram (Fig. S2c, g).
Accordingly, the proposed two-dimensional offset boosting in both the _x_ and _w_ directions is established (Fig. 2l).

Considering the linear terms and nonlinear terms in system (11), special nonlinearity offers the merit of
complete amplitude control, including amplitude rescaling and easy offset boosting. Where _x_ → _mx_, _y_ → _my_, _z_ →
_mz_, _w_ → _mw_, _t_ → _t_ ( _k_ - 0), system (11) becomes:

Fig. 1 . Construction method for chaotic systems with independent one-dimensional offset functionality with
initial values (0.1 0.1 0.1) for 3D-systems, and (0.1 0.1 0.1 0.1) for 4D-systems. Chaotic attractors in the _x-z_
plane of the three-dimensional ( a ) VB system, ( b ) Lorenz system, ( c ) Chen system; Chaotic attractors in the
_x-z_ plane for the reconstructed four-dimensional ( d ) VB system, ( e ) Lorenz system, ( f ) Chen system; Influence
of parameter _c_ on the structures of ( g ) VB system, ( h ) Lorenz system, ( i ) Chen system; Influence of parameter
_d_ on the structures of ( j ) VB system, ( k ) Lorenz system, ( l ) Chen system; Influence of parameter _d_ 2 on the
structures of ( m ) VB system, ( n ) Lorenz system, ( o ) Chen system.

Fig. 2 . Dynamic evolution and complete control of a chaotic system. ( a ) Bifurcation diagram showing system
evolution as a function of varying parameter _b_ and the corresponding Lyapunov exponents. ( b ) Period 1 state
when parameter _b_ = 3. ( c ) Period 2 state when parameter _b_ = 2.9. ( d ) One form of chaotic state when parameter
_b_ = 2.75. ( e ) One form of chaotic state when parameter _b_ = 2.7. ( f ) Coexisting attractors of the system when
_b_ = 2.85. ( g ) Corresponding basin of attraction of the system on the _x_ - _z_ plane when _b_ = 2.85. ( h ) Coexisting
attractors of the system when _b_ = 2.86. ( i ) Corresponding basin of attraction of the system on the _x_ - _z_ plane
when _b_ = 2.86. ( j ) Coexisting attractors of the system when _b_ = 2.87. ( k ) Corresponding basin of attraction of
the system on the _x_ - _z_ plane when _b_ = 2.87. ( l ) Amplitude control in the _x_ - _w_ direction. ( m ) Two-dimensional
offset boosting of the attractor in the _x_ - _w_ direction. ( n ) Complete control of both the amplitude and offset
boosting.






_dxdt_ = _y_ + _z_

_dx_

_dydt_ = _my_ 2 _−_ _az_

_dy_





_dzdt_ = _x_ + _d_ 1 + _by_

_dz_

(12)

_dwdt_ = _cz −_ _d_ ( _w_ + _d_ 2)

_dw_

The coefficient _m_ in the last dimension controls the amplitude of the system state variables _x_, _y_, _z_ and _w_ without
revising the internal variable. The rescaling of the hyperchaotic attractor is realized by the parameter _m_ . The
change of _m_ results in a scaling of the state variables _x_, _y_, _z_ and _w_ (Fig. 2m). Application of system (12) offers
a complete control for system states, enabling precise manipulation of the amplitude and offset through the
corresponding non-bifurcation parameters (Fig. 2n). The Lyapunov exponents of the system remain constant
when _m_ is varied between [0, 3] (Fig. S2i). These results demonstrate that the variation of parameter _m_ does not
change the dynamic behavior of system (12).

To further investigate the effects of amplitude control and offset boosting on the basin of attraction structure
of coexistence chaotic attractors, calculations were performed across varying control parameters. These results
indicate that, as the amplitude control parameter _m_ is increased, the structure of the attractors will remain
unchanged while their amplitudes will be progressively reduced (Fig. 3a-c). Correspondingly, the basin structures
underwent gradual compression while preserving their morphological characteristics, indicating that amplitude
control can scale the basin without altering its inherent structure (Fig. 3d-f).

In this study, two offset boosting parameters, _d_ 1 and _d_ 2, were employed to independently control the arbitrary
shifts of the corresponding variables _x_ and _w_, respectively. The computational results pertaining to their
associated basins of attraction revealed that as _d_ 1 increased, the basin gradually shifted towards negative values,
with no variations for its structure (Fig. 3g-i). A reversal in the direction of this shift was observed when the sign
of _d_ 1 was altered, confirming that offset boosting can achieve an overall displacement of the attraction basin.

Furthermore, the basin structure on the _x-w_ plane exhibited a stripe pattern, suggesting that once the
parameters _x_, _y_, and _z_ are fixed, the value of _w_ has no influence on the structure of coexistence chaotic attractors.
This implies that the initial value sensitivity of the system is primarily determined by parameters _x_, _y_, and _z_
(Fig. 3j). It is also observed that variations in parameter _d_ 1 can result in a significant _x_ -direction shift of the
attraction basin, accompanied by notable structural variations (Fig. 3k). Because the basin of attraction in the _w_ direction is striped, the overall axial displacement cannot reveal any structural differences in the basin. Therefore,
although parameter _d_ 2 actually causes a shift in the system, it appears unchanged due to the consistent structure
(Fig. 3l). Overall, amplitude control and offset boosting techniques are found to enable arbitrary scaling and
shifting of the basin of attraction without altering the dynamical characteristics of chaotic systems.

Circuit implementation
Chaotic circuits were found to be the most effective means for transforming mathematical chaotic systems into
practical electrical signals. In this process, the constants within the differential equation systems were mapped
to the resistance values in the circuit, and the bias parameters were controlled by adjusting the DC power supply
voltages 53,54 . The analog electronic circuit of system (12) can be designed for circuit system (13). It is composed
of four adders, one multiplier, four inverters and four integrators.






_dx_

_dt_ = C 1 1

_dx_

C 1 1 ( _R_ _y_

_R_ _z_ 2 )

_dy_

_dt_ = C 1 2

_R_ _y_ 1 + _R_ _z_

_R_ _y_ 2 6 )

_dy_

C 1 2 ( _−_ _R_ _z_ 5

(13)





_dz_

_dt_ = C 1 3

_dz_

C 1 3 ( _R_ _x_

_dw_

_dt_ = C 1

_−_ _R_ _z_ 5 + _K_ _R_ _y_ 2 6

_R_ _x_ 9 + _Ry_ 11 )

_z_ _Rw_ 15 )

_R_ 14 _−_

_dw_

C 1 4 ( _z_

_R_

where _K_ is the gain coefficient which is equal to 0.1 in system (13). The parameters of the corresponding
electronic devices are _C_ 1 _= C_ 2 _= C_ 3 _= C_ 4 = 10 nF, _R_ 1 = _R_ 2 = _R_ 5 = _R_ 9 = _R_ 10 = _R_ 16 = 100 kΩ, _R_ 6 = 0 ~ 50 kΩ, _R_ 11 = 0 ~ 50
kΩ, _R_ 14 = 8.33 kΩ, _R_ 15 = 10 kΩ, and _R_ 7 = _R_ 8 = 10 kΩ.

The actual chaotic circuit is based on the simulation circuit (Fig. 4a, and Fig. S14), and the corresponding
voltage (detected using an oscilloscope) is shown in Fig. 4b-m, and Fig. S3. It is clear that the structure of the
electrical signal attractor is fully consistent with the theoretical results calculated by system (12). The system
realizes amplitude control (Fig. 4b-j) and offset boosting (Fig. 4k-m). Moreover, the state of the chaotic system
can also be regulated by adjustment of the resistance value of resistor _R_ 11, which includes period 1 (Fig. 4b-d),
period 2 (Fig. 4e-g) and chaos (Fig. 4h-j). It should be noted that an accurate control of CC can be achieved by
offset boosting, and the regulation of current density in all the electrodeposition experiments that are presented
have been conducted in accordance with this methodology.

Fig. 3 . Attraction basins of the coexisting attractors of cycle 1 and cycle 2 when _a_ = 1, _b_ = 2.86, _c_ = 12, _d_ = 10.
Amplitude control for the coexisting attractors of _x_ - _y_ where ( a ) m = 0.5, ( b ) m = 1, ( c ) m = 2; Effect of amplitude
control on the configuration of the attraction basins of _x_ - _y_ with initial values ( _x y_ 0.1 0.1) where ( d ) m = 0.5,
( e ) m = 1, ( f ) m = 2; Effect of offset boosting on attraction basins of _x_ - _y_ where ( g ) _d_ 1 = 0.03, ( h ) _d_ 1 = 0.1, ( i ) _d_ 1 =
-0.1; Effect of offset boosting on attraction basins of _x_ - _w_ with initial values ( _x_ 0.1 0.1 _w_ ) where ( j ) _d_ 1 = _d_ 2 = 0, ( k )
_d_ 1 = 0.01 and _d_ 2 = 0, ( l ) _d_ 1 = 0 and _d_ 2 = 0.1.

Dynamic regulation of electrodeposition
Electrochemical oscillations
Based on the previous work, the CC manganese metal electrodeposition device has been further improved 38–40 .
Firstly, the overall offset of the voltage waveform on an arbitrary scale is achieved using offset boosting, which
in turn enables precise control of the current magnitude, replacing the traditional module of controlling current
magnitude by series resistance voltage division (Fig. 5a, and Fig. S15). Secondly, the conventional current and
voltage sensing device has been replaced with a new multimeter (UT61E+, Uni-Trend Technology Co., Ltd.,
China), increasing the sensing range of the current and voltage signals.

The phenomenon of electrochemical oscillations is a typical example of nonlinear kinetic behavior, which
reflects the complex physicochemical variations that occur in the system 55,56 . Consequently, the regulation of
electrochemical oscillations represents a regulation of the kinetic behavior of the entire system. Significant
electrochemical oscillation electrical signals were detected in the 0.03–0.09 A interval with DC (Fig. 5b-e).

No electrochemical oscillation signals were detected with the use of period 1 (Fig. 5i), period 2 (Fig. 5h),
chaos-a (Fig. 5f), and chaos-b (Fig. 5g) currents. This serves to establish that the time-varying current can
regulate kinetic behavior during electrodeposition. Furthermore, synchronizing the voltage and current during

Fig. 4 . Complete control of an actual chaotic circuit electrical signal. ( a ) Electronic schematic of system (13).
( b ) Amplitude control of system (13) electrical signal waveform in period 1 state when _R_ 11 = 30 kΩ, _R_ 6 = 40
kΩ. ( c ) _R_ 11 = 30 kΩ, _R_ 6 = 30 kΩ. ( d ) _R_ 11 = 30 kΩ, _R_ 6 = 20 kΩ. ( e ) Amplitude control of system electrical signal
waveform in period 2 state when _R_ 11 = 35 kΩ, _R_ 6 = 40 kΩ. ( f ) _R_ 11 = 35 kΩ, _R_ 6 = 30 kΩ. ( g ) _R_ 11 = 35 kΩ, _R_ 6 = 20
kΩ. ( h ) Amplitude control of system electrical signal waveform in chaotic state when _R_ 11 = 37 kΩ, _R_ 6 = 40 kΩ.
( i ) _R_ 11 = 37 kΩ, _R_ 6 = 30 kΩ. ( j ) _R_ 11 = 37 kΩ, _R_ 6 = 20 kΩ. ( k ) Offset boosting of system electrical signal waveform
in chaotic state when _R_ 11 = 37 kΩ, _R_ 6 = 40 kΩ with 0 V offset voltage. ( l ) 2 V offset voltage. ( m ) 4 V offset
voltage.

the electrochemical oscillation test has revealed that the time-varying current can reduce the voltage to current
ratio when compared with DC. Moreover, the voltage-to-current ratio can be decreased with an increased
current density, consistent with the non-linear nature of electrodeposition (Fig. 5j). It can be concluded that the
voltage/current ratio of the chaotic state is lower than the period state.

Effect of current density
As the ions in solution are consumed continuously during the electrodeposition of manganese metal, the
properties of the electrode plate are also transformed. The current (Fig. 6a, d) and voltage (Fig. 6b, e) signals
generated during electrodeposition were collected and the average value calculated to ensure accurate calculated
CE and EC data. The results indicate that the fluctuation range of current and voltage signals in the chaotic circuit
is clearly greater (Fig. 6g, h). The CC voltages and currents employed in these experiments exhibit an overall
shift to the upper portion of the slope 39 (Fig. 6c). This response demonstrates that an overall shift of the currents
and voltages can be achieved by offset boosting. Furthermore, the voltage-current signals through phase-space
reconstruction of the DC electrolysis experiments with various current densities demonstrate that the range of

Fig. 5 . Chaotic power integration and kinetic behavior regulation for manganese electrodeposition employing
various currents. ( a ) Experimental set-up for electrodeposition by chaotic power. ( b ) Electrochemical
oscillatory signals with DC for 0.03 A. ( c ) DC for 0.05 A. ( d ) DC for 0.07 A. ( e ) DC for 0.09 A. ( f )
Electrochemical oscillatory signals with chaotic-a current for 0.07 A. ( g ) Chaotic-b current for 0.07 A. ( h )
Period 2 current for 0.07 A. ( i ) Period 1 current for 0.07 A. ( j ) Ratio of the voltage to current during the entire
electrochemical oscillation testing process.

voltage fluctuations under the influence of elevated current densities is amplified (Fig. 6f). This phenomenon
may be attributed to an intensified reaction and accelerated rate of ion depletion at higher currents. To accurately
quantify the fluctuation ranges of currents and voltages, the range of electrical signals at various current densities
was calculated. This has substantiated the exceptionally wide fluctuation range of chaotic currents (Fig. 6g, h).

Fig. 6 . Signal characteristics of various current densities in manganese metal electrodeposition systems.
( a ) Chaotic current signals. ( b ) Chaotic voltage signals. ( c ) Correlation of the current and voltage of chaotic
power. ( d ) Direct current signals. ( e ) Direct voltage signals. ( f ) Correlation of the current and voltage of DC.
( g ) Current fluctuation range for the two forms of power. ( h ) Voltage fluctuation range for thee two forms of
power.

In terms of cost reduction, it is significant that the CE can be improved and EC reduced by CC over a
wide range (1110 A/m 2 -1575 A/m 2 ) of current densities (Fig. 7a, b). The voltage/current ratio during actual
manganese metal electrodeposition has demonstrated that the increased current efficiency of chaotic currents
can be attributed to a reduction of the cell voltage (Fig. 7c). The results of the area distribution of manganese
nodules calculated using image processing have revealed that CC can achieve a significant suppression of
nodules over a wide range of current density variations (Fig. S4a-d). The reduced area ratio of nodules (AR-N)
was found to exceed 40%, with a maximum of 48.8% (Fig. 7d).

To more accurately evaluate the specific variations of improved CE and enhanced EC, the reduced ratio of
voltage to current and area of nodules was calculated (Fig. 7e, f). The results have shown that the trends with
respect to CE and EC exhibit peaks, indicating an optimal range of current density for chaotic electrolysis. When
the average current under CC was set to 0.508 A, the CE can be improved by 6.08%, the EC can be lowered by
0.84 kWh/kg, the ratio of voltage to current can be reduced by 0.4, and the area ratio of nodules can be inhibited
by 40.5%, which represents the best result reported to date 39,40 .

Effect of the waveform
The amplitude of the chaotic signal can also be controlled by adjustment of the amplification factor of the power
amplifier. The amplification factor of the output voltage is represented by 14 dB (voltage amplification by 14 times)
and 4 dB (4 times). The influence of voltage fluctuation amplitude and the waveform on the electrodeposition
process of manganese metal was evaluated at 4 dB and 14 dB with five types of waveforms (period 1, period 2,
chaos-a, chaos-b, and DC).

The current signals and voltage signals are illustrated in Fig. 8a, b, respectively, with a gain of 14 dB. When the
gain was 4 dB, the corresponding current and voltage signals are depicted in Fig. 8d, e. These results demonstrated
that the amplitude of the current and voltage can be modulated by power amplifiers, and the pattern of voltage

Fig. 7 . The relationship between current efficiency, energy consumption, ratio of voltage to current, and
area ratio of nodules with 2 × 2 cm 2 plates for various current densities. ( a ) Current efficiency. ( b ) Energy
consumption. ( c ) Ratio of voltage to current. ( d ) Area ratio of nodules. ( e ) Improved current efficiency and
lower energy consumption. ( f ) Reduced ratio of voltage to current and the reduced area ratio of nodules.

fluctuation range variation aligns consistently with the current variation. To better compare the fluctuation
ranges and correlations of current signals and voltage signals under various types of current excitations, the
reconstructed phase space of the signals was plotted with the current signals taken as the abscissa and the voltage
signals as the ordinate (Fig. 8c, f). The impact of the electrical signal waveform on the final output working
current is significant. It can be concluded that chaotic signals result in greater current fluctuations, while the
fluctuation range of periodic current is smaller. This response provides an additional rationale for the greater
efficiency of CC in the electrodeposition of manganese when compared with DC.

For a more precise assessment of current and voltage fluctuation ranges, the ranges (The differences between
maximum and minimum values) of both current and voltage signals were calculated during the electrodeposition
process. It is clear that, under constant direct current (DC) mode, the fluctuation range of the current signal
remains below 0.01 A during electrodeposition (Fig. 9a). Conversely, when a chaotic current is employed, the
current fluctuation range can extend to 0.25 A, with a corresponding voltage fluctuation of 2.03 V (Fig. 9a, b).
Significantly, this fluctuation range can be finely tuned by adjusting the gain factor of the power amplifier. The
influence of the voltage fluctuation amplitude on CE is greater than the impact of the waveform (Fig. 9c, and Fig.
S5c). A wider voltage fluctuation range is more conducive to manganese electrodeposition. On the other hand,
a chaotic waveform with a more complex structure can result in a larger fluctuation range of the output current,

Fig. 8 . Signal characteristics of various currents in manganese metal electrodeposition systems. ( a ) Chaotic
current signals amplified by a 14 dB power amplifier. ( b ) Chaotic voltage signals amplified by a 14 dB power
amplifier. ( c ) Correlation of the current and voltage of chaotic power amplified by a 14 dB power amplifier.
( d ) Chaotic current signals amplified by a 4 dB power amplifier. ( e ) Chaotic voltage signals amplified by a 4
dB power amplifier. ( f ) Correlation of the current and voltage of chaotic power amplified by a 4 dB power
amplifier.

which improves CE relative to a periodic electrical signal. The energy consumption should consider both current
efficiency and cell voltage, where the chaotic electrical signal with a large fluctuation range demonstrates the
most favorable effect (Fig. 9c-e, and Fig. S5d, e). The value of reduced EC can reach 0.58 kWh/kg compared with
DC electrolysis.

To demonstrate the relationship between the growth of manganese nodules and CE, nodule distribution area
and microscopic tests were carried out on manganese metal products under the action of different waveforms
and amplitudes (Fig. 9g-j). It is evident that the manganese nodules can be almost completely inhibited by the
current with a high fluctuation range (Fig. 9g, h), which serves to reduce the ratio of nodules by at least 26%
(Fig. 9f, and Fig. S5a, b, f).

This effect is corroborated by microscopic observations, where there were no discernible spherical manganese
nodules in all regions of the deposited manganese metal in the case of large fluctuating ranges of currents
(Fig. 9i, j, and Fig. S6). In contrast, a considerable number of spherical manganese nodules were observed in all
regions of the manganese metal following electrolysis by DC or CC with low fluctuating ranges. These findings
are consistent with the proposal that manganese nodules are the main cause of energy dissipation during
electrodeposition.

Effect of electrodeposition time
The effect of time was assessed in comparative tests with various electrodeposition times. The fluctuation range of
current signals showed little variation with the duration of electroplating (Fig. S7a, d). Moreover, the fluctuation
range of chaotic current-voltage signals and the attractors identified in phase space reconstruction consistently
demonstrated the stability of the fluctuation range of chaotic electrical signals (Fig. S7b, c). The cell voltage
decreased continuously with increasing electrodeposition time by DC (Fig. S7e, f). The increasing specific
surface area of the manganese metal as it is continuously deposited on the stainless-steel electrode plate may
account for this effect. It should be noted that the amplitude of the CC and voltage fluctuations also increased
with time (Fig. S7g, h, and Fig. S8).

The CE results demonstrate that the electrodeposition time exerts a significant influence on the manganese
metal electrodeposition process. The CE shows a decreasing tendency with increased electrodeposition time.
This may be attributed to a continuous depletion of manganese ions. The current efficiency of CC electrolysis
is 6.13% higher than DC electrolysis at an electrodeposition time of 30 min. In contrast, the current efficiency
of CC electrolysis is 4.34% lower than DC electrolysis at an electrodeposition time of 120 min (Fig. 10a, d). The
data for energy consumption align with the CE results (Fig. 10b, e). The application of CC can reduce energy
consumption by 0.95 kWh/kg for an electrodeposition time of 30 min. The ratio of voltage to current (R-VC)
results indicate that the chaotic power can reduce the cell voltage (Fig. 10c).

It is evident that the growth of manganese metal nodules is significantly inhibited by the CC, with a minimum
of 26% inhibition when compared with DC electrolysis, and a 47% inhibition for an electrodeposition time of
120 min (Fig. 10f).

Fig. 9 . Signal characteristics, current efficiency, energy consumption and the distribution of nodules for
current waveforms. ( a ) Current fluctuation range for the two forms of power. ( b ) Voltage fluctuation range. ( c )
Enhanced current efficiency by CC compared with DC. ( d ) Reduced energy consumption compared with DC.
( e ) Reduced ratio of the voltage to current. ( f ) Reduced area ratio of nodules. ( g ) Distribution of manganese
nodules by time-varying current amplification to 14 dB. ( h ) Distribution of manganese nodules by the timevarying current amplification to 4 dB. ( i ) Microscopy images of manganese metal with chaotic-a current
amplified to 14 dB. ( j ) Microscopy images of manganese metal with chaotic-a current amplified to 4 dB.

The results of the distribution of manganese nodules after the binarized extraction (Fig. 10g, h), and the
corresponding optical microscopy analysis have corroborated the findings of the area calculations for the
manganese nodules (Fig. 10i, j). A considerable number of nodules are distributed throughout the manganese
metal electrode plates by DC.

Microscopic analysis was conducted at higher magnitude to obtain greater detail regarding the growth,
morphology and evolution of manganese nodules with time in the middle part of the electrode plate (Fig. 11). The

Fig. 10 . Energy consumption during the electrodeposition of manganese metal and associated current
efficiency, voltage-current ratio, and the nodule distribution for various times. ( a ) Current efficiency. ( b )
Energy consumption. ( c ) Ratio of voltage to current. ( d ) Improved current efficiency by CC compared with
DC. ( e ) Reduced energy consumption. ( f ) Reduced area ratio of nodules. ( g ) Distribution of manganese
nodules with CC. ( h ) Distribution of manganese nodules with DC. ( i ) Microscopy images of manganese metal
by CC after 120 min. ( j ) Microscopy images of manganese metal by DC after 120 min.

results of a statistical analysis of nodule particle size have demonstrated that the manganese nodules exhibited a
continuous increase from a value of 36.6 μm (20 min) to a final 142.6 μm (120 min) with DC (Fig. 11c, f, i, l, o).

Scanning electron microscopy (SEM) analysis of the manganese metal deposited at different times on a
smaller scale was also conducted. It can be seen that the manganese metal deposited by CC consists of smallscale spherical particles that accumulate as a very dense crystal growth (Fig. 12a, c, e, g, i). Conversely, the surface
of manganese metal deposited by DC electrodeposition is characterized by the presence of a considerable largerscale spherical manganese nodule component (Fig. 12b, d, f, h, j). Moreover, the structural units of manganese
metal constituting the spherical manganese nodules are demonstrably distinct from the manganese metal
deposited by CC electrodeposition. Furthermore, the SEM analysis has established that the spherical manganese
nodules were formed over time.

Fig. 11 . Morphology of the manganese metal from optical microscopy analysis of the central part of the
electrode plate. ( a ) Chaos-a current for 20 min. ( b ) DC for 20 min. ( c ) Particle size distribution of manganese
nodules with DC for 20 min (AD denotes average diameter). ( d ) Chaos-a current for 30 min. ( e ) DC for
30 min. ( f ) Particle size distribution with DC for 30 min. ( g ) Chaos-a current for 60 min. ( h ) DC for 60 min.
( i ) Particle size distribution with DC for 60 min. ( j ) Chaos-a current for 90 min. ( k ) DC for 90 min. ( l ) Particle
size distribution with DC for 90 min. ( m ) Chaos-a current for 120 min. ( n ) DC for 120 min. ( o ) Particle size
distribution with DC for 120 min.

The phase composition of electrodeposited manganese metal at different times was assessed by XRD analysis
(Fig. 12k). The predominant physical phase was manganese metal, with the most prominent characteristic peak
attributed to the (330) crystal plane. It should be noted that the intensity of the (330) diffraction peak decreased
under the influence of a chaotic power supply, suggesting that the physical phase composition of the manganese
metal has undergone a transformation. In contrast, there was no significant difference in the intensity of the
diffraction peaks under DC electrolysis (Fig. 12l). In addition, the intensity of all the characteristic peaks before

Fig. 12 . Morphology and physical composition of manganese metal from SEM and XRD analysis. ( a ) SEM
images of manganese metal by CC for 20 min. ( b ) SEM images of manganese metal by DC for 20 min. ( c ) SEM
images of manganese metal by CC for 30 min. ( d ) SEM images of manganese metal by DC for 30 min. ( e ) SEM
images of manganese metal by CC for 60 min. ( f ) SEM images of manganese metal by DC for 60 min. ( g ) SEM
images of manganese metal by CC for 90 min. ( h ) SEM images of manganese metal by DC for 90 min. ( i ) SEM
images of manganese metal by CC for 120 min. ( j ) SEM images of manganese metal by DC for 120 min. ( k )
XRD patterns of manganese metal by CC for various times. ( l ) XRD patterns of manganese metal by DC for
various times.

90 min was lower than that associated with manganese metal electrodeposited by CC. This result provides
further evidence that the crystal structure of manganese nodules differs from that of manganese metal, and
demonstrates that a chaotic power supply can be used to modulate the crystal structure of manganese metal.

A denser Mn metal is more resistant to oxidation, increasing purity and reducing the necessity for passivation
solutions. X-ray photoelectron spectroscopy (XPS) analysis was conducted on the manganese metal products
deposited at various times. The results have indicated that the principal impurity elements present are N, S, and
Cr (Fig. 13a-d). The manganese element content of the product generated by CC is higher than DC. Chromium
was not detected for the CC electrodeposition products (Fig. 13c). In contrast, the presence of chromium was
identified at 20 min, 30 min and 120 min by DC deposition (Fig. 13d). This indicates that the CC deposited
manganese metal is denser, which subsequently reduces the residual chromium. It is well established that
chromium is a highly toxic pollutant 57 . The development of chromium-free passivation techniques represents
a significant challenge in the electrodeposition industry. The advent of CC electrodeposition offers a promising
avenue for the advancement of chromium-free passivation research 58 .

The XPS peaks due to surface oxygen on the manganese metal surfaces were deconvoluted and fitted. It is
evident that the intensity of the surface oxygen peaks at lower energy levels diminishes gradually with prolonged
chaotic circuit electrodeposition (Fig. 13e). In contrast, the surface oxygen peaks for DC electrodeposited

Fig. 13 . Elemental distribution and valence distribution of manganese metal generated at various times
and currents. ( a ) XPS full spectrum of manganese metal by CC for various times. ( b ) XPS full spectrum of
manganese metal by DC for various times. ( c ) Elemental composition of Mn metal by CC. ( d ) Elemental
composition of Mn metal by DC. ( e ) XPS spectra for oxygen with peak deconvolution and fitting associated
with Mn metal by CC. ( f ) XPS spectra for oxygen with peak deconvolution and fitting associated with Mn
metal by DC.

manganese metal were consistently low in intensity, appreciably lower than those obtained from CC
electrodeposition (Fig. 13f).

Electrodeposition amplification experiment
This study has also considered the potential applications of this technology in industry, employing a high-power
current amplifier to achieve CC amplification on a larger scale (1 A). The current and voltage signals over the
entire electrolysis process have also been collected (Fig. S9). The range of current and voltage fluctuations of the
chaotic power under high current was much greater than that of DC power (Fig. 14a, b). The ratio of voltage to
current with CC was lower relative to DC (Fig. 14c).

The application of CC enabled an increase in current efficiency with energy saving (Fig. 14d, e). It is also
evident that CC resulted in a reduction in nodule content, where microscopy analysis has established a close
relationship between energy consumption and nodule formation (Fig. 14f-j, Fig. S10, and Fig. S11). An analysis
of manganese metal cross-sections by SEM has established enhanced metal deposition by CC, which may be
linked to the energy component dissipated by the manganese nodules in the case of DC (Fig. 14k, l, and Fig. S12).

Fig. 14 . Energy consumption and the relationship with current efficiency, signal properties, nodule
distributions, and thickness of Mn metal generated for larger current scale using the power amplifier (FPA
101). ( a ) Current fluctuation for the two forms of current. ( b ) Voltage fluctuation. ( c ) Comparison of the ratio
of voltage to current. ( d ) Current efficiency. ( e ) Energy consumption. ( f ) Area ratio of nodules. ( g ) Nodules
distribution with CC. ( h ) Nodules distribution with DC. ( i ) Morphology of manganese metal with CC using
optical microscopy. ( j ) Morphology of manganese metal with DC. ( k ) Cross-section SEM images of manganese
metal by CC. ( l ) Cross-section SEM images of manganese metal by DC.

Effect of electrodeposition time on anode mud
To ascertain whether the physicochemical properties of the by-products formed during electrodeposition
undergo a similar transformation, a SEM analysis was conducted on the anode residue or “mud”. The
microstructural composition of the anode mud is complex, but some obvious variations are apparent with
increased electrodeposition time (Fig. S13) 59 . Applying a 40 μm scale, it can be seen that the anode mud consists
of a large number of spherical particles and manganese dioxide particles characterized by a much smaller
spherical structure. It should be noted that the anodic mud deposited under chaotic currents appears to contain
a greater proportion of spherical manganese dioxide.

The electrodeposition products at the anode were subjected to TEM analysis (Fig. 15a-f). The high-resolution
TEM (HRTEM) images reveal that the anodic by-products are very complex with a range of lattice spacings,
suggesting at least three crystal structures (Fig. 15b, e). A spacing of 0.297 nm may be ascribed to β − MnO2

Fig. 15 . Physicochemical properties of the anode mud for various times and current types. ( a ) TEM image of
the anode mud with CC for 120 min. ( b ) High-resolution TEM image of the anode mud with CC. ( c ) Broad
halo ring in the SAED pattern indicating amorphous character the anode mud with CC. ( d ) TEM image of the
anode mud with DC for 120 min. ( e ) High-resolution TEM image of the anode mud with DC. ( f ) Broad halo
ring in the SAED pattern indicating polycrystalline character of the anode mud with DC. ( g ) XRD patterns
of the anode mud with CC. ( h ) XRD patterns of the anode mud with DC. ( i ) Raman spectra of the anode
mud with CC. ( j ) Raman spectra of the anode mud with DC. ( k ) XPS survey scan of the anode mud with CC.
( l ) XPS survey scan of the anode mud with DC. ( m ) Elemental distribution in the anode mud. ( n ) XPS O1s
spectrum with peak fitting.

or δ − MnO2 according to previous research 60,61 . The crystal spacings of 0.336 nm and 0.339 nm are between
β − MnO2 and γ − MnO259. Two distinct lattice fringes of 0.280 nm and 0.281 nm are consistent with the

β − MnO2 and γ − MnO259. Two distinct lattice fringes of 0.280 nm and 0.281 nm are consistent with the

lattice framework of Mn3O462,63. In addition, the lattice fringes of 0.239 nm and 0.244 nm may be attributed

to β − MnO261. The broad halo ring in the selected area electron diffraction (SAED) pattern provides evidence

that the sample of CC anode mud includes an amorphous component (Fig. 15c). The halo ring observed in the
SAED pattern for the DC anode mud is indicative of polycrystalline material (Fig. 15f). These results confirm
that hyperchaotic current can regulate the crystal structure of the anode electrodeposition products.

The results of XRD and Raman spectroscopy analyses have also illustrated the complex phase composition
and chemical structure of the anode mud (Fig. 15g-j). The phase compositions generated using the two power
sources are predominantly MnO2, Mn2O3, Mn3O4 and SeO264–68.

sources are predominantly MnO2, Mn2O3, Mn3O4 and SeO264–68.

The XPS survey scan of the anode mud has identified the principal elemental composition as Mn, O, N, S, and
Pb (Fig. 15k, l). The distribution of these elements is shown as a function of time in Fig. 15m. The XPS spectrum
for O1s was subjected to a peak fitting procedure 67 . The results have indicated that the surface oxygen content
of anode mud generated by CC is higher than the DC electrolysis by-products. Furthermore, the surface oxygen
content exhibited a declining trend over time. The complexity of the anode mud poses serious challenges in
terms of accurately quantifying physical composition while ensuring experimental reproducibility.

Discussion
The electrodeposition of manganese metal is a complex kinetic system involving a number of interrelated
variables, including time, space, energy, electrochemical reactions and ion migration in the flow field. The
spatial variables are primarily reflected in the apparent morphology of the manganese metal. Energy is the most
significant variable in this study, where we have considered the replacement of a conventional power source with
a continuous, uniform input of electrical energy and continuous variable-speed supply. The electrochemical
reactions that occur during the electrodeposition of manganese metal are primarily crystallization reactions
at the electrode plate interface. The resulting concentration changes are a consequence of these reactions.
Ion migration involves the directional migration of ions in solution during the energization process. It can
be demonstrated that CC effectively regulates the kinetic behavior of the manganese metal electrodeposition
process (Fig. 16). The findings of this study support the following three observations.

1. The time-varying currents, which include chaos-a, chaos-b, period 1, and period 2, can impede the electro
chemical oscillation behavior.
2. The size of manganese nodules increases with the duration of electrodeposition by DC, demonstrating a clear

spatial variation in the electrodeposition process. Application of CC promotes energy saving in the initial
stages (30 min) of electrodeposition.
3. The occurrence of nodules results in a reduction in current efficiency and increased energy consumption.

Increasing the thickness of the deposited manganese metal reduces the energy consumption associated with
the time-varying current. Operation of CC at the optimal current density (1270 A/m 2 (0.508 A)) can improve
current efficiency by 6.08%. Nodules are not produced during DC or CC electrolysis when the current density is too low. However, the introduction of high external energies can result in a destabilization of the kinetic
system, which in turn impedes the suppression of manganese nodules, even in the presence of time-varying
currents 39,40 .

The primary reasons for enhanced energy saving and emission reduction associated with Mn metal
electrodeposition using a chaotic power supply can be attributed to the following factors: (1) The distribution
of input energy across the electrode plate is more uniform, circumventing the build-up of electrons at the edge
for an extended period; (2) The regulation of electrical energy input controls the electrochemical reaction at the
interface of the electrode plate, which alters the migration and transformation of the Mn 2+ ions in the electrolytic
cell.

Conclusions
Previous studies on the electrodeposition of manganese metal by CC have noted that the effects of waveform
structure and electrodeposition time on current efficiency warrant further investigation. The main conclusions
of this study are as follows:

(1) A novel methodology for constructing an offset boosting function for any chaotic systems was proposed.

Through the synergistic integration of this proposed approach with traditional offset boosting, a novel
four-dimensional chaotic system, equipped with two-dimensional offset boosting capabilities, was successfully developed. In addition, this 4D chaotic system was successfully transferred to a real chaotic circuit
and was used for in the manganese metal electrolysis system, supplanting conventional DC. The amplitude
and waveform of the voltage for the chaotic circuit were established. In addition, a two-dimensional offset boosting function was implemented, which can be employed to control the amount of current while
expanding the range of applications for electrical signals. The chaotic circuit comprises a single nonlinear
term (a single multiplier) that further reduces the cost of implementation.
(2) An effective coupling of electrical energy and chemical reactions was achieved through the use of time-var
ying current, which resulted in the complete suppression of electrochemical oscillations. All four types of
waveforms (period 1, period 2, chaos-a, and chaos-b) have been demonstrated to fully inhibit electrochemical oscillations of manganese metal. The ratio of voltage to current indicates that the time-varying current
electrodeposition can reduce the cell voltage, achieving energy savings compared with DC. The extent of the
reduction in cell voltage from a high to a low level is as follows: chaos-a > chaos-b > period 2 > period 1.

Fig. 16 . Schematic representation of the manganese metal deposition mechanism.

(3) Both the amplitude fluctuation and the waveforms can influence current efficiency. The thickness of the

deposited manganese metal can be enhanced, thereby reducing the energy consumption associated with the
time-varying current. Moreover, the CC has an optimal current density at 1270 A/m 2 (0.508 A), which can
improve current efficiency by 6.08%. Furthermore, the chaos-a current can increase the current efficiency
by 5.83% and with an energy consumption saving of 0.58 kWh/kg for an electrodeposition time of 30 min,
which is associated with the largest current fluctuation range.
(4) Current efficiency decreased with increasing electrodeposition time. Furthermore, the current efficiency

can be increased by 6.13%, the area ratio of the manganese nodules reduced by 28%, and the energy consumption reduced by 0.95 kWh/kg compared with DC when the electrodeposition time is 30 min.

This study offers new insights and results related to the use of time-varying currents to deliver an efficient
electrodeposition process applied to manganese metal. Nevertheless, the frequency effects of the electrical
signals in the electrodeposition process, and the industrial amplification challenges of CC with associated cost
considerations, are of critical importance in future research.

Data availability
All data generated or analysed during this study are included in this published article and its supplementary
information files.

Received: 28 September 2024; Accepted: 17 December 2024

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Acknowledgements
This work is supported by the Fundamental Research Funds for the Central Universities (Grant number 2022CDJQY − 005, 2024CDJXY010), Guangxi Science and Technology Program (AB24010229), Natural Science Foundation of Chongqing (2024NSCQMSX0756), and the Graduate Scientific Research and Innovation Foundation
of Chongqing (Grant number CYB23015).

Author contributions
Author contribution statementJie Yang: Experiments and writing-original draft preparation. Chunbiao Li: Investigation, Methodology, Experiments, and Visualization. Qian Zhang: Conceptualization, and Methodology. Zhihao Wu: Methodology and Experiments. Xin Zhang: Theoretical and computational support. Peiqiao Liu: Investigation and Data processing. Zuohua Liu*: Conceptualization, Methodology, Project administration, Funding
acquisition. Changyuan Tao: Methodology. Guocan Zheng: Testing and Data Processing. Yong Yang: Research
and industrial ideas provide. Hanke Wei: Research and industrial ideas provide. All authors contributed to the
discussions and manuscript preparation.

Declarations

Competing interests
The authors declare no competing interests.

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