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Generating Multi-Scroll Chua's Attractors via Simplified Piecewise-Linear Chua's Diode
Ning Wang, Chengqing Li, Han Bao, Mo Chen, Bocheng Bao
TL;DR
Multi-scroll Chua circuits are difficult to implement compactly because multi-segment nonlinear resistors require intricate breakpoints and slopes. The paper synthesizes a simplified multi-segment Chua’s diode from op-amp-based nonlinear-resistor cells, then combines it with an active Sallen-Key high-pass filter. Simulations and hardware experiments verify multi-scroll generation, parameter-controlled scroll variation, and operation under different power supplies.
Problem
Multi-scroll Chua circuits face implementation complexity because their multi-segment piecewise-linear resistors require intricate breakpoints and slopes.
Method
The paper cascades op-amp-based passive and active nonlinear-resistor cells through a negative impedance converter and uses the resulting diode with a second-order active Sallen-Key high-pass filter.
Results
4-, 6-, 8-, and 10-scroll chaotic attractors are observed as resistance R varies from 1.385 kΩ to 1.515 kΩ.
Takeaways & Limitations
The proposed design supports adjustable multi-scroll attractors, lower implementation complexity, and operation under different power supplies, with feasibility verified in simulations and hardware.
Abstract
from arXiv · showhide
High implementation complexity of multi-scroll circuit is a bottleneck problem in real chaos-based communication. Especially, in multi-scroll Chua's circuit, the simplified implementation of piecewise-linear resistors with multiple segments is difficult due to their intricate irregular breakpoints and slopes. To solve the challenge, this paper presents a systematic scheme for synthesizing a Chua's diode with multi-segment piecewise-linearity, which is achieved by cascading even-numbered passive nonlinear resistors with odd-numbered ones via a negative impedance converter. The traditional voltage mode op-amps are used to implement nonlinear resistors. As no extra DC bias voltage is employed, the scheme can be implemented by much simpler circuits. The voltage-current characteristics of the obtained Chua's diode are analyzed theoretically and verified by numerical simulations. Using the Chua's diode and a second-order active Sallen-Key high-pass filter, a new inductor-free Chua's circuit is then constructed to generate multi-scroll chaotic attractors. Different number of scrolls can be generated by changing the number of passive nonlinear resistor cells or adjusting two coupling parameters. Besides, the system can be scaled by using different power supplies, satisfying the low-voltage low-power requirement of integrated circuit design. The circuit simulations and hardware experiments both confirmed the feasibility of the designed system.
I. INTRODUCTION
The paper targets simpler, lower-complexity implementation of multi-scroll Chua circuits for practical chaotic-signal applications. It introduces op-amp-based nonlinear-resistor cells and combines them into a multi-segment Chua’s diode without extra DC bias voltages.
- Motivation: Prior multi-scroll constructions can require many components, extra DC bias voltages, or pulse excitations, increasing hardware complexity and energy use.These constraints are especially relevant to IoT, wireless communication, and integrated-circuit applications.
- Objectives: The paper focuses on simplifying multi-piecewise Chua’s-diode implementation and extending the family of multi-scroll, inductor-free Chua circuits.The authors validate the proposed system through theoretical analysis, numerical simulation, and experiments.
- Demonstration: The paper presents 17-segment and 19-segment voltage-current characteristics for synthesized Chua’s diodes.These examples demonstrate how cascaded nonlinear-resistor cells produce multi-segment characteristics.
- Approach: The proposed diode combines multiple op-amp-based nonlinear resistors with different breakpoints to realize multi-segment piecewise linearity.The approach defines passive type-I and active type-II nonlinear-resistor cells for the synthesis scheme.
- Type-I passive nonlinear resistor: The type-I nonlinear resistor uses one op-amp and two resistors, with positive inner and outer slopes and passive power behavior.Its voltage-current curve has three regions determined by op-amp saturation and the resulting breakpoints.
2) Type-II Active Nonlinear Resistor:
The type-II active nonlinear resistor is formed by adding a negative impedance converter to the passive type-I cell. Its negative-slope characteristic reproduces the essential behavior of a classical Chua’s diode with fewer resistors.
- Construction: A negative impedance converter connected to the type-I resistor produces the type-II active nonlinear resistor.The resulting device is analyzed through its inner slope, outer slope, and breakpoint parameters.
- Electrical characteristic: The type-II resistor has negative inner and outer slopes, with the inner slope magnitude exceeding the outer slope magnitude.This gives it the characteristic shape associated with the classical Chua’s diode.
- Implementation: The type-II implementation uses four resistors, compared with six in the cited classical Chua’s-diode design.The paper presents this as a component-count reduction for the active nonlinear-resistor cell.
B. Multi-Segment Piecewise-Linear Chua’s Diode
The multi-segment Chua’s diode is synthesized by cascading nonlinear-resistor cells with systematic breakpoint and slope selection. Symmetric breakpoints from each cell generate a regular multi-segment voltage-current characteristic.
- Synthesis structure: The synthesis combines type-I cells with different breakpoints, while the negative impedance converter reverses current so odd-numbered cells act as type-II resistors.This cascading structure creates alternating contributions to the overall Chua’s-diode characteristic.
- Breakpoint generation: Each cell generates two symmetric breakpoints, so M cells produce 2M breakpoints and 2M+1 piecewise-linear segments.The breakpoints are determined by the cell resistances and op-amp saturation voltage.
- Parameter selection: The design procedure fixes the outermost breakpoints, divides the interval between them into equal segments, and keeps the inner slope unchanged across cells.These steps provide systematic parameter selection for the cascaded diode.
C. The v–i Characteristic of the Proposed Chua’s Diode
The proposed Chua’s diodes produce multi-segment piecewise-linear voltage-current curves while reducing implementation components and eliminating extra DC bias voltages.
- The 8-NRI and 9-NRI designs generate 17-segment and 19-segment piecewise-linear v-i curves, respectively.The curves are shown for fixed outermost breakpoints of ±4 V and op-amp saturation voltage Esat = 13 V.
- Compared with earlier construction methods, the proposed approach removes extra DC bias voltages and uses fewer resistors for the same number of segments.For 17 segments, it uses 18 resistors versus 27 and 48 in the cited alternatives.
- The section presents the resulting diode characteristics alongside the Sallen-Key high-pass-filter-based circuit used for subsequent system construction.The circuit schematic is identified as the Sallen-Key HPF-based Chua’s circuit.
III. SALLEN-KEY HPF-BASED CHUA’S CIRCUIT
The paper replaces the classical Chua circuit’s parallel LC network with a second-order active Sallen-Key high-pass filter, yielding an inductor-free third-order autonomous circuit.
- The proposed circuit is a third-order inductor-free Chua’s circuit designed to generate multi-scroll chaotic attractors.Its three dynamical elements are capacitors C1, C2, and C3.
- A second-order active Sallen-Key high-pass filter replaces the parallel LC network of the classical Chua’s circuit.The proposed Chua’s diode occupies one side of the circuit, while the Sallen-Key HPF occupies the other.
- The Sallen-Key HPF uses equal component pairs C2 = C3 and R3 = R4.
- The circuit dynamics are represented by three coupled first-order autonomous differential equations in node voltages V1, V2, and V3.The Chua’s diode contributes the voltage-current characteristic h(V1).
B. Equilibrium Point and Its Stability
The equilibrium analysis obtains operating points from intersections of two curves and classifies their stability through Jacobian eigenvalues, revealing two saddle-focus types.
- Setting the relevant time derivatives to zero gives V3 = 0, while V1 and V2 are obtained from intersections of the curves defined by Eqs. (6) and (7).
- For an M-NRI-based Chua’s diode, the inductor-free circuit generates (2M+1) intersection points and corresponding equilibrium points.The paper illustrates this construction using 8-NRI and 9-NRI-based diodes.
- The equilibrium points contain unstable index-1 and index-2 saddle-foci distinguished by their eigenvalue sign patterns.Index-1 points have one positive real root and two complex roots with negative real parts; index-2 points have two complex roots with positive real parts and one negative real root.
- Under the stated Shil’nikov eigenvalue conditions, bond orbits and scrolls may form near unstable index-1 and index-2 equilibria, respectively.
C. Multi-Scroll Chaotic Attractor
The proposed circuit produces multi-scroll chaotic attractors whose number, dynamics, and amplitude can be varied through diode configuration, coupling parameters, and op-amp supply voltage.
- Numerical experiments show that modifying the Chua’s diode can generate no less than 20 scrolls, with 9-scroll and 10-scroll cases demonstrated.The 9-scroll and 10-scroll attractors emerge near unstable index-2 saddle-foci, while bond orbits arise near unstable index-1 saddle-foci.
- A (2M + 1)-segment piecewise-linear curve can theoretically produce an (M + 1)-scroll attractor when suitable parameters are configured.
- Changing R and C1 produces different dynamical regions and scroll types, as characterized by dynamics maps and Lyapunov-exponent maps.
- With C1 = 6.8 nF, positive largest Lyapunov exponents occur across R ∈ [1.385 kΩ, 1.515 kΩ], where 4-, 6-, 8-, and 10-scroll attractors are observed.Representative resistance values are 1.44 kΩ, 1.47 kΩ, 1.483 kΩ, and 1.5 kΩ.
- Changing the op-amp saturation voltage Esat scales attractor amplitude while leaving the slopes of the diode characteristic unchanged.The maxima and minima of V1, V2, and V3 increase linearly with Esat.
E. Comparison of Related Chua’s Ciruits
The comparison places the proposed multi-scroll Chua’s circuit among related implementations and highlights its reduced component burden. The paper also describes an image-encryption application using the proposed chaotic system.
- Table III compares implementations of related multi-scroll Chua’s circuits.
- The proposed design uses ten op-amps for a 19-segment Chua’s diode and removes 10 resistors and 8 DC bias voltages.
- The proposed inductor-free circuit uses fewer circuit elements than several compared methods, although implementation complexity lacks a unified standard.
- The encryption application uses 19- and 21-segment piecewise-linear Chua’s diodes.
- The encryption process generates chaotic sequences, preprocesses one sequence into a pseudo-random sequence, and applies XOR bit by bit.
B. Simulation Results and Performance Analyses
The paper evaluates the proposed encryption scheme with statistical, histogram, and image-based analyses. Reported results include strong randomness-test performance and a uniform cipher-image histogram.
- The evaluation measures randomness, histogram behavior, adjacent-pixel correlation, and information entropy.
- U-values for sequences generated with 10-scroll and 11-scroll attractors are all larger than 0.0001.
- For 100 samples from the 10-scroll attractor, two of 18 sub-tests fail to meet the stated passing ratio slightly.
- The cipher-image histogram is reported as very uniform, while the original and decrypted images are also shown.
- Cipher-image adjacent-pixel correlations are much lower than those of the plain image.
3) Correlation Analysis:
The correlation analysis defines adjacent-pixel correlation and compares plain and cipher images across horizontal, vertical, and diagonal directions. The section also reports the cipher-image entropy result and summarizes circuit verification.
- Figure 12 compares horizontal, vertical, and diagonal adjacent-pixel correlations for the original and cipher images.
- Correlation is calculated from paired adjacent-pixel intensities, with x and y denoting intensities and N the number of pixels.
- The plain image has high adjacent-pixel correlations, whereas the cipher image has much lower values.
- The cipher-image entropy is 7.9889, and the 11-scroll sequence produces 7.9898 for the same plain image.
- Circuit simulations and hardware experiments are used to verify the system’s real performance, including phase portraits of 9-scroll and 10-scroll attractors.
B. Hardware Experiments
Hardware experiments measured the proposed diode’s piecewise-linear characteristics and 9-scroll and 10-scroll attractors, with results consistent with numerical simulations and supporting the scheme’s feasibility.
- Experimental setup: The hardware setup used a printed circuit board with adjustable resistors, ceramic capacitors, and TL082CP op-amps.
- Experimental setup: The experiment used ±14.6 V DC supplies, with oscillator saturation near ±13 V.
- Diode characteristics: With Vm = 5V and f = 100 Hz, measured v−i loci were obtained for the 17-segment and 19-segment piecewise-linear curves.
- Chaotic attractors: Hardware phase portraits measured 9-scroll and 10-scroll chaotic attractors in the V1−V3 plane.
- Validation: Circuit simulations and hardware experiments were consistent with numerical simulations, demonstrating the feasibility of the proposed scheme.
- Design implications: The proposed design supports modular diode cells, lower implementation complexity, inductor-free autonomous operation, and operation with different power supplies.