Multi-scroll chaos generation via linear systems and hysteresis function series

Anti-control of chaos has attracted a lot of attention recently due to its potential applications in science and engineering. How to generate useful chaos that is also practically implementable and useful is a current focus of research. This research aims at developing new chaos generation schemes which demonstrate complex dynamical behaviours using simple linear systems with hysteresis function series. A continuous-time linear unstable second-order system with a feedback of hysteresis function is first proposed for generating chaos. The design for chaos generation is studied theoretically. A Poincaré map is used to demonstrate the dynamical behaviour of the system. The existence and the analytic solution of the limit cycle that bounds the basin of attraction of the chaotic attractor are derived. Conditions for the existence of chaotic attractors are studied. A hysteresis based system with a maximum chaotic stability margin is designed. Second, systematic methods for generating 1D n-scroll chaotic attractors in the directions of the state variables and 2D nxm-grid scroll chaotic attractors in the phase plane via continuous-time linear unstable second-order systems with a feedback of hysteresis function series are proposed. Furthermore, systematic methods for generating 1D n-scroll, 2D nxm-grid scroll and 3D nxmxl-space scroll chaotic attractors via continuous-time linear unstable third-order systems using hysteresis function series feedback are also presented in this thesis. Simulation results are presented to demonstrate effectiveness of the schemes. It is shown that the multi-scroll chaos generation systems can be represented in Lur'e form, and as a result it may be used within synchronization schemes for secure communication. Third, the limit cycle that bounds the basin of attraction in the multi-scroll chaos generation with second-order systems case is studied. The relationship of the size of the basin of attraction with the numbers of hysteresis function series is studied. The multi-scroll chaos generation mechanism is then further explored by analyzing the system trajectories; the switching boundaries, switching rules and the trajectories on each subspace. The chaotic behaviours are confirmed theoretically and it is proved that a non-ordinary attractor exists in the multi-scroll chaotic attractor of the second-order systems case. The abundant dynamical behaviour of the multi-scroll chaos generation systems using different hysteresis feedback are demonstrated. A double-hysteresis function, which is the superimposition of two basic hysteresis functions, is proposed for the implementation of the hysteresis based chaotic system. In this design, the double-hysteresis block and its series are constructed via a systematic method. The ideal hysteresis function series can be implemented easily with the proposed double-hysteresis function. The number of scroll attractors can be designed arbitrarily, and the multi-scroll chaotic attractors can be located anywhere and cover any chosen area of the phase plane. The circuitry implementation for generating 1D n-scroll, 2D nxm-grid scroll chaotic attractors with linear second-order systems and hysteresis function series is given. And the oscilloscope illustrated waveforms which included as many as 9x9=81 scrolls chaotic attractor are presented. The experimental results confirmed the theoretical analysis very well and validated the effectiveness as well as the feasibility of the proposed multi-scroll chaos generation schemes. This research may find potential engineering applications in areas such as digital coding and image processing, etc

A family of hyperchaotic multi-scroll attractors in Rn

Investigació

Construction of classes of circuit-independent chaotic oscillatorsusing passive-only nonlinear devices

Two generic classes of chaotic oscillators comprising four different configurations are constructed. The proposed structures are based on the simplest possible abstract models of generic second-order RC sinusoidal oscillators that satisfy the basic condition for oscillation and the frequency of oscillation formulas. By linking these sinusoidal oscillator engines to simple passive first-order or second-order nonlinear composites, chaos is generated and the evolution of the two-dimensional sinusoidal oscillator dynamics into a higher dimensional state space is clearly recognized. We further discuss three architectures into which autonomous chaotic oscillators can be decomposed. Based on one of these architectures we classify a large number of the available chaotic oscillators and propose a novel reconstruction of the classical Chua’s circuit. The well-known Lorenz system of equations is also studied and a simplified model with equivalent dynamics, but containing no multipliers, is introduced

Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices

Two generic classes of chaotic oscillators comprising four different configurations are constructed. The proposed structures are based on the simplest possible abstract models of generic second-order RC sinusoidal oscillators that satisfy the basic condition for oscillation and the frequency of oscillation formulas. By linking these sinusoidal oscillator engines to simple passive first-order or second-order nonlinear composites, chaos is generated and the evolution of the two-dimensional sinusoidal oscillator dynamics into a higher dimensional state space is clearly recognized. We further discuss three architectures into which autonomous chaotic oscillators can be decomposed. Based on one of these architectures we classify a large number of the available chaotic oscillators and propose a novel reconstruction of the classical Chua's circuit. The well-known Lorenz system of equations is also studied and a simplified model with equivalent dynamics, but containing no multipliers, is introduce

Chaotic attractors based on unstable dissipative systems via third-order differential equation

"In this paper, we present an approach how to yield 1D, 2D and 3D-grid multi-scroll chaotic systems in R3 based on unstable dissipative systems via third-order differential equation. This class of systems is constructed by a switching control law(SCL) changing the equilibrium point of an unstable dissipative system. The switching control law that governs the position of the equilibrium point varies according to the number of scrolls displayed in the attractor.

Maximal unstable dissipative interval to preserve multi-scroll attractors via multi-saturated functions

"In this paper, we present families of piecewise linear systems which are controlled by a continuous piecewise monoparametric control function for the generation of monoparametric families of multi-scroll attractors. Thus, the maximum range of values that the parameter set can take in order to preserve the useful dynamics for generating of multi-scroll attractors is found and it will be called maximal robust dynamics interval. This class of dynamical systems is the result of combining two or more unstable “one-spiral” trajectories. We give necessary and sufficient conditions in order to preserve multi-scroll attractors in terms of a parameter, i.e., a family of multi-scroll attractors is generated by means of a family of switching systems with multiple monoparametric companion matrices. Lastly, we provide an example to show how the developed theory works.

A Triple-Memristor Hopfield Neural Network With Space Multi-Structure Attractors And Space Initial-Offset Behaviors

© 2023 IEEE. This is the accepted manuscript version of an article which has been published in final form at https://doi.org/10.1109/TCAD.2023.3287760Memristors have recently demonstrated great promise in constructing memristive neural networks with complex dynamics. This paper proposes a memristive Hopfield neural network with three memristive coupling synaptic weights. The complex dynamical behaviors of the triple-memristor Hopfield neural network (TM-HNN), which have never been observed in previous Hopfield-type neural networks, include space multi-structure chaotic attractors and space initial-offset coexisting behaviors. Bifurcation diagrams, Lyapunov exponents, phase portraits, Poincaré maps, and basins of attraction are used to reveal and examine the specific dynamics. Theoretical analysis and numerical simulation show that the number of space multi-structure attractors can be adjusted by changing the control parameters of the memristors, and the position of space coexisting attractors can be changed by switching the initial states of the memristors. Extreme multistability emerges as a result of the TM-HNN’s unique dynamical behaviors, making it more suitable for applications based on chaos. Moreover, a digital hardware platform is developed and the space multi-structure attractors as well as the space coexisting attractors are experimentally demonstrated. Finally, we design a pseudo-random number generator to explore the potential application of the proposed TM-HNN.Peer reviewe