By utilizing a memristor to substitute a coupling resistor in the realization circuit of a three-dimensional chaotic system having one saddle and two stable node-foci, a novel memristive hyperchaotic system with coexisting infinitely many hidden attractors is presented. The memristive system does not display any equilibrium, but can exhibit hyperchaotic, chaotic, and periodic dynamics as well as transient hyperchaos. In particular, the phenomenon of extreme multistability with hidden oscillation is revealed and the coexistence of infinitely many hidden attractors is observed. The results illustrate that the long term dynamical behavior closely depends on the memristor initial condition thus leading to the emergence of hidden extreme multistability in the memristive hyperchaotic system. Additionally, hardware experiments and PSIM circuit simulations are performed to verify numerical simulations.

Hidden extreme multistability in memristive hyperchaotic system

This tutorial clarifies the axiomatic definition of (v(α); i(β)) circuit elements via a lookup table dubbed an A-pad, of admissible (v; i) signals measured via Gedanken probing circuits. The (v(α); i(β)) elements are ordered via a complexity metric. Under this metric, the memristor emerges naturally as the fourth element, characterized by a state-dependent Ohm's law. A logical generalization to memristive devices reveals a common fingerprint consisting of a dense continuum of pinched hysteresis loops whose area decreases with the frequency ω and tends to a straight line as ω ~ ∞, for all bipolar periodic signals and for all initial conditions. This common fingerprint suggests that the term memristor be used hence-forth as a moniker for memristive devices.

The Fourth Element.

Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria

This paper presents a novel fifth-order two-memristor-based Chua’s hyperchaotic circuit, which is synthesized from an active band pass filter-based Chua’s circuit through replacing a nonlinear resistor and a linear resistor with two different memristors. This physical circuit has a plane equilibrium and therefore emerges complex phenomenon of extreme multistability. Based on the mathematical model, stability distributions of three nonzero eigenvalues in the equilibrium plane are exhibited, from which it is observed that four different stability regions with unstable saddle-focus, and stable and unstable node-focus are distributed, thereby leading to coexisting phenomenon of infinitely many attractors. Furthermore, extreme multistability depending on two-memristor initial conditions is investigated by bifurcation diagrams and Lyapunov exponent spectra and coexisting infinitely many attractors’ behavior is revealed by phase portraits and attraction basins. At last, a hardware circuit is fabricated and some experimental observations are captured to verify that extreme multistability indeed exists in the two-memristor-based Chua’s hyperchaotic circuit.

Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability

When possessing a potential difference between two neurons, an electromagnetic induction current appears in the Hopfield neural network (HNN), which can be emulated by a flux-controlled memristor synapse. Thus, a three-order two-neuron-based autonomous memristive HNN is presented in this paper, which is the lowest order and has not been reported in the previous studies. With the mathematical model, the detailed stability analyses for the line equilibrium are executed, so that the fold and Hopf bifurcation sets and stability region distributions in the parameter plane are obtained. Furthermore, numerical results of coexisting bifurcation patterns are investigated, which are confirmed effectively by local basins of attraction and phase plane plots. The numerical results demonstrate coexisting multi-stable patterns of the spiral chaotic patterns with different dynamic amplitudes, periodic patterns with different periodicities, and stable resting patterns with different positions in the memristive HNN. Besides, the circuit synthesis and breadboard experiments are performed to well validate the numerical simulations.

Coexisting multi-stable patterns in memristor synapse-coupled Hopfield neural network with two neurons

This paper presents an inductor-free memristive circuit, which is implemented by linearly coupling an active band-pass filter (BPF) with a parallel memristor and capacitor filter. Mathematical model is established, and numerical simulations are performed. The results verified by hardware experiments show that the active BPF-based memristive circuit exhibits the dynamical behaviors of point, period, chaos, and period-doubling bifurcation route. Most important of all, the newly proposed memristive circuit has a line equilibrium and its stability closely relies on memristor initial condition, which results in the emergence of extreme multistability. Stability distribution related to memristor initial condition is numerically estimated and the coexistence of infinitely many attractors is intuitively captured by numerical simulations and PSIM circuit simulations.

Coexisting infinitely many attractors in active band-pass filter-based memristive circuit

This paper investigates extreme multistability and its controllability for an ideal voltage-controlled memristor emulator-based canonical Chua’s circuit. With the voltage-current model, the initial condition-dependent extreme multistability is explored through analyzing the stability distribution of line equilibrium point and then the coexisting infinitely many attractors are numerically uncovered in such a memristive circuit by the attraction basin and phase portraits. Furthermore, based on the accurate constitutive relation of the memristor emulator, a set of incremental flux-charge describing equations for the memristor-based canonical Chua’s circuit are formulated and a dimensionality reduction model is thus established. As a result, the initial condition-dependent dynamics in the voltage-current domain is converted into the system parameter-associated dynamics in the flux-charge domain, which is confirmed by numerical simulations and circuit simulations. Therefore, a controllable strategy for extreme multistability can be expediently implemented, which is greatly significant for seeking chaos-based engineering applications of multistable memristive circuits.

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Memristor-Based Canonical Chua’s Circuit: Extreme Multistability in Voltage-Current Domain and Its Controllability in Flux-Charge Domain

It is known that dynamical behaviors of memristive circuit are significantly affected by its initial states, which are difficult to be explicitly analyzed or controlled in voltage–current domain an...

Flux-Charge Analysis of Initial State-Dependent Dynamical Behaviors of a Memristor Emulator-Based Chua’s Circuit

The purpose of this paper is to develop a simple chaotic circuit. The circuit can be fabricated by less discrete electronic components, within which complex dynamical behaviors can be generated.,A second-order non-autonomous inductor-free chaotic circuit is presented, which is obtained by introducing a sinusoidal voltage stimulus into the classical Wien-bridge oscillator. The proposed circuit only has two dynamic elements, and its nonlinearity is realized by the saturation characteristic of the operational amplifier in the classical Wien-bridge oscillator. After that, its dynamical behaviors are revealed by means of bifurcation diagram, Lyapunov exponent and phase portrait and further confirmed using the 0-1 test method. Moreover, an analog circuit using less discrete electronic components is implemented, and its experimental results are measured to verify the numerical simulations.,The equilibrium point located in a line segment varies with time evolution, which leads to the occurrence of periodic, quasi-periodic and chaotic behaviors in the proposed circuit.,Unlike the previously published works, the significant values of the proposed circuit with simple topology are inductor-free realization and without extra nonlinearity, which make the circuit can be used as a paradigm for academic teaching and experimental illustraction for chaos.

Tao Jiang

Papers

Two-memristor-based Chua’s hyperchaotic circuit with plane equilibrium and its extreme multistability

Coexisting infinitely many attractors in active band-pass filter-based memristive circuit

Memristor-Based Canonical Chua’s Circuit: Extreme Multistability in Voltage-Current Domain and Its Controllability in Flux-Charge Domain

Flux-Charge Analysis of Initial State-Dependent Dynamical Behaviors of a Memristor Emulator-Based Chua’s Circuit

Chaos in a second-order non-autonomous Wien-bridge oscillator without extra nonlinearity