Bocheng Bao

Self-excited and hidden attractors found simultaneously in a modified Chua's circuit

By replacing the Chuas diode in Chuas circuit with a first-order hybrid diode circuit, a fourth-order modified Chuas circuit is presented. The circuit has an unstable zero saddle point and two nonzero saddle-foci. By Routh–Hurwitz criterion, it is found that in a narrow parameter region, the two nonzero saddle-foci have a transition from unstable to stable saddle-foci, leading to generations of self-excited and hidden attractors in the modified Chuas circuit simultaneously, which have not been previously reported. Complex dynamical behaviors are investigated both numerically and experimentally. The results indicate that the proposed circuit exhibits complicated nonlinear phenomena including self-excited attractors, coexisting self-excited attractors, hidden attractors, and coexisting hidden attractors.

A Simple Third-Order Memristive Band Pass Filter Chaotic Circuit

This brief presents a simple third-order inductor-free memristive chaotic circuit, which is derived from a second-order active band pass filter (BPF) by replacing a resistor with an improved memristor and has only three op-amps, two multipliers, three capacitors, and six resistors. The circuit has three unstable saddle-foci and exhibits complex dynamical behaviors, including period, chaos, period doubling bifurcation, coexisting bifurcation modes, and constant Lyapunov exponents (CLEs). Especially, the property of CLEs leads to that the amplitudes of the chaotic signals are linearly controlled by a potentiometer without changing system’s essences. Moreover, hardware circuit using less discrete components is fabricated and experimental verifications are performed, from which the existence of chaos is validated. Compared with other memristive chaotic circuits reported before, the proposed memristive BPF chaotic circuit is inductor-free and topologically simplified, which is only third-order, and much simpler and more intuitive in practical realization.

Non-Autonomous Second-Order Memristive Chaotic Circuit

A non-autonomous second-order memristive chaotic circuit is considered in this paper, which is comparatively simple, only consisting of a memristor, a capacitor, a resistor, and a sinusoidal voltage source. Based on the descriptive equation of the memristive circuit, the dynamical behaviors are investigated by theoretical analyses and numerical simulations. It is noted that the number of AC equilibrium points changes with the evolution of the time and the circuit exhibits striking dynamical features, including period, chaos, forward period-doubling, reverse period-doubling, tangent bifurcation, and crisis scenarios. Furthermore, a hardware circuit is set up by off-the-shelf discrete components, where hardware experiments are performed to verify the numerical results. The most significant feature of the proposed memristive circuit is the inductor-free realization with simplified topology, which makes the circuit much simpler and more intuitive in physical realization.

Bi-Stability in an Improved Memristor-Based Third-Order Wien-Bridge Oscillator

ABSTRACT This paper reports the emergence of bi-stability in a third-order memristive Wien-bridge oscillator. By transforming the parallel RC feedback network into a series resistor and capacitor (RC) feedback network in the previous non-ideal voltage-controlled memristor emulator, an improved memristor emulator is proposed and its fingerprints are exhibited. And then through introducing the proposed emulator to replace the parallel resistor in the conventional Wien-bridge oscillator, an improved memristor-based third-order Wien-bridge oscillator is constructed. Based on its mathematical model, the initial condition-dependent dynamical behaviour is revealed by theoretical analyses and numerical simulations and further verified by hardware experiments, from which the emergence of bi-stability in the memristive oscillator is well demonstrated.

Three-Dimensional Memristive Hindmarsh–Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors

Since the electrical activities of neurons are closely related to complex electrophysiological environment in neuronal system, a novel three-dimensional memristive Hindmarsh–Rose (HR) neuron model is presented in this paper to describe complex dynamics of neuronal activities with electromagnetic induction. The proposed memristive HR neuron model has no equilibrium point but can show hidden dynamical behaviors of coexisting asymmetric attractors, which has not been reported in the previous references for the HR neuron model. Mathematical model based numerical simulations for hidden coexisting asymmetric attractors are performed by bifurcation analyses, phase portraits, attraction basins, and dynamical maps, which just demonstrate the occurrence of complex dynamical behaviors of electrical activities in neuron with electromagnetic induction. Additionally, circuit breadboard based experimental results well confirm the numerical simulations.

Coexistence of multiple bifurcation modes in memristive diode-bridge-based canonical Chua’s circuit

ABSTRACT Based on a memristive diode bridge cascaded with series resistor and inductor filter, a modified memristive canonical Chua’s circuit is presented in this paper. With the modelling of the memristive circuit, a normalised system model is built. Stability analyses of the equilibrium points are performed and bifurcation behaviours are investigated by numerical simulations and hardware experiments. Most extraordinary in the memristive circuit is that within a parameter region, coexisting phenomenon of multiple bifurcation modes is emerged under six sets of different initial values, resulting in the coexistence of four sets of topologically different and disconnected attractors. These coexisting attractors are easily captured by repeatedly switching on and off the circuit power supplies, which well verify the numerical simulations.

Coexistence of Multiple Attractors in an Active Diode Pair Based Chua’s Circuit

This paper focuses on the coexistence of multiple attractors in an active diode pair based Chua’s circuit with smooth nonlinearity. With dimensionless equations, dynamical properties, including boundness of system orbits and stability distributions of two nonzero equilibrium points, are investigated, and complex coexisting behaviors of multiple kinds of disconnected attractors of stable point attractors, limit cycles and chaotic attractors are numerically revealed. The results show that unlike the classical Chua’s circuit, the proposed circuit has two stable nonzero node-foci for the specified circuit parameters, thereby resulting in the emergence of multistability phenomenon. Based on two general impedance converters, the active diode pair based Chua’s circuit with an adjustable inductor and an adjustable capacitor is made in hardware, from which coexisting multiple attractors are conveniently captured.

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

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.

An Improved Memristive Diode Bridge-Based Band Pass Filter Chaotic Circuit

By replacing a series resistor in active band pass filter (BPF) with an improved memristive diode bridge emulator, a third-order memristive BPF chaotic circuit is presented. The improved memristive diode bridge emulator without grounded limitation is equivalently achieved by a diode bridge cascaded with only one inductor, whose fingerprints of pinched hysteresis loop are examined by numerical simulations and hardware experiments. The memristive BPF chaotic circuit has only one zero unstable saddle point but causes complex dynamical behaviors including period, chaos, period doubling bifurcation, and coexisting bifurcation modes. Specially, it should be highly significant that two kinds of bifurcation routes are displayed under different initial conditions and the coexistence of three different topological attractors is found in a narrow parameter range. Moreover, hardware circuit using discrete components is fabricated and experimental measurements are performed, upon which the numerical simulations are validated. Notably, the proposed memristive BPF chaotic circuit is only third-order and has simple topological structure.

Parameter-Independent Dynamical Behaviors in Memristor-Based Wien-Bridge Oscillator

This paper presents a novel memristor-based Wien-bridge oscillator and investigates its parameter-independent dynamical behaviors. The newly proposed memristive chaotic oscillator is constructed by linearly coupling a nonlinear active filter composed of memristor and capacitor to a Wien-bridge oscillator. For a set of circuit parameters, phase portraits of a double-scroll chaotic attractor are obtained by numerical simulations and then validated by hardware experiments. With a dimensionless system model and the determined system parameters, the initial condition-dependent dynamical behaviors are explored through bifurcation diagrams, Lyapunov exponents, and phase portraits, upon which the coexisting infinitely many attractors and transient chaos related to initial conditions are perfectly offered. These results are well verified by PSIM circuit simulations.