Period-doubling Bifurcation Route Research Articles

Hidden firing patterns and memristor initial condition-offset boosting behavior in a memristive Hindmarsh-Rose neuron model

Electromagnetic induction can effectively induce abundant firing patterns in neurons. In modeling a neuron model with the electromagnetic induction effect, an electromagnetic induction current is frequently added to the state equation of membrane potential. To more properly reflect the non-uniform distribution of the ions inside and outside the neuron membrane, an ideal flux-controlled memristor with sinusoidal memductance function and non-linearly modulated input is raised to depict an electromagnetic induction effect on a Hindmarsh–Rose neuron model, and thereby, a three-dimensional (3D) memristive Hindmarsh–Rose (mHR) neuron model is built in this paper. The proposed mHR neuron model possesses no equilibrium point since the involvement of the ideal flux-controlled memristor, which induces the generation of hidden dynamics. Numerical results declare that the mHR neuron model can generate abundant hidden dynamics, i.e., periodic spiking, chaotic spiking, period-doubling bifurcation route, tangent bifurcation, and chaos crisis. These hidden dynamics are much related to the memristor coupling strength and externally applied stimulus. Afterward, the memristor initial condition-offset boosting behavior is revealed. This can trigger the generation of infinite multiple coexisting firing patterns along the memristor variable coordinate. These coexisting firing patterns have identical attractor topology but different locations in the phase plane. Finally, an analog circuit is designed for implementing the mHR neuron model, and PSIM-based circuit simulation is executed. The circuit-simulated results perfectly verify the generation of hidden infinite multiple coexisting initial condition-offset boosting firing patterns in the proposed mHR neuron model.

Dynamical analysis, FPGA implementation and synchronization for secure communication of new chaotic system with hidden and coexisting attractors

This paper proposes an interesting autonomous chaotic system with hidden attractors and coexisting attractors. The system has no equilibrium, one equilibrium, three equilibria and line equilibria for different parameter regions. The existence of hidden attractors and coexisting attractors of the system has been revealed by using simulation analysis. The bifurcation diagram shows the period-doubling bifurcation route to chaos with the variation of parameters. The analog circuit and FPGA implementation of the system are presented. The synchronization for secure communication of the system is investigated. The synchronization conditions are established by using the adaptive control method.

A novel current-controlled memristor-based chaotic circuit

In this paper, a novel third-order autonomous memristor-based chaotic circuit is proposed. The circuit has simple topology and contains only four elements including one linear negative impedance converter-based resistor, one linear capacitor, one linear inductor, and one nonlinear current-controlled memristor. Firstly, the voltage-current characteristic analysis of the memristor emulator for different driving amplitudes and frequencies are presented. With dimensionless system, the symmetry, equilibrium point and its stability are analysed. It is shown that the system has two unstable saddle-foci and one unstable saddle. A set of typical parameters are chosen for the generation of chaotic attractor. Differing from the common period-doubling bifurcation route in smooth dynamical systems, this memristive system shows abrupt transition from the coexisting period-1 limit cycles to robust chaos when varying system parameters. Various dynamical behaviors are analysed using the numerical simulations and circuit verifications.

Bifurcation and Chaos in indirect field-oriented controlled induction motor drive system

In this paper, the higher-order three-phase indirect field-oriented controlled (IFOC) induction motor is modeled in synchronous reference frame. The IFOC induction motor is analyzed numerically to investigate the system behavior due to control parameters change. The slip speed compensator, integral, and proportional gains variations of the speed loop controller are used to confirm the system dynamics. The simulation results show that the chaos behavior is noted in small region of the slip speed compensator gain which is difficult to be observed through the dominant limit cycles and fixed-point solutions. The effects of the integral and proportional gains change on the system dynamic are verified. Also, the system exhibits period-doubling bifurcation (period-2, period-4, period-8, and period-16) route to chaos. The bifurcation diagram and Lyapunov exponent spectrum assign these situations. The phase portrait and time response results are also presented. The system has multistability and coexistence of different attractors for the same system parameters as represented in the basins of attractions plots.

Arch microbeam bifurcation gas sensors

We investigate the potential of electrostatic initially curved microbeams to serve as bifurcation gas sensors. Toward that end, we develop static and dynamic reduced-order models of those beams and investigate their nonlinear response. Unlike many models, ours takes into account naturally occurring asymmetries present in fabricated microbeams. We conduct a detailed analysis of the nonlinear dynamics of arch beams focused on their exploitation for inertial sensing applications. The static response reveals that accounting for asymmetry replaces the saddle-node bifurcation, where snap-back occurs, with a symmetry-breaking bifurcation and reduced the voltage range for bistability. The dynamic analysis shows that a symmetry-breaking bifurcation precludes dynamic snap-through in the vicinity of superharmonic resonance, thereby significantly reducing the amplitude of those oscillations. It also shows evidence of a period-doubling bifurcation route to chaos in the vicinity of primary resonance. Based on these findings, we present a novel phase-based bifurcation gas sensor. The proposed detection mechanism allows, for the first time, the use of transition from regular periodic to chaotic motions in inertial sensing of gases. The sensor operation point is set close to the cyclic-fold bifurcation in the vicinity of primary resonance. When mass added by gas immobilization on the detector layer exceeds a threshold, the sensor oscillations abruptly transition from regular periodic motions to chaotic motions. This change can be detected by monitoring the response phase angle as it undergoes a major shift from slow variation within a limited range to fast variation over the full range due to the stretching and folding of the chaotic attractor. The proposed detection mechanism allows the sensor to operate in binary (digital) and analog modes. This is achieved by evaluating the RMS of the response phase angle $$\bar{\varphi }$$ and using it to either detect a gas concentration in excess of a safe threshold as an abrupt jump in $$\bar{\varphi }$$ or via a calibration curve relating $$\bar{\varphi }$$ to the gas concentration. The minimum detectable mass of the sensor is found to be 120 pg.

Hyperchaos from DTC Induction Motor Drive System

Abstract In the present work, a 3-phase induction machine model has been derived in a stationary-reference-frame. The strategy of the direct torque control (DTC) with a constant voltage-to-frequency ratio is used to control the obtained model. The obtained driven machine system involves nine nonlinear first-order differential equalities. The nine-dimensional system performance is investigated by using the numerical analysis due to changes in the parameter of the control part. The gain of the integral in the speed part of the controller is designated to investigate the dynamics of the modeled system. The results illustrate that the system has a period-doubling (period-1, period-2, and period-4) bifurcation route to chaotic oscillation. The bifurcation diagram and Lyapunov exponent spectrum assign these situations. Period-5 is noted in a window inside the bifurcation diagram. The large positive magnitudes of two of Lyapunov exponents show that the system is hyperchaos, which denotes the system shows very strong randomness and a high degree of disturbance. The system generates different coexisting attractors (periodic and chaotic attractors) for different values of the integral gain of the speed control loop.

Cosmological consequences of a scalar field with oscillating equation of state. II. Oscillating scaling and chaotic accelerating solutions

Multiacceleration scenario can be used to solve the cosmological coincidence problem. In this paper, after considering the early radiation era, we revisit the cosmological dynamics of the oscillating dark energy model proposed in [https://doi.org/10.1103/PhysRevD.101.063531, Phys. Rev. D {\bf 101}, 063531 (2020)]. We find this model allows the Universe evolves as oscillating scaling solution (OSS) in the radiation era and as chaotic accelerating solution (CAS) in the matter era. Mathematically, the transition from OSS to CAS is a route of period-doubling bifurcation to chaos. Physically, there are two reasons convince us that this scenario can be a nice picture to describe the real Universe. One is the global cosmological parameter constraints are practicable if the Universe evolves as OSS in the radiation era. The other is the late-time Universe described by CAS can successfully explain the observed cosmic acceleration at low redshifts.

Riddled Attraction Basin and Multistability in Three-Element-Based Memristive Circuit

By coupling a diode bridge-based second-order memristor and an active voltage-controlled memristor with a capacitor, a three-element-based memristive circuit is synthesized and its system model is then built. The boundedness of the three-element-based memristive circuit is theoretically proved by employing the contraction mapping principle. Besides, the stability distributions of equilibrium points are theoretically and numerically expounded in a 2D parameter plane. The results imply the memristive circuit has a zero unstable saddle focus and a pair of nonzero stable node-foci or unstable saddle-foci depending on the considered parameters. The dynamical behaviors include point attractor, period, chaos, coexisting bifurcation mode, period-doubling bifurcation route, and crisis scenarios, which are explored using some common dynamical methods. Of particular concern, riddled attraction basins and multistability are uncovered under two sets of specified model parameters nearing the tiny neighborhood of crisis scenarios by local attraction basins and phase plane plots. The riddled attraction basins with island-like structure demonstrate that their dynamical behaviors are extremely sensitive to the initial conditions, resulting in the coexistence of limit cycles with period-2 and period-6, as well as the coexistence of period-1 limit cycles and single-scroll chaotic attractors. Moreover, a feasible on-breadboard hardware circuit is manually made and the experimental measurements are executed, upon which phase plane trajectories for some discrete model parameters are captured to further confirm the numerically simulated ones.

Hidden attractors and multistability in a modified Chua’s circuit

The first hidden chaotic attractor was discovered in a dimensionless piecewise-linear Chua’s system with a special Chua’s diode. But designing such physical Chua’s circuit is a challenging task due to the distinct slopes of Chua’s diode. In this paper, a modified Chua’s circuit is implemented using a 5-segment piecewise-linear Chua’s diode. In particular, the coexisting phenomena of hidden attractors and three point attractors are noticed in the entire period-doubling bifurcation route. Attraction basins of different coexisting attractors are explored. It is demonstrated that the hidden attractors have very small basins of attraction not being connected with any fixed point. The PSIM circuit simulations and DSP-assisted experiments are presented to illustrate the existence of hidden attractors and coexisting attractors.

A Unique Jerk System with Abundant Dynamics: Symmetric or Asymmetric Bistability, Tristability, and Coexisting Bubbles

Finding and revealing new features and behaviors of simple chaotic systems have always been an important and attractive research topic. This paper aims to introduce a new 3D autonomous jerk chaotic system and explore its rich dynamical behaviors including period-doubling bifurcation and reverse period-doubling bifurcation routes to chaos, crisis and internal crisis, multiple symmetric coexisting attractors, and antimonotonicity. Especially, the phenomena of asymmetric bistability (e.g., coexistence of a point attractor and chaotic attractor or coexistence of a point attractor and period-5 limit cycle), tristability (e.g., coexistence of a point attractor and a pair of symmetric chaotic or periodic attractors), and coexisting bubbles are found, which have been rarely reported before. By using standard nonlinear analysis methods such as bifurcation diagrams, the largest Lyapunov exponent, phase portraits, Poincare sections, 0–1 test chart and the basin of attraction for an attractor, and the complex dynamical behaviors of the system are investigated in detail. Furthermore, a corresponding hardware electronic circuit is designed to verify the numerical simulations.

Bifurcations Caused by Feedback between Voltage and Intracellular Ion Concentrations in Ventricular Myocytes.

We develop an iterated map model to describe the bifurcations and complex dynamics caused by the feedback between voltage and intracellular Ca^{2+} and Na^{+} concentrations in paced ventricular myocytes. Voltage and Ca^{2+} can form either a positive or a negative feedback loop, while voltage and Na^{+} form a negative feedback loop. Under certain diseased conditions, when the feedback between voltage and Ca^{2+} is positive, Hopf bifurcations occur, leading to periodic oscillatory behaviors. When this feedback is negative, period-doubling bifurcation routes to alternans and chaos occur.

Numerical analyses and experimental validations of coexisting multiple attractors in Hopfield neural network

By simplifying connection topology of Hopfield neural network (HNN) with three neurons, a kind of HNN-based nonlinear system is proposed. Taking a coupling-connection weight as unique adjusting parameter and utilizing conventional dynamical analysis methods, dynamical behaviors with the variation of the adjusting parameter are discussed and coexisting multiple attractors’ behavior under different state initial values are investigated. The results imply that the HNN-based system displays point, periodic, and chaotic behaviors as well as period-doubling and tangent bifurcation routes; particularly, this system exhibits some striking phenomena of coexisting multiple attractors, such as, a pair of single-scroll chaotic attractors accompanied with a pair of periodic attractors, a pair of periodic attractors with two periodicities, and so on. Of particular interest, it should be highly significant that a hardware circuit of the HNN-based system is developed by using commercially available electronic components and many kinds of coexisting multiple attractors are captured from the hardware experiments. The results of the experimental measurements have well consistency to those of MATLAB and PSpice simulations.

Chaotic Resonance in Typical Routes to Chaos in the Izhikevich Neuron Model

Chaotic resonance (CR), in which a system responds to a weak signal through the effects of chaotic activities, is a known function of chaos in neural systems. The current belief suggests that chaotic states are induced by different routes to chaos in spiking neural systems. However, few studies have compared the efficiency of signal responses in CR across the different chaotic states in spiking neural systems. We focused herein on the Izhikevich neuron model, comparing the characteristics of CR in the chaotic states arising through the period-doubling or tangent bifurcation routes. We found that the signal response in CR had a unimodal maximum with respect to the stability of chaotic orbits in the tested chaotic states. Furthermore, the efficiency of signal responses at the edge of chaos became especially high as a result of synchronization between the input signal and the periodic component in chaotic spiking activity.

Memory-Induced Chaos in Cardiac Excitation.

Excitable systems display memory, but how memory affects the excitation dynamics of such systems remains to be elucidated. Here we use computer simulation of cardiac action potential models to demonstrate that memory can cause dynamical instabilities that result in complex excitation dynamics and chaos. We develop an iterated map model that correctly describes these dynamics and show that memory converts a monotonic first return map of action potential duration into a nonmonotonic one, resulting in a period-doubling bifurcation route to chaos.

Period-Timing Bifurcations in a Dispersion-Managed Fiber Laser With Zero Group Velocity Dispersion

We report on the numerical results of period timing bifurcations in a dispersion-managed (DM) fiber laser with zero group velocity dispersion (GVD).The fiber laser is passively mode-locked by using the nonlinear polarization rotation technique. Period-tripling bifurcation route to chaos is numerically observed apart from the conventional period-doubling bifurcation route to chaos. The small signal gain is the only varying parameter, while all the other cavity parameters are fixed. Different from the conventional period-doubling bifurcation route to chaos where the route begins from order, the period-tripling bifurcation route to chaos is led by a direct route from period one to chaos and then to a period-tripling state. The appearance of nonlinear window of period-tripling suggests that the ultrashort pulses generated from the DM fiber laser with zero GVD have rich nonlinear dynamics.

Asynchronous-Switching Map-Based Stability Effects of Circuit Parameters in Fixed Off-Time Controlled Buck Converter

Both constant on-time (COT) and fixed off-time (FOT) control techniques are suitable for various applications requiring fast transient response. However, the discrete-time model of COT controlled buck converter only has two switched borderlines, whereas that of FOT controlled buck converter has four switched borderlines. Based on the derivations of these borderlines, an asynchronous-switching map of FOT controlled buck converter is established. With the decrease of equivalent series resistance (ESR) of output capacitor, instability and mode shifting from continuous conduction mode (CCM) to discontinuous conduction mode (DCM) are discussed. Furthermore, with small ESR of output capacitor, stability effects of load resistance and inductance on dynamical behaviors are investigated, and the approximate stability criteria and the corresponding normalized critical conditions are obtained. The theoretical analyses and experimental results show that the converter operates in DCM chaos via period-doubling and border-collision bifurcation routes, and its instability caused by small ESR can be removed by choosing appropriate load resistance, inductance, and voltage transfer ratio, which are very suitable for the circuit design of FOT controlled buck converter.

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

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.

Chaos suppression for a four-dimensional fundamental power system model using adaptive feedback control

In this paper, the problem of chaos suppression for a four-dimensional fundamental power system (FDFPS) model is considered via the design of a novel adaptive feedback controller. The period doubling bifurcation route to chaos and some dynamical behaviors of the power system are investigated in detail. Based on stability analysis using an energy-type Lyapunov function, a single adaptive feedback controller is derived to suppress chaotic oscillation in four-dimensional fundamental power systems. The proposed controller simplifies the design of power system stabilizer and provides an easy way to implement in practical power system control. In addition, effectiveness of damping out chaotic oscillation and robustness against parameter uncertainty and external disturbance also make the proposed control scheme applicable for industrial application. Simulation results illustrate the effectiveness, the robustness and the superiority of proposed control method.

Propagation of envelope bright breather coupled wave modes in Ablowitz–Ladik chains

We investigate a model of a two coupled Ablowitz–Ladik (AL) lattices which admits the integrable AL model, when it is reduced to symmetric states. We compute the discrete modulational instability gain analytically and verified numerically. The play-role of the coupling parameter ϵ which is the back-bone of the two coupled AL chain is analyzed. We perform molecular dynamics (MD) simulations and elucidate the short time as well as the long time instability of AL chain. We demonstrate that the AL chain supports the formation of bright breather coupled wave modes spanning over 36 lattice sites on either side of the bond-center. We explore a kink solitary profile of solutions for the AL chains by invoking the extended tangent hyperbolic function method (ETHF) embedded with symbolic computation. Spectral stability of the exact solutions is determined by the eigenvalues of the discrete spectrum. We demonstrate the onset of chaos that leads the quasi-periodic dynamics of AL chains through the period-doubling bifurcation route to chaos.

Periodic solution and bifurcation of a suspension vibration system by incremental harmonic balance and continuation method

A compressed air generator hang under vehicle is simplified as a suspension mass connected to a vertical spring and two horizontal springs. It is configured generally as a geometrical negative stiffness to reduce dynamic stiffness. The periodic motion, chaotic motion and bifurcation of the compressed air generator model are investigated using the incremental harmonic balance method in combination with arc length continuation technique. The stability and bifurcation route are also distinguished with Floquet theory. The system exhibits a period doubling bifurcation route to chaos in different regions of excitation frequency. The stiffness ratio of the vertical spring and the horizontal spring has a significant influence on the dynamic response. When the vertical stiffness is close to the stiffness at horizontal direction, resonance occurs with the emergence of the chaotic motion. The dynamic response of the vibration system can be improved by reducing the stiffness in the horizontal direction to increase the stiffness ratio.