A new hyperchaotic system and its circuit implementation

In this paper, a new hyperchaotic system is presented by adding a nonlinear controller to the three-dimensional autonomous chaotic system. The generated hyperchaotic system undergoes hyperchaos, chaos, and some different periodic orbits with control parameters changed. The complex dynamic behaviors are verified by means of Lyapunov exponent spectrum, bifurcation analysis, phase portraits and circuit realization. The Multisim results of the hyperchaotic circuit were well agreed with the simulation results.     

Dynamical behavior,chaos control and synchronization of a memristor-based ADVP circuit

This paper is devoted to study the dynamical behavior of a modified Autonomous Van der Pol-Duffing (ADVP) circuit when its nonlinear element is replaced by a flux controlled memristor. The bifurcation diagrams, Lyapunov exponents, and phase portraits of the state variables are presented. Then, the chaos which appears at certain values of the system’s parameters is controlled using linear feedback control. Finally, the synchronization between two chaotic modified ADVP circuits is achieved in the case of fully unknown parameters of the system using adaptive synchronization.     

Bifurcations and synchronization of the fractional-order simplified Lorenz hyperchaotic systems

In this paper, dynamics of the fractional-order simplied Lorenz hyperchaotic system is investigated. Modied Adams-Bashforth-Moulton method is applied for numerical simulation. Chaotic regions and periodic windows are identied. Dierent types of motions are shown along the routes to chaos by means of phase portraits, bifurcation diagrams, and the largest Lyapunov exponent. The lowest fractional order to generate chaos is 3.8584. Synchronization between two fractional-order simplied Lorenz hyperchaotic systems is achieved by using active control method. The synchronization performances are studied by changing the fractional order, eigenvalues and eigenvalue standard deviation of the error system.     

Hopf bifurcations, Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system

In this paper, complex dynamical behavior of a class of centrifugal flywheel governor system is studied. These systems have a rich variety of nonlinear behavior, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Bubbles of periodic orbits may also occur within the bifurcation sequence. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. This paper proposes a parametric open-plus-closed-loop approach to controlling chaos, which is capable of switching from chaotic motion to any desired periodic orbit. The theoretical work and numerical simulations of this paper can be extended to other systems. Finally, the results of this paper are of practical utility to designers of rotational machines.     

Control, anticontrol and synchronization of chaos for an autonomous rotational machine system with time-delay

Chaos, control, anticontrol and synchronization of chaos for an autonomous rotational machine system with a hexagonal centrifugal governor and spring for which time-delay effect is considered are studied in the paper. By applying numerical results, phase diagram and power spectrum are presented to observe periodic and chaotic motions. Linear feedback control and adaptive control algorithm are used to control chaos effectively. Linear and nonlinear feedback synchronization and phase synchronization for the coupled systems are presented. Finally, anticontrol of chaos for this system is also studied.     

Chaos anticontrol and synchronization of three time scales brushless DC motor system

Chaos anticontrol of three time scale brushless dc motors and chaos synchronization of different order systems are studied. Nondimensional dynamic equations of three time scale brushless DC motor system are presented. Using numerical results, such as phase diagram, bifurcation diagram, and Lyapunov exponent, periodic and chaotic motions can be observed. By adding constant term, periodic square wave, the periodic triangle wave, the periodic sawtooth wave, and kx|x| term, to achieve anticontrol of chaotic or periodic systems, it is found that more chaotic phenomena of the system can be observed. Then, by coupled terms and linearization of error dynamics, we obtain the partial synchronization of two different order systems, i.e. brushless DC motor system and rate gyroscope system.     

Bifurcations and chaos of a discrete mathematical model for respiratory process in bacterial culture

The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.     

Analysis of bifurcations for non-linear stochastic non-smooth vibro-impact system via top Lyapunov exponent

Bifurcations are discussed by the criterion of top Lyapunov exponent. Based on the local map and Kaminski’s algorithms, a general formulation of the top Lyapunov exponents is proposed for non-linear vibro-impact oscillators with Gaussian white noise perturbation. The analytical results are verified by phase portraits and bifurcation diagrams for a classical stochastic Duffing vibro-impact oscillator. Both results are consistent.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator

Systems constituted by moving components that make intermittent contacts with each other can be modelled by a system of ordinary differential equations containing piecewise linear terms. We consider a soft impact bilinear oscillator for which we obtain bifurcation diagrams, Lyapunov coefficients, return maps and phase portraits of the response. Besides Lyapunov coefficients diagrams, bifurcation diagrams are represented in terms of both non-dimensional time instants of contact (when the mass impacts the obstacle) and of portions of contact duration (the percentage-time interval when the material point is inside the obstacle) vs. non-dimensional external force frequency (or amplitude). The second kind of diagrams is needed because the contact duration (or the complementary flight time duration) are quantities that can easily be measured in an experiment aiming at confirming the validity of the present model. Lyapunov coefficients are evaluated converting the piecewise linear system of ordinary differential equations into a map, the so-called impact map, where time and velocity corresponding to a given impact are evaluated as functions of time and velocity corresponding to the previous impact. Thus, the usual methods related to this last map are used. The trajectories are represented in terms of return maps (all points in the time-velocity plane involved in the impact events) and phase portraits (the trajectory-itself in the displacement-velocity plane). In the bifurcation diagrams, transition between different responses is evidenced and a perfect correlation between chaotic (periodic) attractors and positive (negative) values of the maximum Lyapunov coefficient is found.     

旋转流动的低模分析及仿真研究

为了探讨Couette-Taylor流从层流到湍流过渡的方式以及流动发展到湍流之后混沌吸引子的某些特征等问题,采用低模分析方法研究了Couette-Taylor流的部分动力学行为及仿真问题,讨论了Couette-Taylor流三模态类Lorenz型方程组的动力学行为,包括定态的失稳、极限环的出现、分岔与混沌的演变和全局稳定性分析等。通过线性稳定性分析和数值模拟等方法给出了此三维模型分岔与混沌等动力学行为及其演化历程,并借此解释了Couette-Taylor流试验中观察到的部分涡流的演化过程.基于系统的分岔图、Lyapunov指数谱、功率谱、Poincaré(庞加莱)截面和返回映射等揭示了系统混沌行为的普适特征.     

Chaos in a modified van der Pol system and in its fractional order systems

Chaos in a modified van der Pol system and in its fractional order systems is studied in this paper. It is found that chaos exists both in the system and in the fractional order systems with order from 1.8 down to 0.8 much less than the number of states of the system, two. By phase portraits, Poincaré maps and bifurcation diagrams, the chaotic behaviors of fractional order modified van der Pol systems are presented.     

Bifurcation and chaos in a discrete genetic toggle switch system

A discrete genetic toggle switch system obtained by Euler method is first investigated. The conditions of existence for fold bifurcation and flip bifurcation are derived by using center manifold theorem and bifurcation theory. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behavior. We show the period 3 to 13 windows in different chaotic regions, period-doubling bifurcation or inverse period-doubling bifurcation from period-2 to 12 orbits leading to chaos, different kind of interior crisis and boundary crisis, intermittency behavior, chaotic set, chaotic non-attracting set, coexistence of period points with invariant cycles, and so on. The influence of the amplitude and frequency of excitable forcing on the system are also first considered by using numerical simulation. A different type of quasiperiodic orbits, jumping behaviors of quasiperiodic set from one set to another set, and the processes from quasiperiodic orbits to strange non-chaotic attractor are found.     

Hopf bifurcation in an hexagonal governor system with a spring

The complex dynamical behaviors of the hexagonal governor system with a spring are studied in this paper. We go deeper investigating the stability of the equilibrium points in the hexagonal governor system with a spring. These systems have a rich variety of nonlinear behaviors, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. By studying numerical simulations, it is possible to provide reliable theory and effective numerical method for other systems.     

Generation of hyperchaos from the Chen–Lee system via sinusoidal perturbation

A system with more than one positive Lyapunov exponent can be classified as a hyperchaotic system. In this study, a sinusoidal perturbation was designed for generating hyperchaos from the Chen–Lee chaotic system. The hyperchaos was identified by the existence of two positive Lyapunov exponents and bifurcation diagrams. The system is hyperchaotic in several different regions of the parameters c, ε, and ω. It was found that this method not only can enhance or suppress chaotic behavior, but also induces chaos in non-chaotic parameter ranges. In addition, two interesting dynamical behaviors, Hopf bifurcation and intermittency, were also found in this study.     

Chaos control by harmonic excitation with proper random phase

Chaos control may have a dual function: to suppress chaos or to generate it. We are interested in a kind of chaos control by exerting a weak harmonic excitation with random phase. The dual function of chaos control in a nonlinear dynamic system, whether a suppressing one or a generating one, can be realized by properly adjusting the level of random phase and determined by the sign of the top Lyapunov exponent of the system response. Two illustrative examples, a Duffing oscillator subject to a harmonic parametric control and a driven Murali-Lakshmanan-Chua (MLC) circuit imposed with a weak harmonic control, are presented here to show that the random phase plays a decisive role for control function. The method for computing the top Lyapunov exponent is based on Khasminskii's formulation for linearized systems. Then, the obtained results are further verified by the Poincare map analysis on dynamical behavior of the system, such as stability, bifurcation and chaos. Both two methods lead to fully consistent results.     

A new hyperchaotic system and its synchronization

In this paper, a four-dimensional (4D) continuous autonomous hyperchaotic system is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the 3D autonomous chaotic system with a reverse butterfly-shape attractor. Some of its basic dynamical properties, such as Lyapunov exponents, Poincare section, bifurcation diagram and the periodic orbits evolving into chaotic, hyperchaotic dynamical behavior by varying parameter d are studied. Furthermore, the full state hybrid projective synchronization (FSHPS) of new hyperchaotic system with unknown parameters including the unknown coefficients of nonlinear terms is studied by using adaptive control. Numerical simulations are presented to show the effective of the proposed chaos synchronization scheme.     

Chaos control for the plates subjected to subsonic flow

The suppression of chaotic motion in viscoelastic plates driven by external subsonic air flow is studied. Nonlinear oscillation of the plate is modeled by the von-Kármán plate theory. The fluid-solid interaction is taken into account. Galerkin’s approach is employed to transform the partial differential equations of the system into the time domain. The corresponding homoclinic orbits of the unperturbed Hamiltonian system are obtained. In order to study the chaotic behavior of the plate, Melnikov’s integral is analytically applied and the threshold of the excitation amplitude and frequency for the occurrence of chaos is presented. It is found that adding a parametric perturbation to the system in terms of an excitation with the same frequency of the external force can lead to eliminate chaos. Variations of the Lyapunov exponent and bifurcation diagrams are provided to analyze the chaotic and periodic responses. Two perturbation-based control strategies are proposed. In the first scenario, the amplitude of control forces reads a constant value that should be precisely determined. In the second strategy, this amplitude can be proportional to the deflection of the plate. The performance of each controller is investigated and it is found that the second scenario would be more efficient.     

Dynamic analysis, controlling chaos and chaotification of a SMIB power system

The dynamic behaviors of a SMIB power system are studied in this paper. A single modal equation is used to analyze the qualitative behaviors of the system. The famous equation of motion is called “swing equation”. The Lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of the system. The bifurcation of the parameter dependent system is studied numerically. Besides, the phase portraits, the Poincaré maps, and the Lyapunov exponents are presented to observe periodic and chaotic motions. Further, the addition of periodic force and the feedback control are used to control chaos effectively. Finally, the chaotification problem of the SMIB power system is also issued.