Periodic Parametric Perturbations Research Articles

Periodic parametric perturbation control for a 3D autonomous chaotic system and its dynamics at infinity

Periodic parametric perturbation control and dynamics at infinity for a 3D autonomous quadratic chaotic system are studied in this paper. Using the Melnikov's method, the existence of homoclinic orbits, oscillating periodic orbits and rotating periodic orbits are discussed after transferring the 3D autonomous chaotic system to a slowly varying oscillator. Moreover, the parameter bifurcation conditions of these orbits are obtained. In order to study the global structure, the dynamics at infinity of this system are analyzed through Poincare compactification. The simulation results demonstrate feasibility of periodic parametric perturbation control technology and correctness of the theoretical results.

Computional Dynamics for Diffusionless Lorenz Equations with Periodic Parametric Perturbation

The dynamics of diffusionless Lorenz equations (DLE) with periodic parametric perturbation is studied through numerical and experimental investigations in this paper. A method for calculating Lyapunov exponents (LEs), Lyapunov dimension (LD) from time series is presented. Furthermore, bifurcation and some complex dynamic behaviors such as periodic, quasi-periodic motion and chaos which occurred in the system are analyzed. And an algorithm for detecting unstable periodic orbits (UPOs) is presented. Also, give some numerical simulation studies of the system in order to verify the analytic results.

Homoclinic orbits analysis of T chaotic system with periodic parametric perturbation

Using Melnikov method we have analysed and calculated the homoclinic orbits of a slowly varying oscillator, derived from the T chaotic system with generalized Hamiltonian structure under periodic parametric perturbation. Also the parameter bifurcation conditions of homoclinic orbits are obtained. The simulation results demonstrate the feasibility of periodic parametric perturbation control technology, and the correctness of the discussion in this paper.

CHAOTIC CONTROL OF LIU SYSTEM WITH PERIODIC PARAMETRIC PERTURBATIONS

In this paper, periodic parametric perturbations are used to control chaos in Liu system. By changing the frequency of the perturbation signal, Liu system can be guided to not only periodic motion but also hyperchaos. Phase diagram, bifurcation diagram, Lyapunov exponents spectrum and Poincáre map are used to analyze the dynamic behavior of the controlled system in the numerical simulations, which show the effectiveness of our method.

Stabilization of chaotic oscillations in systems with a hyperbolic-type attractor

It has been shown that the chaotic dynamics of systems with nearly hyperbolic-type attractors can be stabilized by periodic parametric perturbations.

Chaos Control in Duffing System Using Impulsive Parametric Perturbations

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this brief, a new scheme is developed for chaos control by applying periodic impulsive parametric perturbations. Based on Melnikov's condition for the existence of chaos, it is mathematically proven that, in a neighborhood of a homoclinic orbit of the Duffing system, chaos can be suppressed. A sufficient condition is also established, serving as the design criterion for the amplitude and the width of the impulsive control signal. Finally, the control effect will clearly be demonstrated with simulation results. </para>

CONTROLLING LIU SYSTEM WITH DIFFERENT METHODS

In this paper, two different kinds of methods are adopted to control Liu system — feedback method and nonfeedback method. On the one hand, direct feedback and adaptive time-delayed feedback are taken as examples for the study of feedback control. In the direct feedback method, Liu system can be stabilized at one equilibrium point or a limit cycle surrounding its equilibrium. In the adaptive time-delayed feedback method, feedback coefficient and delay time can be adjusted adaptively to stabilize Liu system at its original unstable periodic orbit. On the other hand, periodic parametric perturbation is used to control chaos in Liu system as a typical nonfeedback method. By changing the frequency of the perturbation signal, Liu system can be guided to not only periodic motion but also hyperchaos. Numerical simulations show the effectiveness of our methods.

DYNAMICAL STATES IN A RING OF FOUR MUTUALLY COUPLED SELF-SUSTAINED ELECTRICAL SYSTEMS WITH TIME PERIODIC COUPLING

We investigate in this Letter different dynamical states in the ring of four mutually coupled self-sustained electrical systems with time periodic coupling. The transition boundaries that can occur between instability and complete synchronization states when the coupling strength varies are derived using the Floquet theory and the Whittaker method. The effects of the amplitude of the periodic parametric perturbations of the coupling parameter on the stability boundaries are analyzed. Numerical simulations are then performed to complement the analytical results.

Controlling chaos with periodic parametric perturbations in Lorenz system

Abstract Using periodic parametric perturbation to control chaos in Lorenz system will be finished in this paper. Based on the chaotic perturbation criteria—Melnikov method, a appropriate parametric condition for controlling chaos will be given out, which guide the system from chaotic motion to low-periodic motion. Some results of the numerical simulation are also explained clearly by rigorous analysis.

Role of nonlinear dissipation in the suppression of chaotic escape from a potential well

The inhibition of chaotic escape from a universal escape oscillator due to a periodic parametric perturbation of the quadratic potential term is studied theoretically by means of Poincaré–Melnikov–Arnold analysis, and the predictions are tested against numerical simulations based on a high-resolution grid of initial conditions. It is shown that chaotic escape suppression is impossible under period-1 and period-2 parametric perturbations. The role of a nonlinear damping term, proportional to the nth power of the velocity, on the inhibition scenario is also discussed.

COMPARATIVE ANALYSIS OF A HEURISTIC CONTROL OF CHAOS ALGORITHM IN SOME MODEL SYSTEMS

In this work we propose a Heuristic algorithm based on parametric perturbation for control of chaos in systems of ODEs. This method appears to be the first to apply a perturbation to a parameter in a system of ODEs that is active over a finite length of time only. The method proposed is comparatively easy to implement and needs almost no information about the system beyond the existence of a system-wide controllable parameter. We demonstrate certain advantages of this technique over two well-known algorithms, namely, control by periodic parametric perturbation and control by addition of a second weak periodic force, such as the possibility of switching behavior, pretargetting the period of the controlled orbit, stabilizing high period orbits etc. We also demonstrate the applicability of the technique in certain numerical models of physical systems and also in a rather difficult model problem, viz. the control of the rheological parameters of periodically forced suspensions of slender rods in simple shear flow.

Controlling chaos in low- and high-dimensional systems with periodic parametric perturbations.

The effect of applying a periodic perturbation to an accessible parameter of various chaotic systems is examined. Numerical results indicate that perturbation frequencies near the natural frequencies of the unstable periodic orbits of the chaotic systems can result in limit cycles for relatively small perturbations. Such perturbations can also control or significantly reduce the dimension of high-dimensional systems. Initial application to the control of fluctuations in a prototypical magnetic fusion plasma device will be reviewed.

Controlling chaos in a high dimensional system with periodic parametric perturbations

The effect of applying a periodic perturbation to an accessible parameter of a high-dimensional (coupled-Lorenz) chaotic system is examined. Numerical results indicate that perturbation frequencies near the natural frequencies of the unstable periodic orbits of the chaotic system can result in limit cycles or significantly reduced dimension for relatively small perturbations.

Synchronization of chaotic oscillators by periodic parametric perturbations

We show that the synchronization of coupled chaotic oscillators can be achieved by means of periodic parametric perturbations of the coupling element. The possibility of synchronization is demonstrated for the simple case of two identical nonautonomous oscillators with piecewise linear characteristics both analytically and numerically. Using linear analysis we have determined the stability conditions for symmetric oscillations.

Suppression of chaos by selective resonant parametric perturbations.

It is shown that, depending on its amplitude, frequency, and initial phase, a time-dependent periodic parametric perturbation can suppress chaos in nonlinear oscillators. The example of the Duffing-Holmes oscillator is used to demonstrate that all the numerically and experimentally observed phenomenology is theoretically explained by using the Melnikov-Holmes method and suppression of chaos is seen to be possible when certain resonant frequencies are involved.

Kink stability, propagation, and length-scale competition in the periodically modulated sine-Gordon equation.

We have examined the dynamical behavior of the kink solutions of the one-dimensional sine-Gordon equation in the presence of a spatially periodic parametric perturbation. Our study clarifies and extends the currently available knowledge on this and related nonlinear problems in four directions. First, we present the results of a numerical simulation program that are not compatible with the existence of a radiative threshold predicted by earlier calculations. Second, we carry out a perturbative calculation that helps interpret those previous predictions, enabling us to understand in depth our numerical results. Third, we apply the collective coordinate formalism to this system and demonstrate numerically that it reproduces accurately the observed kink dynamics. Fourth, we report on the occurrence of length-scale competition in this system and show how it can be understood by means of linear stability analysis. Finally, we conclude by summarizing the general physical framework that arises from our study.

Bifurcation, Chaos and Suppression of Chaos in FitzHugh-Nagumo Nerve Conduction Model Equation

We study the effect of constant and periodic membrane currents in neuronal axons described by the FitzHugh-Nagumo equation in its wave form. Linear stability analysis is carried out in the absence of periodic membrane current. Occurrence of chaotic motion, (i) in the absence of both constant and periodic membrane currents, (ii) with constant current only, (iii) with periodic membrane current only, and (iv) with both constant and periodic currents is investigated for specific parametric choices. We show how chaos sets in through a cascade of period doubling bifurcations. We then demonstrate the possibility of control of chaos using various control mechanisms. Specifically, we show the control of chaos by (i) adaptive control mechanism, (ii) periodic parametric perturbation and (iii) stabilization of unstable periodic orbits.

Controlling of chaos by weak periodic perturbations in Duffing-van der Pol oscillator

This paper investigates the possibility of controlling horseshoe and asymptotic chaos in the Duffing-van der Pol oscillator by both periodic parametric perturbation and addition of second periodic force. Using Melnikov method the effect of weak perturbations on horseshoe chaos is studied. Parametric regimes where suppression of horseshoe occurs are predicted. Analytical predictions are demonstrated through direct numerical simulations. Starting from asymptotic chaos we show the recovery of periodic motion for a range of values of amplitude and frequency of the periodic perturbations. Interestingly, suppression of chaos is found in the parametric regimes where the Melnikov function does not change sign.

Resonant parametric perturbations of the Hopf bifurcation

The effects of periodic parametric perturbations on a system undergoing Hopf bifurcation are studied in detail. Of primary interest is the case of resonance between the Hopf bifurcation frequency and the perturbation frequency. Method of alternative problems is used to obtain the small nonlinear periodic solutions. It is shown that for a certain range of parameters, the Hopf solution is modified to give rise to jump response as well as isolated solutions. For some parameter combinations, stable solutions can get unstable and may bifurcate into aperiodic or amplitude modulated motions.