Chaos Suppression Research Articles

Suppression of chaotic behavior in horizontal platform systems based on an adaptive sliding mode control scheme

This work presents an adaptive sliding mode control scheme to elucidate the robust chaos suppression control of non-autonomous chaotic systems. The proposed control scheme utilizes extended systems to ensure that continuous control input is obtained in order to avoid chattering phenomenon as frequently in conventional sliding mode control systems. A switching surface is adopted to ensure the relative ease in stabilizing the extended error dynamics in the sliding mode. An adaptive sliding mode controller (ASMC) is then derived to guarantee the occurrence of the sliding motion, even when the chaotic horizontal platform system (HPS) is undergoing parametric uncertainties. Based on Lyapunov stability theorem, control laws are derived. In addition to guaranteeing that uncertain horizontal platform chaotic systems can be stabilized to a steady state, the proposed control scheme ensures asymptotically tracking of any desired trajectory. Furthermore, the numerical simulations verify the accuracy of the proposed control scheme, which is applicable to another chaotic system based on the same design scheme.

Suppression of Chaos in a Bose–Einstein Condensate Loaded into a Moving Optical Superlattice Potential

We have shown that the application of modulating the secondary lattice is an efficient route to suppressing the generation of chaotic traveling waves of a Bose–Einstein Condensate with attractive interatomic interaction loaded into a moving optical superlattice consisting of two lattices. With the Melnikov method, we obtain the optimal value of the relative phase between the two lattice harmonics for the control of chaos. We also find that the regularization route as the potential depth of the secondary lattice is varied and fairly rich, including the period-doubling bifurcations.

Suppressing spiral waves and spatiotemporal chaos in cardiac tissue by using calcium channel agonist

Suppression of spiral waves and spatiotemporal chaos in cardiac tissue has been considered based on the LuoRudy 91 model. To suppress spiral waves and spatiotemporal chaos, the control strategy which employs calcium channel agonist to enhance the maximum conductivity of the slow inward calcium current is proposed. The numerical results show that the method can effectively suppress the spiral wave and spatiotemporal chaos in cardiac tissue even if the distribution of the time-dependent potassium current is non-uniform. The control mechanism has been analyzed.

Suppressing spiral waves and spatiotemporal chaos in cardiac tissue using local feedback

Suppression of spiral waves and spatiotemporal chaos are considered based on the LuoRud91model. We apply local feedback to suppress spiral wave and spatiotemporal chaos. Two control strategies are proposed, namely the static and the moving control methods. It is found that when related parameters are properly chosen spiral wave can be suppressed by static controller, whereas spatiotemporal chaos cannot be suppressed effectively. However, spiral wave and spatiotemporal chaos can be suppressed effectively by moving controller. The corresponding control speed depends on the traveling speed of controller. Spiral wave and spatiotemporal chaos can be suppressed in a short time when the control parameters are suitably chosen.

Synchronization and suppression of chaos in non-locally coupled map lattices

We considered coupled map lattices with long-range interactions to study the spatiotemporal behaviour of spatially extended dynamical systems. Coupled map lattices have been intensively investigated as models to understand many spatiotemporal phenomena observed in extended system, and consequently spatiotemporal chaos. We used the complex order parameter to quantify chaos synchronization for a one-dimensional chain of coupled logistic maps with a coupling strength which varies with the lattice in a power-law fashion. Depending on the range of the interactions, complete chaos synchronization and chaos suppression may be attained. Furthermore, we also calculated the Lyapunov dimension and the transversal distance to the synchronization manifold.

Effect of rectified and modulated sine forces on chaos in Duffing oscillator

The effect of rectified and modulated sine forces on the onset of horseshoe chaos is studied both analytically and numerically in the Duffing oscillator. With single force analytical threshold condition for the onset of horseshoe chaos is obtained using the Melnikov method. The Melnikov threshold curve is drawn in a parameter space. For the rectified sine wave, onset of cross-well asymptotic chaos is observed just above the Melnikov threshold curve. For the modulus of sine wave long time transient motion followed by a periodic attractor is realized. The possibility of controlling of chaos by the addition of second modulated force is then analyzed. Parametric regimes where suppression of horseshoe chaos occurs are predicted analytically and verified numerically. Interestingly, suppression of chaos is found in the parametric regimes where the Melnikov function does not change sign.

Synchronization regimes in conjugate coupled chaotic oscillators

Nonlinear oscillators that are mutually coupled via dissimilar (or conjugate) variables display distinct regimes of synchronous behavior. In identical chaotic oscillators diffusively coupled in this manner, complete synchronization occurs only by chaos suppression when the coupled subsystems drive each other into a regime of periodic dynamics. Furthermore, the coupling does not vanish but acts as an "internal" drive. When the oscillators are mismatched, phase synchronization occurs, while in a master slave configuration, generalized synchrony results. These effects are demonstrated in a system of coupled chaotic Rossler oscillators.

Spatiotemporal dynamics and transport reduction in helical magnetic configuration

Effects of multihelicity confinement magnetic fields on turbulent transport and zonal flows are investigated by means of spatiotemporal analysis of gyrokinetic Vlasov simulation results for the ion temperature gradient turbulence, where the standard and the inward-shifted configurations of the Large Helical Device are considered. The analysis of simulation results demonstrates that fluctuations of electrostatic potential for zonal flows exhibit spatiotemporal chaos in both configurations. However, the intensity of chaos found is considerably decreased in the inward-shifted configuration consistent with improved confinement. Enhanced zonal flow generation in the inward shifted case is accompanied by transport reduction which may be a direct consequence of chaos suppression.

Dynamical control of chaos by slave–master feedback

Techniques for stabilizing unstable state in nonlinear dynamical systems using small perturbations fall into three general categories: feedback, non-feedback schemes, and a combination of feedback and non-feedback. However, the general problem of finding conditions for creation or suppression of chaos still remains open. We describe a method for dynamical control of chaos. This method is based on a definition of the hierarchy of solvable chaotic maps with dynamical parameter as a control parameter. In order to study the new mechanism of control of chaotic process, Kolmogorov–Sinai entropy of the chaotic map with dynamical parameter based on discussion the properties of invariant measure have been calculated and confirmed by calculation of Lyapunov exponents. The introduced chaotic maps can be used as dynamical control.

CHAOS CONTROL IMPLEMENTATION BY INTERRUPTED ELECTRIC CIRCUIT WITH SWITCHING DELAY

We have demonstrated that the chaotic circuit with a switching delay is modeled by a return map, and a controller for the suppression of chaos is proposed. A circuit representing a controller stabilizing a period-1 unstable periodic orbit in an interrupted electric circuit with a certain switching delay is also discussed.

Control of the new 4th-order hyper-chaotic system with one input

We solve the problem of chaos suppression of Lu's hyper-chaotic system via feedback control. We use only one control input and moreover the controller is a simple proportional feedback and uses the measurement of only one variable. We show that this simple control law suffices to stabilize the hyper-chaotic system to the zero equilibrium globally and asymptotically. We present stability proofs based on Lyapunov's direct method and integration of solutions. As a corollary of our main result we draw the conclusion that the system is globally stabilizable by simply varying one parameter, when possible. Simulation experiments that show the effectiveness of our method are also presented.

Chaos control in AFM system using sliding mode control by backstepping design

This paper presents a robust algorithm to control the chaotic atomic force microscope system (AFMs) by backstepping design procedure. The proposed feedback controller is composed by a sliding mode control (SMC) and a backstepping feedback, so its implementation is quite simple and can be made on the basis of the measured signal. The developed control scheme allows chaos suppression despite uncertainties in the model as well as system external disturbances. The concept of extended system is used such that a continuous sliding mode control effort is generated using backstepping scheme. It is guaranteed that under the proposed control law, uncertain AFMs can asymptotically track target orbits. The converging speed of error states can be arbitrary turned by assigning the corresponding dynamics of the sliding surfaces. Numerical simulations demonstrate its advantages by stabilizing the unstable periodic orbits of the AFMs and this method can also be easily extended to elimination chaotic motion in any types of chaotic AFMs.

Chaos suppression and desynchronization phenomena in periodically coupled pendula subjected to localized heterogeneous forces

The suppressory effects of localized heterogeneous periodic pulses on the chaotic behaviour of sinusoidally coupled nonlinear pendula are studied. We assume that when the pendula are driven synchronously, i.e., all driving pulses have the same waveform, the chains display chaotic dynamics. It is shown that decreasing the impulse transmitted by the pulses of a minimal number of pendula results in regularization with the whole array exhibiting frequency synchronization over a wide range of coupling periods. These findings demonstrate that decreasing the impulse transmitted by localized external forces can tame chaos and lead to frequency-locked states in networks of periodically coupled dissipative systems.

Stochastic resonance in extended systems

Here, three highly nontrivial instances are briefly discussed, showing both the degree of maturity achieved by the field of stochastic resonance (SR) and the wealth of potential realizations of SR in extended systems: (1) a “bound” KPZ system, for which the knowledge of its nonequilibrium potential (NEP) makes simple the solution of a very complex problem; (2) the Ising-Bloch front bifurcation in an activator-inhibitor system, a case of SR between moving fronts of opposite chirality; (3) a spatiotemporally chaotic “toy model”, useful for climate research, exhibiting internal SR due to chaos suppression.

HYPERCHAOTIC SYNCHRONIZATION UNDER SQUARE SYMMETRY

We study two identical hyperchaotic oscillators symmetrically coupled. Each oscillator represents a codimension-2 Takens–Bogdanov bifurcation under square symmetry, and was used to model a convection experiment in a time-dependent state. In the coupled system, the Lyapunov exponents behavior against the coupling parameter is used to detect changes in the dynamics, and the synchronization state is controlled by checking the phase planes. Complete synchronization is achieved without chaos suppression in a coupling parameter interval. Outside this window, complete synchronization cannot be generally achieved. As a consequence of a bubbling transition, synchronization is obtained only for some particular values of the initial conditions.

Accounting the control effort to improve chaos suppression via robust adaptive feedback

This contribution deals with the optimisation of the chaos suppression via robust (low-parameterised) adaptive feedback. The robust adaptive feedback (RAF) has been proved to be capable of inducing the chaos suppression in face of uncertainties and exogenous disturbances. In this contribution the RAF scheme is accomplished with an optimal tuning scheme based on Riccatti equations. The underlying idea is to optimise the robust chaos suppression by accounting the control effort despite the saturation in electronic components. A finite horizon can be arbitrarily established by ensuring that the chaos suppression is achieved at established time. It should be noted that there is not exact convergence to the non-chaotic reference. As a matter of fact, only the suppression of the chaotic behaviour is aimed. An advantage is that the proposed scheme accounts the energy wasted by the controller and the closed-loop performance on suppression. Numerical and experimental results show the performance of the optimisation strategy.

Dynamical response of the non-linear vibration of single-walled carbon nanotubes

Dynamical response in single-walled carbon nanotubes (SWNTs) has been investigated through analytical and computational techniques. Analytically, an explicit expression for a formal control term has been estimated by using the Lie transforms. Computational studies reveal the suppression of chaos, which is in good agreement with the analytical investigations. And, our system follows regular patterns after the inclusion of control term.

Adaptive variable structure control for chaos suppression of unified chaotic systems

This paper investigates the chaos suppression problem for a class of unified chaotic systems subject to uncertainties and control input with dead-zone nonlinearity. Using the variable structure control technique, a robust adaptive control law is established to suppress chaos in a class of unified chaotic systems even with unknown uncertainties and dead-zone nonlinearity in input. In particular, a switching surface is newly proposed to simplify the task of ensuring the stability of the closed-loop system in the sliding mode. Numerical results validate the effectiveness of the proposed adaptive control scheme.

Control of stochastic chaos using sliding mode method

Stabilizing unstable periodic orbits of a deterministic chaotic system which is perturbed by a stochastic process is studied in this paper. The stochastic chaos is modeled by exciting a deterministic chaotic system with a white noise obtained from derivative of a Wiener process which eventually generates an Ito differential equation. It is also assumed that the chaotic system being studied has some model uncertainties which are not random. The sliding mode controller with some modifications is used for stochastic chaos suppression. It is shown that the system states converge to the desired orbit in such a way that the error covariance converges to an arbitrarily small bound around zero. As some case studies, the stabilization of 1-cycle and 2-cycle orbits of chaotic Duffing and Φ 6 Van der Pol systems is investigated by applying the proposed method to their corresponding stochastically perturbed systems. Simulation results show the effectiveness of the method and the accuracy of the statements proved in the paper.

Control of chaos in a three-well duffing system

Analytical and numerical results concerning control of chaos in a three-well duffing system with two external excitations are given by using the Melnikov methods proposed by Chacón et al. [Chacón R. General results on chaos suppression for biharmonically driven dissipative systems. Phys Lett A 1999;257:293–300, Chacón R, Palmero F, Balibrea F. Taming chaos in a driven Josephson Junction. Int J Bifurc Chaos 2001;11(7):1897–909, Chacón R. Role of ultrasubharmonic resonances in taming chaos by weak harmonic perturbations. Europhys Lett 2001;54(2):148C153]. We theoretically give the parameter-space region and intervals of initial phase difference for primary and subharmonic resonance and the necessary condition for the superharmonic and supersubharmonic resonance, where homoclinic chaos or heteroclinic chaos can be suppressed. Numerical simulations show the consistency and difference with theoretical analysis and the chaotic behavior can be converted to periodic orbits by adjusting amplitude and phase-difference of inhibiting excitation. Moreover, we consider the influence of parametric frequency on maximum Lyapunov exponent (LE) for different phase-differences, and give the distribution of maximum Lyapunov exponents in parameter-plane, which indicates the regions of non-chaotic states (non-positive LE) and chaotic states (positive LE).