Control Of Spatiotemporal Chaos Research Articles

Prediction and control of spatiotemporal chaos by learning conjugate tubular neighborhoods

I present a data-driven predictive modeling tool that is applicable to high-dimensional chaotic systems with unstable periodic orbits. The basic idea is using deep neural networks to learn coordinate transformations between the trajectories in the periodic orbits’ neighborhoods and those of low-dimensional linear systems in a latent space. I argue that the resulting models are partially interpretable since their latent-space dynamics is fully understood. To illustrate the method, I apply it to the numerical solutions of the Kuramoto–Sivashinsky partial differential equation in one dimension. Besides the forward-time predictions, I also show that these models can be leveraged for control.

Data-driven control of spatiotemporal chaos with reduced-order neural ODE-based models and reinforcement learning

Deep reinforcement learning (RL) is a data-driven method capable of discovering complex control strategies for high-dimensional systems, making it promising for flow control applications. In particular, the present work is motivated by the goal of reducing energy dissipation in turbulent flows, and the example considered is the spatiotemporally chaotic dynamics of the Kuramoto–Sivashinsky equation (KSE). A major challenge associated with RL is that substantial training data must be generated by repeatedly interacting with the target system, making it costly when the system is computationally or experimentally expensive. We mitigate this challenge in a data-driven manner by combining dimensionality reduction via an autoencoder with a neural ODE framework to obtain a low-dimensional dynamical model from just a limited data set. We substitute this data-driven reduced-order model (ROM) in place of the true system during RL training to efficiently estimate the optimal policy, which can then be deployed on the true system. For the KSE actuated with localized forcing (‘jets’) at four locations, we demonstrate that we are able to learn a ROM that accurately captures the actuated dynamics as well as the underlying natural dynamics just from snapshots of the KSE experiencing random actuations. Using this ROM and a control objective of minimizing dissipation and power cost, we extract a control policy from it using deep RL. We show that the ROM-based control strategy translates well to the true KSE and highlight that the RL agent discovers and stabilizes an underlying forced equilibrium solution of the KSE system. We show that this forced equilibrium captured in the ROM and discovered through RL is related to an existing known equilibrium solution of the natural KSE.

Pattern formation and control of spatiotemporal chaos in a reaction diffusion prey–predator system supplying additional food

Spatiotemporal dynamics of a predator–prey system in presence of spatial diffusion is investigated in presence of additional food exists for predators. Conditions for stability of Hopf as well as Turing patterns in a spatial domain are determined by making use of the linear stability analysis. Impact of additional food is clear from these conditions. Numerical simulation results are presented in order to validate the analytical findings. Finally numerical simulations are carried out around the steady state under zero flux boundary conditions. With the help of numerical simulations, the different types of spatial patterns (including stationary spatial pattern, oscillatory pattern, and spatiotemporal chaos) are identified in this diffusive predator–prey system in presence of additional food, depending on the quantity, quality of the additional food and the spatial domain and other parameters of the model. The key observation is that spatiotemporal chaos can be controlled supplying suitable additional food to predator. These investigations may be useful to understand complex spatiotemporal dynamics of population dynamical models in presence of additional food.

Nonlinear diffusion control of defect turbulence in cubic-quintic complex Ginzburg-Landau equation

The control of spatiotemporal chaos generated by traveling holes in spatially extended oscillatory media near a subcritical Hopf bifurcation is investigated. Analytical and numerical approaches are proposed for the purpose of this study. This work is done by using the nonlinear diffusion parameter control. We show that the unstable traveling hole in the one-dimensional cubic-quintic complex Ginzburg-Landau equation can be effectively stabilized in the chaotic regime.

Low-energy control of electrical turbulence in the heart

Controlling the complex spatio-temporal dynamics underlying life-threatening cardiac arrhythmias such as fibrillation is extremely difficult due to the nonlinear interaction of excitation waves within a heterogeneous anatomical substrate1–4. Lacking a better strategy, strong, globally resetting electrical shocks remain the only reliable treatment for cardiac fibrillation5–7. Here, we establish the relation between the response of the tissue to an electric field and the spatial distribution of heterogeneities of the scale-free coronary vascular structure. We show that in response to a pulsed electric field E, these heterogeneities serve as nucleation sites for the generation of intramural electrical waves with a source density ρ(E), and a characteristic time τ for tissue depolarization that obeys a power law τ∝Eα. These intramural wave sources permit targeting of electrical turbulence near the cores of the vortices of electrical activity that drive complex fibrillatory dynamics. We show in vitro that simultaneous and direct access to multiple vortex cores results in rapid synchronization of cardiac tissue and therefore efficient termination of fibrillation. Using this novel control strategy, we demonstrate, for the first time, low-energy termination of fibrillation in vivo. Our results give new insights into the mechanisms and dynamics underlying the control of spatio-temporal chaos in heterogeneous excitable media and at the same time provide new research perspectives towards alternative, life-saving low-energy defibrillation techniques.

Transition zone in controlling spatiotemporal chaos

The controllability of spatiotemporal chaos is investigated. Contrary to our common sense that there is only one transition point (i.e., a critical control strength) for successful control, we find that actually a transition zone exists, connecting two transition points for the local and global stabilities of the controlled state, respectively. Within the zone, the controllable probability increases from zero to one for random initial conditions. This behavior is found to be very generic and is expected to have a severe consequence in realistic applications in the control of spatiotemporal chaos.

Spatiotemporal chaos in capacitively coupled intrinsic Josephson junction arrays and control by ashunted LCR series resonance circuit

The dynamics of capacitively coupled intrinsic Josephson junction arrays with a McCumber parameterβc<1 driven by a dc current are investigated by using numerical simulations. For the first time,spatiotemporal chaotic behaviors are found. Analysis of the dynamics of individualjunctions shows that the chaos is caused by the strong diffusive coupling between thejunctions in the array. The chaotic regions of different parameters are also presented, fromwhich the conditions for chaos are deduced. The properties of the array shunted by anLCR series resonance circuit are also investigated. The results show thatthe spatiotemporal chaos can be effectively controlled by the shuntedLCR circuit. This presents a new spatiotemporal chaos control method.

The coupled feedback control of spatiotemporal chaos based on the flocking algorithms

Spatiotemporal chaos control in a one-dimensional nonlinear drift-wave equation is considered. We propose a method of coupled feedback to suppress the spatiotemporal chaos of the drift-wave,based on the flocking algorithms. By using real agents and virtual agents as our target, we show numerically that the spatiotemporal chaos can be controlled to a regular state if appropriate control strength is chosen. The physical mechanism is analyzed based on the correlation coefficient

Controll of spatiotemporal chaos by applying feedback method based on the flocking algorithms

Spatiotemporal chaos control in one-dimensional complex Ginzburg-Landau equation was described. We proposed a method of coupled feedback based on the flocking algorithms. By using the homogeneous-periodic solutions and the traveling wave solutions as our target, we showed numerically that the spatiotemporal chaos can be controlled to a regular state if appropriate control strength was chosen. The physical mechanism was analyzed based on the correlation function.

CONTROLLING TURBULENCE VIA TARGET WAVES GENERATED BY LOCAL PHASE SPACE COMPRESSION

The control of spatiotemporal chaos by developing target waves via local phase space compression in dynamical systems described by partial differential equation is reported. By considering spatially one-dimensional (1D) and two-dimensional (2D) complex Ginzburg–Landau equations (CGLE), we find that by using local system perturbation and confining variable in a finite range within a controlled area, periodic signals are generated and spread into an uncontrolled area in 1D spatiotemporal systems with weak turbulence. In 2D systems, target waves can be developed to suppress spatiotemporal chaos by using this compression technique in the convective instability area. The controllable parameters are investigated by performing numerical simulations.

Spatiotemporal chaos control in two-wave driven systems

Spatiotemporal chaos control is considered by taking a one-dimensional driven/damped nonlinear drift-wave equation as a model. We apply an additional sinusoidal wave to suppress spatiotemporal chaos, and the system becomes a two-sinusoidal-wave driven system (the original driving wave with frequency ω and an additional controlling wave with frequency Ω). Numerical simulations show that when the frequency of the controlling wave is in the proper range, spatiotemporal chaos can be modified into a regular state where the amplitudes of all modes vary periodically with frequency Ω-ω while the phases of all modes evolve quasi-periodically with a running frequency Ω overlapped by a small modulation of frequency Ω-ω. The physical reason for this peculiar phenomenon is attributed to a frequency entrainment in the competition of the two external waves.

Control of spatiotemporal chaos in catalytic CO oxidation by laser-induced pacemakers

Control of spatiotemporal chaos is achieved in the catalytic oxidation of CO on Pt(110) by localized modification of the kinetic properties of the surface chemical reaction. In the experiment, a small temperature heterogeneity is created on the surface by a focused laser beam. This heterogeneity constitutes a pacemaker and starts to emit target waves. These waves slowly entrain the medium and suppress the spatiotemporal chaos that is present in the absence of control. We compare this experimental result with a numerical study of the Krischer-Eiswirth-Ertl model for CO oxidation on Pt(110). We confirm the experimental findings and identify regimes where complete and partial controls are possible.

Control of spatiotemporal chaos in coupled map lattice by discrete-time variable structure control

A discrete-time sliding mode controller is introduced for controlling spatiotemporal chaos modeled by coupled map lattices. The controller which is designed based on Lyapunov function satisfies reaching law such that tracking problem is solved. This method can apply for all types of coupled map lattices. Simulation results reveal the feasibility of the control method.

Control of spatiotemporal chaos: dependence of the minimum pinning distance on the spatial measure entropy

We investigate the control of spatiotemporal chaos by external forcing at equidistant points (pinning sites) in one-dimensional systems. A monotonic decrease of the minimum distance between pinning sites versus the spatial measure entropy (in the absence of forcing) can be obtained for an appropriate choice of the forcing procedure. Such a relation between a feature for control and the disorder of the uncontrolled system is shown for four systems: binary cellular automata, coupled logistic equations, a stick-slip model and coupled differential equations.

Impulsive control and synchronization of spatiotemporal chaos

Abstract The impulsive control of spatiotemporal chaos of a particular type of non-linear partial differential equations has been investigated. A criterion for the solutions of these partial differential equations to be equi-attractive in the large is determined and an estimate for the basin of attraction is given in terms of the impulse durations and the magnitude of the impulses. Extending these results to impulsively synchronize spatiotemporal chaos of the same type of partial differential equations is explored. A proof for the existence of a certain kind of impulses for synchronization such that the error dynamics is equi-attractive in the large, is established. A comparison of the developed theoretical model with other existent numerical models available in the literature has been studied. Several simulation results are given to confirm the theoretical results. Moreover, an investigation of the Lyapunov exponents of the error dynamics between impulsively synchronized spatiotemporal chaotic systems, is done to further confirm the theoretical results.

INVESTIGATION ON EVOLUTIONARY DETERMINISTIC CHAOS CONTROL

This contribution introduces an investigation on deterministic spatiotemporal chaos control based on evolutionary algorithms use. Two evolutionary algorithms are used for chaos control here: differential evolution and self-organizing migrating algorithm. Like a model of spatiotemporal chaos so called coupled map lattices are used. Main aim of this investigation was to show that evolutionary algorithms are capable of deterministic chaos control when cost function is properly defined. Investigation consists of four different case studies with increasing calculation complexity. For both algorithms each simulation was 50 times repeated to show and check robustness of used methods.

Control of spatio-temporal chaos in coupled map lattices by state feedback

In this paper, the stable control of spatiotemporal chaos in coupled map lattices is realized by state feedback method. The feedback mode can be either delayed or without delay. The control mode can be either continuous or pulsate. The numerical simulation results demonstrate that this method is an effective one.

Giant improvement of time-delayed feedback control by spatio-temporal filtering.

Control of spatio-temporal chaos by the time-delay autosynchronization method is improved by several orders of magnitude. Unstable time periodic patterns are efficiently stabilized if one employs filters and couplings which originate from the Floquet eigenvalue problem of the unstable orbit. We illustrate our scheme by an application to a globally coupled reaction-diffusion model which describes charge transport in semiconductor devices.

Controlling chemical turbulence by global delayed feedback: pattern formation in catalytic CO oxidation on Pt(110).

Control of spatiotemporal chaos is one of the central problems of nonlinear dynamics. We report on suppression of chemical turbulence by global delayed feedback using, as an example, catalytic carbon monoxide oxidation on a platinum (110) single-crystal surface and carbon monoxide partial pressure as the controlled feedback variable. When feedback intensity was increased, spiral-wave turbulence was transformed into new intermittent chaotic regimes with cascades of reproducing and annihilating local structures on the background of uniform oscillations. The global feedback further led to the development of cluster patterns and standing waves and to the stabilization of uniform oscillations. These findings are reproduced by theoretical simulations.

Analytical study of spatiotemporal chaos control by applying local injections

Spatiotemporal chaos control by applying local feedback injections is investigated analytically. The influence of gradient force on the controllability is investigated. It is shown that as the gradient force of the system is larger than a critical value, local control can reach very high efficiency to drive the turbulent system of infinite size to a regular target state by using a single control signal. The complex Ginzburg-Landau equation is used as a model to confirm the above analysis, and a four-wave-mixing mode is revealed to determine the dynamical behavior of the controlled system at the onset of instability.