Learning to control a complex multistable system

Gadaleta, Sabino; Dangelmayr, Gerhard

doi:10.1103/physreve.63.036217

Cited by 21 publications

(12 citation statements)

References 29 publications

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“…For example, it was shown recently [Feudel & Grebogi, 1997;Feudel et al, 1998;Kraut et al, 1999;Gadaleta & Dangelmayr, 2001] that the task of controlling complexity in multistable systems goes far beyond the original [Shinbrot et al, 1990] control-of-chaos idea. As well as stabilising the system on one of the attractors, one needs to control switching between them.…”

Section: Discussionmentioning

confidence: 99%

Optimal Fluctuations and the Control of Chaos

Luchinsky

Beri

Mannella

et al. 2002

Int. J. Bifurcation Chaos

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The energy-optimal migration of a chaotic oscillator from one attractor to another coexisting attractor is investigated via an analogy between the Hamiltonian theory of fluctuations and Hamiltonian formulation of the control problem. We demonstrate both on physical grounds and rigorously that the Wentzel-Freidlin Hamiltonian arising in the analysis of fluctuations is equivalent to Pontryagin's Hamiltonian in the control problem with an additive linear unrestricted control. The deterministic optimal control function is identified with the optimal fluctuational force. Numerical and analogue experiments undertaken to verify these ideas demonstrate that, in the limit of small noise intensity, fluctuational escape from the chaotic attractor occurs via a unique (optimal) path corresponding to a unique (optimal) fluctuational force. Initial conditions on the chaotic attractor are identified. The solution of the boundary value control problem for the Pontryagin Hamiltonian is found numerically. It is shown that this solution is approximated very accurately by the optimal fluctuational force found using statistical analysis of the escape trajectories. A second series of numerical experiments on the deterministic system (i.e. in the absence of noise) show that a control function of precisely the same shape and magnitude is indeed able to instigate escape. It is demonstrated that this control function minimizes the cost functional and the corresponding energy is found to be smaller than that obtained with some earlier adaptive control algorithms.

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Section: Discussionmentioning

confidence: 99%

Optimal Fluctuations and the Control of Chaos

Luchinsky

Beri

Mannella

et al. 2002

Int. J. Bifurcation Chaos

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“…A similar situation occurs when multistable systems require stabilization with respect to small perturbations like noise [Feudel & Grebogi, 1997 or when a control strategy is needed to switch from one desired attractor to another. Multistability is a fragile phenomenon: it can be suppressed by weak perturbations [Pisarchik & Corbalán, 1999;Pisarchik & Goswami, 2000;Chizhevsky, 2001], a feature that can be exploited in control techniques [Triandaf & Schwartz, 2000;Gadaleta & Dangelmayr, 2001;Pisarchik, 2001]. Our study is thus relevant in designing control strategies since these typically require knowledge about the basins of attraction: the asymptotic dynamics in the system without control depend crucially both on initial conditions as well as on the extent of the basin of the given attractor.…”

Section: Introductionmentioning

confidence: 99%

The Nature of Attractor Basins in Multistable Systems

Shrimali

Prasad

Ramaswamy

et al. 2008

Int. J. Bifurcation Chaos

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In systems that exhibit multistability, namely those that have more than one coexisting attractor, the basins of attraction evolve in specific ways with the creation of each new attractor. These multiple attractors can be created via different mechanisms. When an attractor is formed via a saddle-node bifurcation, the size of its basin increases as a power-law in the bifurcation parameter. In systems with weak dissipation, the basins of low-order periodic attractors increase linearly, while those of high-order periodic attractors decay exponentially as the dissipation is increased. These general features are illustrated for autonomous as well as driven mappings. In addition, the boundaries of the basins can also change from being smooth to fractal when a new attractor appears. Transitions in the basin boundary morphology are reflected in abrupt changes in the dependence of the uncertainty exponent on the bifurcation parameter.

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“…By introducing pseudoperiodic driving [14,15] or harmonic disturbance [16,17], the undesirable attractor types could be eliminated, and then, the system could be controlled in a certain stable state. In order to stabilize the system in a certain desired state, feedback control strategy was usually adopted [18], such as periodic driving [19] and time-delay feedback [13]. Yet these control strategies cannot achieve the multistable control.…”

Section: Multistability Generic Control Strategymentioning

confidence: 99%

“…Traditional control strategies usually adopted nonfeedback control strategy to convert a multistable system to a mono-stable system [13][14][15][16][17] or adopted feedback control strategy to stabilize the system in a certain desired state [13,18,19]. But these control strategies cannot achieve the multistable control.…”

Section: Introductionmentioning

confidence: 99%

Recent Advances in Dimensionality Reduction Modeling and Multistability Reconstitution of Memristive Circuit

2021

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Due to the introduction of memristors, the memristor-based nonlinear oscillator circuits readily present the state initial-dependent multistability (or extreme multistability), i.e., coexisting multiple attractors (or coexisting infinitely many attractors). The dimensionality reduction modeling for a memristive circuit is carried out to realize accurate prediction, quantitative analysis, and physical control of its multistability, which has become one of the hottest research topics in the field of information science. Based on these considerations, this paper briefly reviews the specific multistability phenomenon generating from the memristive circuit in the voltage-current domain and expounds the multistability control strategy. Then, this paper introduces the accurate flux-charge constitutive relation of memristors. Afterwards, the dimensionality reduction modeling method of the memristive circuits, i.e., the incremental flux-charge analysis method, is emphatically introduced, whose core idea is to implement the explicit expressions of the initial conditions in the flux-charge model and to discuss the feasibility and effectiveness of the multistability reconstitution of the memristive circuits using their flux-charge models. Furthermore, the incremental integral transformation method for modeling of the memristive system is reviewed by following the idea of the incremental flux-charge analysis method. The theory and application promotion of the dimensionality reduction modeling and multistability reconstitution are proceeded, and the application prospect is prospected by taking the synchronization application of the memristor-coupled system as an example.

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Learning to control a complex multistable system

Cited by 21 publications

References 29 publications

Optimal Fluctuations and the Control of Chaos

Optimal Fluctuations and the Control of Chaos

The Nature of Attractor Basins in Multistable Systems

Recent Advances in Dimensionality Reduction Modeling and Multistability Reconstitution of Memristive Circuit

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