Finding unstable periodic orbits: A hybrid approach with polynomial optimization

Lakshmi, Mayur; Fantuzzi, Giovanni; Chernyshenko, S. I.; Lasagna, Davide

doi:10.1016/j.physd.2021.133009

Cited by 4 publications

(2 citation statements)

References 42 publications

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“…This strategy outperforms data-driven methods that first identify a model for the Poincaré map data [14][15][16], because small inaccuracies in the identified model propagate to forward iterations and make it difficult to discover long-period UPOs. There are also advantages compared to related methods for extracting UPOs from approximate invariant measures in continuoustime [17,18], as these rely on the target UPO being well approximated by the zero level set of a nonnegative polynomial. This is usually not the case unless the polynomial degree is large, because continuous-time UPOs are not algebraic curves in general.…”

Section: Introductionmentioning

confidence: 99%

Data-driven discovery of invariant measures

Bramburger,

Fantuzzi

2024

Proc. R. Soc. A.

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Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification.

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Section: Introductionmentioning

confidence: 99%

Data-driven discovery of invariant measures

Bramburger,

Fantuzzi

2024

Proc. R. Soc. A.

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“…Many systems in reality exhibit chaotic behavior for certain values of the parameters that characterize them. In these dynamic regimes, systems have a wide variety of different behaviors [5,6]. In principle, a chaotic system can take an unlimited number of states, which are unstable, and which are adopted by the system in an unpredictable manner.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Time Series Analysis in Unstable Periodic Orbits Identification-Control Methods of Nonlinear Systems

Ivan

Arva

2022

Electronics

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The main purpose of this paper is to present a solution to the well-known problems generated by classical control methods through the analysis of nonlinear time series. Among the problems analyzed, for which an explanation has been sought for a long time, we list the significant reduction in control power and the identification of unstable periodic orbits (UPOs) in chaotic time series. To accurately identify the type of behavior of complex systems, a new solution is presented that involves a method of two-dimensional representation specific to the graphical point of view, and in particular the recurrence plot (RP). An example of the issue studied is presented by applying the recurrence graph to identify the UPO in a chaotic attractor. To identify a certain type of behavior in the numerical data of chaotic systems, nonlinear time series will be used, as a novelty element, to locate unstable periodic orbits. Another area of use for the theories presented above, following the application of these methods, is related to the control of chaotic dynamical systems by using RP in control techniques. Thus, the authors’ contributions are outlined by using the recurrence graph, which is used to identify the UPO from a chaotic attractor, in the control techniques that modify a system variable. These control techniques are part of the closed loop or feedback strategies that describe control as a function of the current state of the UPO stabilization system. To exemplify the advantages of the methods presented above, the use of the recurrence graph in the control of a buck converter through the application of a phase difference signal was analyzed. The study on the command of a direct current motor using a buck converter shows, through a final concrete application, the advantages of using these analysis methods in controlling dynamic systems.

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Stability of a parametrically driven, coupled oscillator system: An auxiliary function method approach

McMillan

Young

2022

Journal of Applied Physics

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Coupled, parametric oscillators are often studied in applied biology, physics, fluids, and many other disciplines. In this paper, we study a parametrically driven, coupled oscillator system where the individual oscillators are subjected to varying frequency and phase with a focus on the influence of the damping and coupling parameters away from parametric resonance frequencies. In particular, we study the long-term statistics of the oscillator system’s trajectories and stability. We present a novel, robust, and computationally efficient method, which has come to be known as an auxiliary function method for long-time averages, and we pair this method with classical, perturbative-asymptotic analysis to corroborate the results of this auxiliary function method. These paired methods are then used to compute the regions of stability for a coupled oscillator system. The objective is to explore the influence of higher order, coupling effects on the stability region across a broad range of modulation frequencies, including frequencies away from parametric resonances. We show that both simplified and more general asymptotic methods can be dangerously un-conservative in predicting the true regions of stability due to high order effects caused by coupling parameters. The differences between the true stability region and the approximate stability region can occur at physically relevant parameter values in regions away from parametric resonance. As an alternative to asymptotic methods, we show that the auxiliary function method for long-time averages is an efficient and robust means of computing true regions of stability across all possible initial conditions.

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Finding unstable periodic orbits: A hybrid approach with polynomial optimization

Cited by 4 publications

References 42 publications

Data-driven discovery of invariant measures

Data-driven discovery of invariant measures

Nonlinear Time Series Analysis in Unstable Periodic Orbits Identification-Control Methods of Nonlinear Systems

Stability of a parametrically driven, coupled oscillator system: An auxiliary function method approach

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