Kernel methods for center manifold approximation and a weak data-based version of the Center Manifold Theorem

Haasdonk, Bernard; Hamzi, Boumediene; Santin, Gabriele; Wittwar, Dominik

doi:10.1016/j.physd.2021.133007

Cited by 14 publications

(3 citation statements)

References 18 publications

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“…Therefore, equation (15) could be divided to two manifolds: (u, v) forms the self-oscillating pair, and the rest k(t) and j(t) are driven by changes of (u, v). From the center manifold theorem [36], resonance of j(t) is lead by v(t) and the response time is omitted because of equation (8); then, k(t) is a stable manifold that is convergent to the (u, v)manifold exponentially.…”

Section: Model Analysismentioning

confidence: 99%

Analytical and numerical computation of self-oscillating gels driven by the Belousov-Zhabotinsky reaction

Chi

2024

Phys. Scr.

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Self-oscillating gel is a class of deformable polymers driven by Belousov-Zhabotinsky (BZ) reactions, which can form periodic deformations without any external stimuli, and are widely used in the research of biomimetic creeping robots. However, quantitative analysis on the self-oscillating gel from perspective of energy conservation is still limited. This work adopts frequency domain analysis to the chemo-mechanical model, and the basic frequency is obtained to evaluate the maintenance energy of the BZ gel. For accurate computation, boundary value problem with unknown period is formulated. Then, continuation algorithm based on perturbation technique is performed to obtain the oscillating trajectories for varying model parameters. The results could be implemented to design self-oscillating gels with prescribed periodicity.

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Section: Model Analysismentioning

confidence: 99%

Analytical and numerical computation of self-oscillating gels driven by the Belousov-Zhabotinsky reaction

Chi

2024

Phys. Scr.

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“…Inspired by the works of [15], [16] that use the Center Manifold Theorem to construct converse LFs with certain structure, in this paper we propose a new approach for certifying the local asymptotic stability of general high state space nonlinear ODEs by searching for partially quadratic LFs of the form,…”

Section: Introductionmentioning

confidence: 99%

Existence of Partially Quadratic Lyapunov Functions That Can Certify The Local Asymptotic Stability of Nonlinear Systems

Jones¹,

Peet²

2022

Preprint

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This paper proposes a method for certifying the local asymptotic stability of a given nonlinear Ordinary Differential Equation (ODE) by using Sum-of-Squares (SOS) programming to search for a partially quadratic Lyapunov Function (LF). The proposed method is particularly well suited to the stability analysis of ODEs with high dimensional state spaces. This is due to the fact that partially quadratic LFs are parametrized by fewer decision variables when compared with general SOS LFs. The main contribution of this paper is using the Center Manifold Theorem to show that partially quadratic LFs that certify the local asymptotic stability of a given ODE exist under certain conditions.

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“…Such a set of non-isolated equilibria is known as a manifold of equilibria 12 , 13 . It should be noted that this concept of manifold of equilibria is different from the center manifold of an isolated equilibirum 14 , 15 . Lyapunov’s linearization method is a useful tool to investigate the stability of a single equilibrium point.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium space and a pseudo linearization of nonlinear systems

Sakata

Oshima

Kawai

et al. 2022

Sci Rep

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This paper attempts to extend the concept of the equilibrium point to what is called equilibrium space, which can adapt to a system in which there exists an infinite number of equilibrium points. In the context of Lyapunov’s linearization method extended for the equilibrium space, this paper proposes a pseudo linearization, from which we can derive a linear representation for a nonlinear system. The equilibrium state of this pseudo linearization and its stability are shown to be the same as that of the original nonlinear system. As an example of the applicability, the proposed pseudo linearization is applied to derive a discrete-time model for a control moment gyroscope system from a nonlinear continuous-time model. Simulation results show that the discrete-time model derived using the proposed pseudo linearization yields responses that are closer to that of the continuous-time model than the discrete-time model derived by the well-known forward-difference method and the conventional pseudo linear representation method, even with a large sampling interval.

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Kernel methods for center manifold approximation and a weak data-based version of the Center Manifold Theorem

Cited by 14 publications

References 18 publications

Analytical and numerical computation of self-oscillating gels driven by the Belousov-Zhabotinsky reaction

Analytical and numerical computation of self-oscillating gels driven by the Belousov-Zhabotinsky reaction

Existence of Partially Quadratic Lyapunov Functions That Can Certify The Local Asymptotic Stability of Nonlinear Systems

Equilibrium space and a pseudo linearization of nonlinear systems

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