Stability Of Invariant Sets Research Articles

Stochastic time-varying extremum seeking and its applications

We investigate a stochastic time-varying extremum seeking method based on new stochastic averaging theorems for time-varying systems and its application to distributed source seeking with switching topology. Firstly, we present sufficient conditions for the approximation of the time-varying averaging system to the time-varying original system with stochastic perturbations and obtain the solution property of the original system under the stability of invariant sets of the average system. Then, by using our developed stochastic averaging theory, we prove that the proposed stochastic time-varying extremum seeking algorithm converges to the time-varying optimal solution. Furthermore, based on the method of stochastic time-varying extremum seeking, we propose a distributed stochastic source seeking algorithm which navigates multiple vehicles to seek the source of aggregated multiple signal fields under a switching topology. We can prove the exponential convergence of such a distributed algorithm by our stochastic averaging theorems and a Lyapunov-based approach. Finally, we present detailed simulation results to illustrate the effectiveness of our method.

Stochastic bifurcations and transient dynamics of probability responses with radial basis function neural networks

Stochastic noise is common in many engineering systems. Stochastic noise and nonlinearity of dynamical systems will lead to the occurrence of complex dynamic phenomena, such as stochastic bifurcations and transitions. In this paper, a radial basis function neural networks (RBF-NN) method is applied to study stochastic bifurcations and transient dynamics of probability response. A Duffing oscillator under additive and multiplicative random noise is presented to show the effectiveness of the proposed solution method. A new type of stochastic bifurcations is found with an increase of system control parameter, which is called a stochastic double P-bifurcation. It occurs when a stationary probability density function (PDF) changes from a single peak to another single one through two double peaks with their maximum exchange. It is revealed that such a double P-bifurcation is linked to two catastrophic bifurcations of its deterministic counterpart. Moreover, these different types of transient evolutions of the PDFs are observed with transitions of probability maximum peak to one, and to the other, as well as to the both of stable invariant sets from different initial PDFs. The evolutionary orientation of transient PDFs aligns with unstable invariant manifolds leading to stable invariant sets. The previous conjecture is confirmed that the noise-induced transient dynamics and bifurcations are dominated by the global topology and its sudden changes of corresponding deterministic systems without noise.

Mean-square invariant manifolds for ill-posed stochastic evolution equations driven by nonlinear noise

This paper discerns the invariant manifold of a class of ill-posed stochastic evolution equations driven by a nonlinear multiplicative noise. To be more precise, we establish the existence of mean-square random unstable invariant manifold and only mean-square stable invariant set. Due to the lack of the Hille-Yosida condition, we construct a modified variation of constants formula by the resolvent operator. With the price of imposing an unusual condition involving a non-decreasing map, we set up the Lyapunov-Perron method and derive the required estimates. We also emphasize that the Lyapunov-Perron map in the forward time loses the invariant due to the adaptedness, we alternatively establish the existence of mean-square random stable sets.

On Stability for State Constrained Hybrid Systems via Barrier Lyapunov Functions

In this paper, the stability of familiar state constrained hybrid systems is considered. In the first, we prove the invariant set stability for the state constrained impulsive hybrid systems. Specifically, a robust control feedback method is applied for state constrained uncertain impulsive hybrid systems. With the auxiliary matrix assistance, some convergence criteria are derived to guarantee robust stability for state constrained uncertain hybrid systems with output disturbance by constructing the symmetric and asymmetric barrier Lyapunov functions (BLF), respectively. Finally, two comparative examples with simulations show that the proposed results are effective and superior.

Learning Dynamics Models with Stable Invariant Sets

Invariance and stability are essential notions in dynamical systems study, and thus it is of great interest to learn a dynamics model with a stable invariant set. However, existing methods can only handle the stability of an equilibrium. In this paper, we propose a method to ensure that a dynamics model has a stable invariant set of general classes such as limit cycles and line attractors. We start with the approach by Manek and Kolter (2019), where they use a learnable Lyapunov function to make a model stable with regard to an equilibrium. We generalize it for general sets by introducing projection onto them. To resolve the difficulty of specifying a to-be stable invariant set analytically, we propose defining such a set as a primitive shape (e.g., sphere) in a latent space and learning the transformation between the original and latent spaces. It enables us to compute the projection easily, and at the same time, we can maintain the model's flexibility using various invertible neural networks for the transformation. We present experimental results that show the validity of the proposed method and the usefulness for long-term prediction.

Mean-square random invariant manifolds for stochastic differential equations

We develop a theory of mean-square random invariant manifolds for mean-square random dynamical systems generated by stochastic differential equations. This theory is applicable to stochastic partial differential equations driven by nonlinear noise. The existence of mean-square random invariant unstable manifolds is proved by the Lyapunov-Perron method based on a backward stochastic differential equation involving the conditional expectation with respect to a filtration. The existence of mean-square random stable invariant sets is also established but the existence of mean-square random stable invariant manifolds remains open.

Стiйкiсть iнварiантного многовиду нелiнiйної системи диференцiальних рiвнянь

The theory of extensions of the dynamical equations on the torus is an important section of the theory of ordinary differential equations that is intensively evolving and has an important applied application to various tasks of science and technology. This theory describes processes that have oscillatory character. One of the important questions of mathematical theory of multifrequency oscillations is the problem of the existence and stability of invariant toroidal manifolds, the problem of the structural stability of the invariant manifold, its preservation under small perturbations for the systems of differential equations that are defined in the direct product of a torus and Euclidean space. Such manifolds serve as carriers of multifrequency oscillations in the system. The basics of this theory have been developed by A. M. Samoilenko.In this paper the class of differential equations, defined in the direct product of m-dimensional torus T m and n-dimensional Euclidean space R n for which there exist conditions for the existence of an asymptotically stable invariant toroidal manifold, is investigated. We formulate and prove sufficient conditions for the existence and asymptotic stability of invariant toroidal class of nonlinear extensions of dynamical systems on torus, that has specific properties in ω-limit set Ω of the trajectories ϕ_t (ϕ). Two theorems, that define the conditions for the existence of asymptotically stable invariant sets for linear expansion of the dynamical system on torus and corresponding perturbed system, are given. We search an invariant manifold of a nonlinear system by the iterative method, proposed by A. M. Samoilenko.

On contraction of nonlinear difference equations with time‐varying delays

AbstractGeneral nonlinear difference equations with time‐varying delays are considered. Explicit criteria for contraction of such equations are presented. Then some simple sufficient conditions for global exponential stability of equilibria and for stability of invariant sets are derived. Furthermore, explicit criteria for existence, uniqueness and global exponential stability of periodic solutions are derived. Finally, the obtained results are applied to time‐varying discrete‐time neural networks with delay.

Nonlinear Dynamics of Buck Converter

This paper is concerned with the problem of modeling and simulation of piecewise linear systems with the pulse width modulation control. Models of this kind of systems are widely used in engineering practice, in particular in power electronic converters. The research provides the study of chaos and bifurcations in one of the switching converters – step-down converter under voltage mode control, operating in discontinuous and continuous current modes, using Matlab and Simulink simulation environment. Various types of models, including discrete – time maps, switched state – space models are examined in order to detect their suitability and reliability in numerical investigation of nonlinear phenomena in DC–DC converters. Some analytical methods for the search of periodic regimes and their stability estimation were also used in order to validate numerically obtained results and evaluate the accuracy of models used. As direct simulation does not accurately pinpoint bifurcation points and computes only stable invariant sets, some aspects and advantages of numerical path-following are also discussed.

Synchronization of diffusively coupled systems on compact Riemannian manifolds in the presence of drift

Recently, it has been shown that the synchronization manifold is an asymptotically stable invariant set of diffusively coupled systems on Riemannian manifolds. We regionally investigate the stability properties of the synchronization manifold when the systems are subject to drift. When the drift vector field is quad (i.e. satisfies a certain quadratic inequality) and the underlying Riemannian manifold is compact, we prove that a sufficiently large algebraic connectivity of the underlying graph is sufficient for the synchronization manifold to remain asymptotically stable. For drift vector fields which are quad or contracting, we explicitly characterize the rate at which the solution converges to the synchronization manifold. Our main result is that the synchronization manifold is asymptotically stable even for drift vector fields which are only locally Lipschitz continuous, as long as the algebraic connectivity of the underlying graph is sufficiently large.

Fuzzy Responses and Bifurcations of a Forced Duffing Oscillator with a Triple-Well Potential

Responses and bifurcations of a forced triple-well potential system with fuzzy uncertainty are studied by means of the Fuzzy Generalized Cell Mapping (FGCM) method. A rigorous mathematical foundation of the FGCM is established as a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. The FGCM offers a very effective approach for solutions to the fuzzy master equation based on the min–max operator of fuzzy logic. A fuzzy response is characterized by its topology in the state space and its possibility measure of membership distribution functions (MDFs). A fuzzy bifurcation implies a sudden change both in the topology and in the MDFs. The response topology is obtained based on the qualitative analysis of the FGCM involving the Boolean operation of 0 and 1. The MDFs are determined by the quantitative analysis of the FGCM with the min–max calculations. With an increase of the intensity of fuzzy noise, noise-induced escape from each of the potential wells defines two types of bifurcations, namely catastrophe and explosion. This paper focuses on the evolution of transient and steady-state MDFs of the fuzzy response. As the intensity of fuzzy noise increases, steady-state MDFs cover a bigger area in the state space with higher membership values spreading out to a larger area. The previous conjectures are further confirmed that steady-state MDFs are dependent on initial possibility distributions due to the nonsmooth and nonlinear nature of the min–max operation. It is found that as time goes on, transient MDFs spread around three potential wells. The evolutionary orientation of transient MDFs aligns with unstable invariant manifolds leading to stable invariant sets. Two examples of additive and multiplicative fuzzy noise are given.

Invariant and attractor sets for fuzzy dynamical systems

In this work we establish conditions for the existence and stability of invariant sets for dynamical systems defined on metric space of fuzzy subsets of Rn. We show that the results presented in this work generalize some conditions of existence and stability of equilibrium points and periodicity of fuzzy differential equations solutions.

On the stability of invariant sets of leaves of three-dimensional periodic systems

The small C1 perturbations of differential equations are studied. The concepts of a weakly hyperbolic set K and a leaf ϒ are introduced for a system of ordinary differential equations. The Lipschitz condition is not supposed. It is shown that, if the perturbation is small enough, then there exists a continuous mapping h: ϒ → ϒY, where ϒY is a leaf of the perturbed system.

Model predictive control suitable for closed-loop re-identification

The main problem of a closed-loop re-identification procedure is that, in general, the dynamic control and identification objectives are conflicting. In fact, to perform a suitable identification, a persistent excitation of the system is needed, while the control objective is to stabilize the system at a given equilibrium point. However, a generalization of the concept of stability, from punctual stability to (invariant) set stability, allows for a flexibility that can be used to avoid the conflict between these objectives. Taking into account that an invariant target set includes not only a stationary component, but also a transient one, the system could be excited without deteriorating the stability of the closed-loop. In this work, a MPC controller is proposed that ensures the stability of invariant sets at the same time that a signal suitable for closed-loop re-identification is generated. Several simulation results show the propose controller formulation properties.

Dynamics of the Heisenberg model and a theorem on stability

We consider the general discrete classical Heisenberg model (HM) with z axis anisotropy and external magnetic field and show that its phase space is foliated into a family of invariant manifolds (the leaves) diffeomorphic to (S2)Λ, where Λ is the number of spins. We also show that the flow on each leaf \documentclass[12pt]{minimal}\begin{document}$\mathcal {S}$\end{document}S is Hamiltonian. Subsequently, we focus on the isotropic HM in the absence of external field. We discuss the rotational symmetry of the model and further analyze its phase space structure. We prove that the manifold \documentclass[12pt]{minimal}\begin{document}$\mathcal {F}$\end{document}F of longitudinal fixed points intersects each leaf \documentclass[12pt]{minimal}\begin{document}$\mathcal {S}$\end{document}S orthogonally. For a real local flow with a continuous symmetry, we show that the Lyapunov stability of invariant sets on an invariant subspace can be extended to the whole phase space. This general theorem is the main result of the article. We use it to show that, in the case of the isotropic HM, the ferromagnetic state and the antiferromagnetic state with non-zero total spin are both stable fixed points. The theorem does not apply at total spin zero, and indeed the AF state on an equal-spins leaf is found to be unstable.

CANARD CYCLES IN GLOBAL DYNAMICS

Fast-slow systems are studied usually by "geometrical dissection" [Borisyuk & Rinzel, 2005]. The fast dynamics exhibit attractors which may bifurcate under the influence of the slow dynamics which is seen as a parameter of the fast dynamics. A generic solution comes close to a connected component of the stable invariant sets of the fast dynamics. As the slow dynamics evolves, this attractor may lose its stability and the solution eventually reaches quickly another connected component of attractors of the fast dynamics and the process may repeat. This scenario explains quite well relaxation oscillations and more complicated oscillations like bursting. More recently, in relation both with theory of dynamical systems [Dumortier & Roussarie, 1996] and with applications to physiology [Desroches et al., 2008; Toporikova et al., 2008], a new interest has emerged in canard cycles. These orbits share the property that they remain for a while close to an unstable invariant set (either singular set or periodic orbits of the fast dynamics). Although canards were first discovered when the transition points were folds, in this article, we focus on the case where one or several transition points (or "jumps") are instead transcritical. We present several new surprising effects like the "amplification of canards" or the "exceptionally fast recovery" on both (1 + 1)-systems and (2 + 1)-systems associated with tritrophic food chain dynamics. Finally, we also mention their possible relevance to the notion of resilience which has been coined out in ecology [Holling, 1973; Ludwig et al., 1997; Martin, 2004].

A natural order in dynamical systems based on Conley–Markov matrices

We introduce a natural order to study properties of dynamical systems, especially their invariant sets. The new concept is based on the classical Conley index theory and transition probabilities among neighborhoods of different invariant sets when the dynamical systems are perturbed by white noises. The transition probabilities can be determined by the Fokker–Planck equation and they form a matrix called a Markov matrix. In the limiting case when the random perturbation is reduced to zero, the Markov matrix recovers the information given by the Conley connection matrix. The Markov matrix also produces a natural order from the least to the most stable invariant sets for general dynamical systems. In particular, it gives the order among the local extreme points if the dynamical system is a gradient-like flow of an energy functional. Consequently, the natural order can be used to determine the global minima for gradient-like systems. Some numerical examples are given to illustrate the Markov matrix and its properties.

The gentlest ascent dynamics

Dynamical systems that describe the escape from the basins of attraction of stable invariant sets are presented and analysed. It is shown that the stable fixed points of such dynamical systems are the index-1 saddle points. Generalizations to high index saddle points are discussed. Both gradient and non-gradient systems are considered. Preliminary results on the nature of the dynamical behaviour are presented.

Lyapunov-Based Sufficient Conditions for Nonlinear Impulsive Systems

This paper focuses on the stability analysis and invariant set stability theorems for nonlinear impulsive systems. A set of Lyapunov-based sufficient conditions are established for these convergent properties. These results do not require the Lyapunov function to be positive definite. Inequalities relating the righthandside of the differential equation and the Lyapunov function derivative are involved for these results. These inequalities make it possible to deduce properties of the functions and thus leads to sufficient conditions for convergence and stability.

Output Feedback Stabilization of Invariant Sets for Nonlinear Switched Systems

摘要: 利用一个非负函数 V(x) 提出一个新的输入输出对状态稳定的概念——输入输出对 V(x) 稳定. 基于这一概念并结合系统无源性的结论, 提出了一类非线性切换系统不变集可输出反馈镇定的充分条件. 利用统一 Lyapunov 函数和多 Lyapunov 函数给出了定理的证明. 最后, 详细讨论了输入输出对 V(x) 稳定与输入输出对状态稳定的关系. 关键词: 输入输出对状态稳定 / 输入输出对 V(x) 稳定 / 不变集 / 输出反馈 / 统一 Lyapunov 函数 / 多 Lyapunov 函数