Qualitative Theory of Dynamical Systems最新文献

In this article, we first propose a reasonable definition of Weyl almost automorphic stochastic process in finite-dimensional distributions. Then, efforts were made to investigate the existence and stability of Weyl almost automorphic solutions in finite-dimensional distributions to a class of stochastic shunting inhibitory cellular neural networks (SICNNs) with D operators. Because the space formed by Weyl almost automorphic random processes is not a complete space, in order to overcome this difficulty, firstly, we use Banach’s fixed point theorem on a closed subset of the Banach space composed of (mathcal {L}^p) bounded and (mathcal {L}^p) uniformly continuous random processes to obtain that the network under consideration admits a unique solution in this subset, secondly, based on the definition of Weyl almost automorphic solutions in finite-dimensional distributions, using inequality techniques, we prove that the solution is also Weyl almost automorphic in finite-dimensional distributions, then, the global exponential stability of the Weyl almost automorphic solution is proved using the contradiction method. The results and methods of this paper are new and can be used to study the corresponding problems of other neural network models. Finally, a numerical example is provided to demonstrate the effectiveness of our results.

Let (f:mathbb {M}rightarrow mathbb {M}) be a continuous map on a compact metric space (mathbb {M}) equipped with a fixed metric d, and let (tau ) be the topology on (mathbb {M}) induced by d. We denote by (mathbb {M}(tau )) the set consisting of all metrics on (mathbb {M}) that are equivalent to d. Let ( text {mdim}_{text {M}}(mathbb {M},d, f)) and ( text {mdim}_{text {H}} (mathbb {M},d, f)) be, respectively, the metric mean dimension and mean Hausdorff dimension of f. First, we will establish some fundamental properties of the mean Hausdorff dimension. Furthermore, it is important to note that ( text {mdim}_{text {M}}(mathbb {M},d, f)) and ( text {mdim}_{text {H}} (mathbb {M},d, f)) depend on the metric d chosen for (mathbb {M}). In this work, we will prove that, for a fixed dynamical system (f:mathbb {M}rightarrow mathbb {M}), the functions (text {mdim}_{text {M}} (mathbb {M}, f):mathbb {M}(tau )rightarrow mathbb {R}cup {infty }) and ( text {mdim}_{text {H}}(mathbb {M}, f): mathbb {M}(tau )rightarrow mathbb {R}cup {infty }) are not continuous, where ( text {mdim}_{text {M}}(mathbb {M}, f) (rho )= text {mdim}_{text {M}} (mathbb {M},rho , f)) and ( text {mdim}_{text {H}}(mathbb {M}, f) (rho )= text {mdim}_{text {H}} (mathbb {M},rho , f)) for any (rho in mathbb {M}(tau )). Furthermore, we will present examples of certain classes of metrics for which the metric mean dimension is a continuous function.

Circular network structure is widely used in neural network, image processing, computer vision and bioinformatics. For example, recurrent neural network is a kind of neural network with a circular structure that can be used to process temporal data. It has a wide range of applications in natural language processing, speech recognition, music generation, etc. In this paper, in order to reduce the complexity of the presentation, we study a class of Hilfer-type fractional differential system and differential inclusion with coupled integral boundary value conditions on the simplest circular graph. First, two existence results of Hilfer-type fractional differential system are proved by some known fixed point theorems. Further, the existence results of convex and non-convex multivalued mappings are obtained by using Leray–Schauder nonlinear alternative and Covitz–Nadler fixed point theorem, respectively. At last, two examples are given to verify our theoretical results.

The goal of this paper is to study the number of sliding limit cycles of regularized piecewise linear visible–invisible twofolds using the notion of slow divergence integral. We focus on limit cycles produced by canard cycles located in the half-plane with an invisible fold point. We prove that the integral has at most 1 zero counting multiplicity (when it is not identically zero). This will imply that the canard cycles can produce at most 2 limit cycles. Moreover, we detect regions in the parameter space with 2 limit cycles.

In this work, we analyze a two-dimensional continuous-time differential equations system derived from a Leslie–Gower predator–prey model with a generalist predator and prey group defence. For our model, we fully characterize the existence and quantity of equilibrium points in terms of the parameters, and we use this to provide necessary and sufficient conditions for the existence and the explicit form of two kinds of equilibrium points: both a degenerate one with associated nilpotent Jacobian matrix, and a weak focus. These conditions allows us to determine whether the system undergoes Bogdanov–Takens and Hopf bifurcations. Consequently, we establish the existence of a simultaneous Bogdanov–Taken and Hopf bifurcation. With this double bifurcation, we guarantee the existence of a new Hopf bifurcation curve and two limit cycles on the system: an infinitesimal and another non-infinitesimal.

Linearization remodeling and state-feedback control for a class of autonomous nonlinear systems based on equilibrium manifold expansion (EME) are visited and explicated in this paper, including linearization approximation, state-feedback stabilization and state estimation. More precisely, firstly, EME linearized remodels of nonlinear systems are explained and their existence is validated rigorously; secondly, EME-based state-feedback control and observer design are developed analytically with EME remodeling and gain scheduling; thirdly, stabilization under EME-based state feedback and observers are tackled, respectively; finally, feasibility and efficiency of the EME approach are illustrated by numerical simulations.

This paper considers a class of mixed slow-fast McKean–Vlasov stochastic differential equations that contain the fractional Brownian motion with Hurst parameter (H > 1/2) and the standard Brownian motion. Firstly, we prove an existence and uniqueness theorem for the mixed coupled system. Secondly, under suitable assumptions on the coefficients, using the approach of Khasminskii’s time discretization, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the mean square sense.

While studying gradient dynamical systems, Morse introduced the idea of encoding the qualitative behavior of a dynamical system into a graph. Smale later refined Morse’s idea and extended it to Axiom-A diffeomorphisms on manifolds. In Smale’s vision, nodes are indecomposable closed invariant subsets of the non-wandering set with a dense orbit and there is an edge from node M to node N (we say that N is downstream from M) if the unstable manifold of M intersects the stable manifold of N. Since then, the decomposition of the non-wandering set was studied in many other settings, while the edges component of Smale’s construction has been often overlooked. In the same years, more sophisticated generalizations of the non-wandering set, introduced by Birkhoff in 1920s, were elaborated first by Auslander in early 1960s, by Conley in early 1970s and later by Easton and other authors. In our language, each of these generalizations involves the introduction of a closed and transitive extension of the prolongational relation, that is closed but not transitive. In the present article, we develop a theory that generalizes at the same time both these lines of research. We study the general properties of closed transitive relations (which we call streams) containing the space of orbits of a discrete-time or continuous-time semi-flow and we argue that these relations play a central role in the qualitative study of dynamical systems. All most studied concepts of recurrence currently in literature can be defined in terms of our streams. Finally, we show how to associate to each stream a graph encoding its qualitative properties. Our main general result is that each stream of a semi-flow with “compact dynamics” has a connected graph. The range of semi-flows covered by our theorem goes from 1-dimensional discrete-time systems like the logistic map up to infinite-dimensional continuous-time systems like the semi-flow of quasilinear parabolic reaction–diffusion partial differential equations.

In the present paper, some new generalizations of dynamic inequalities of Hardy-type in two variables on time scales are established. The integral and discrete Hardy-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Jensen inequality and the Fubini theorem on time scales.

In this study, the dynamics of a novel three-species food chain model featuring the Sokol–Howell functional response are explored. The fear of predators is incorporated into prey reproduction, and refuge is integrated into the middle predators within the framework of the Caputo fractional derivative. Theoretical aspects such as the existence and uniqueness of equilibria, their boundedness, and stability analysis are encompassed in the investigation. To examine the existence of chaos, Lyapunov exponents are computed. The optimal control measure concerning the growth of the prey population was considered, and the conditions that must be met for the optimal response to exist in the optimal control issue were determined using Pontryagin’s Maximum Principle. The theoretical outcomes were validated by using numerical simulation powered by the Adams–Bashforth–Moulton type predictor-corrector technique. Numerical justifications are provided for the influences of fear and refuge factors. When fear is absent, a numerical analysis is conducted on the global stability of the system for fractional order derivative.