scholarly journals On the structure of the Levinson center for monotone non-autonomous dynamical systems with a first integral

2021 ◽  
Vol 38 (1) ◽  
pp. 67-94
Author(s):  
DAVID CHEBAN ◽  
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Global Attractor ◽  
Difference Equations ◽  
First Integral ◽  
Almost Periodic ◽  
Almost Automorphic ◽  
Stable Solutions ◽  
Autonomous Dynamical Systems

In this paper we give a description of the structure of compact global attractor (Levinson center) for monotone Bohr/Levitan almost periodic dynamical system $x'=f(t,x)$ (*) with the strictly monotone first integral. It is shown that Levinson center of equation (*) consists of the Bohr/Levitan almost periodic (respectively, almost automorphic, recurrent or Poisson stable) solutions. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We also give some applications of theses results to different classes of differential/difference equations.

scholarly journals On the realization of explicit Runge-Kutta schemes preserving quadratic invariants of dynamical systems

2020 ◽  
Vol 28 (4) ◽  
pp. 327-345
Author(s):  
Yu Ying ◽  
Mikhail D. Malykh
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conservation Law ◽  
Numerical Experiments ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Quadratic Invariant ◽  
Quadratic Invariants ◽  
Runge Kutta ◽  
Autonomous Dynamical Systems

We implement several explicit Runge-Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage. In this paper, we want to present our package ex.sage and the results of our numerical experiments. In the package, the functions rrk_solve, idt_solve and project_1 are constructed for the case when only one given quadratic invariant will be exactly preserved. The function phi_solve_1 allows us to preserve two specified quadratic invariants simultaneously. To solve the equations with respect to parameters determined by the conservation law we use the elimination technique based on Grbner basis implemented in Sage. An elliptic oscillator is used as a test example of the presented package. This dynamical system has two quadratic invariants. Numerical results of the comparing of standard explicit Runge-Kutta method RK(4,4) with rrk_solve are presented. In addition, for the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator. In conclusion, the drawbacks of using these schemes are discussed.

scholarly journals Kosambi–Cartan–Chern (KCC) theory for higher-order dynamical systems

2016 ◽  
Vol 13 (02) ◽  
pp. 1650014 ◽  
Author(s):  
Tiberiu Harko ◽  
Praiboon Pantaragphong ◽  
Sorin V. Sabau
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Cold Dark Matter ◽  
Evolution Equations ◽  
Dynamical Evolution ◽  
Finsler Spaces ◽  
Stability And Instability ◽  
The Relationship ◽  
Autonomous Dynamical Systems

The Kosambi–Cartan–Chern (KCC) theory represents a powerful mathematical method for the investigation of the properties of dynamical systems. The KCC theory introduces a geometric description of the time evolution of a dynamical system, with the solution curves of the dynamical system described by methods inspired by the theory of geodesics in a Finsler spaces. The evolution of a dynamical system is geometrized by introducing a nonlinear connection, which allows the construction of the KCC covariant derivative, and of the deviation curvature tensor. In the KCC theory, the properties of any dynamical system are described in terms of five geometrical invariants, with the second one giving the Jacobi stability of the system. Usually, the KCC theory is formulated by reducing the dynamical evolution equations to a set of second-order differential equations. In this paper, we introduce and develop the KCC approach for dynamical systems described by systems of arbitrary [Formula: see text]-dimensional first-order differential equations. We investigate in detail the properties of the [Formula: see text]-dimensional autonomous dynamical systems, as well as the relationship between the linear stability and the Jacobi stability. As a main result we find that only even-dimensional dynamical systems can exhibit both Jacobi stability and instability behaviors, while odd-dimensional dynamical systems are always Jacobi unstable, no matter their Lyapunov stability. As applications of the developed formalism we consider the geometrization and the study of the Jacobi stability of the complex dynamical networks, and of the [Formula: see text]-Cold Dark Matter ([Formula: see text]CDM) cosmological models, respectively.

Almost Periodic Dynamics of a Class of Delayed Neural Networks with Discontinuous Activations

2008 ◽  
Vol 20 (4) ◽  
pp. 1065-1090 ◽  
Author(s):  
Wenlian Lu ◽  
Tianping Chen
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Existence And Uniqueness ◽  
Network Models ◽  
Almost Periodic ◽  
Filippov Solution ◽  
Neural Network Models ◽  
Delayed Neural Networks ◽  
Special Cases ◽  
Discontinuous Activations

We use the concept of the Filippov solution to study the dynamics of a class of delayed dynamical systems with discontinuous right-hand side, which contains the widely studied delayed neural network models with almost periodic self-inhibitions, interconnection weights, and external inputs. We prove that diagonal-dominant conditions can guarantee the existence and uniqueness of an almost periodic solution, as well as its global exponential stability. As special cases, we derive a series of results on the dynamics of delayed dynamical systems with discontinuous activations and periodic coefficients or constant coefficients, respectively. From the proof of the existence and uniqueness of the solution, we prove that the solution of a delayed dynamical system with high-slope activations approximates to the Filippov solution of the dynamical system with discontinuous activations.

scholarly journals Invariants for a Dynamical System with Strong Random Perturbations

2021 ◽  
Author(s):  
Elena Karachanskaya
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Deterministic Model ◽  
First Integral ◽  
Jump Diffusion ◽  
Invariant Function ◽  
Actual Process ◽  
Stochastic Dynamical Systems ◽  
Stochastic Dynamical System ◽  
Diffusion Function

In this chapter we consider the invariant method for stochastic system with strong perturbations, and its application to many different tasks related to dynamical systems with invariants. This theory allows constructing the mathematical model (deterministic and stochastic) of actual process if it has invariant functions. These models have a kind of jump-diffusion equations system (stochastic differential Itô equations with a Wiener and a Poisson paths). We show that an invariant function (with probability 1) for stochastic dynamical system under strong perturbations exists. We consider a programmed control with Prob. 1 for stochastic dynamical systems – PSP1. We study the construction of stochastic models with invariant function based on deterministic model with invariant one and show the results of numerical simulation. The concept of a first integral for stochastic differential equation Itô introduce by V. Doobko, and the generalized Itô – Wentzell formula for jump-diffusion function proved us, play the key role for this research.

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