In a toroidal plasma confined by a purely toroidal magnetic field with a weak vertical field superimposed a system of convection cells are generated spontaneously, interacting with a background electron density gradient. The dynamics of this interaction is low-dimensional, chaotic, and consistent with solutions of the Lorenz equations in the diffusionless limit.

In this study, it is aimed to use the Lorenz matrix multiplication find n^th powers of some special matrices and reach quadratic equations characteristic roots obtained in way. addition, contribute literature studies field by reaching identities.

The paper deals with the problem of estimation of the topological entropy for non-autonomous systems of differential equations via the second (direct) Lyapunov method. The main result of the paper is illustrated by examples concerning the Lorenz system and Duffing oscillator.

A generalized Lyapunov method is outlined which predicts global stability of a broad class of dissipative dynamical systems. The method is applied to the complex Lorenz model and to the Navier-Stokes equations. In both cases one finds compact domains in phase space which contain the ω sets of all trajectories, in particular the fixed points, limit cycles, and strange attractors.

A new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with computerassisted computations. As an application of these methods it is proven that for an explicit parameter value the Lorenz equations exhibit chaotic dynamics.

Abstract We present a computer-assisted approach to prove the existence of Hopf bubbles and degenerate bifurcations in ordinary delay differential equations. apply method rigorously investigate these nonlocal orbit structures FitzHugh–Nagumo equation, extended Lorenz-84 model time-delay SI model.

Moment vector equations (MVEs) are presented for use in approximating and analyzing multi-dimensional non-linear discreteand continuous-time equations. A non-linear equation is expanded into simultaneous equations of generalized moments and then reduced to an MVE of a coefficient matrix and a moment vector. The MVE can be used to analyze the statistical properties, such as the mean, variance, c...

has been much studied for positive values of its parameters (a, b, c) and has chaotic solutions for typical parameters of (35, 3, 28). Recently, Algaba et al. [2013] have shown that the Chen system has only two independent parameters a/c and b/c and that it is identical to the reversed-time Lorenz system for c > 0 and to the forward-time Lorenz system for c < 0 in the parameter plane ρ + σ = −1...