Complexity factor for charged dissipative dynamical system

In this paper, we examine the complexity factor for a dynamical spherical system with dissipative charged anisotropic fluid. We evaluate the Einstein-Maxwell field equations and structure scalars using Bel’s approach which help to discuss the structure as well as evolution of a self-gravitating system. We measure the complexity factor for the pattern of evolution through the homologous condition and homogeneous expansion. We also analyze the stability of vanishing complexity condition for dissipative and non-dissipative fluids. It is found that the complexity as well as stability of the spherical system increases and decreases, respectively, under the effects of electromagnetic field.

ASYMPTOTICS FOR A DISSIPATIVE DYNAMICAL SYSTEM WITH LINEAR AND GRADIENT-DRIVEN DAMPING

We study, in the setting of a real Hilbert space H, the asymptotic behavior of trajectories of the second-order dissipative dynamical system with linear and gradient-driven nonlinear damping where λ > 0 and f, Φ: H → R are two convex differentiable functions. It is proved that if Φ is coercive and bounded from below, then the trajectory converges weakly towards a minimizer of Φ. In particular, we state that under suitable conditions, the trajectory strongly converges to the minimizer of Φ exponentially or polynomially.

Three Conditions Lyapunov Exponents Should Satisfy

In contrast to the unilateral claim in some papers that a positive Lyapunov exponent means chaos, it was claimed in this paper that this is just one of the three conditions that Lyapunov exponent should satisfy in a dissipative dynamical system when the chaotic motion appears. The other two conditions, any continuous dynamical system without a fixed point has at least one zero exponent, and any dissipative dynamical system has at least one negative exponent and the sum of all of the 1-dimensional Lyapunov exponents id negative, are also discussed. In order to verify the conclusion, a MATLAB scheme was developed for the computation of the 1-dimensional and 3-dimensional Lyapunov exponents of the Duffing system with square and cubic nonlinearity.

THE HEAVY BALL WITH FRICTION METHOD, I. THE CONTINUOUS DYNAMICAL SYSTEM: GLOBAL EXPLORATION OF THE LOCAL MINIMA OF A REAL-VALUED FUNCTION BY ASYMPTOTIC ANALYSIS OF A DISSIPATIVE DYNAMICAL SYSTEM

Let H be a real Hilbert space and Φ:H ↦ R a continuously differentiable function, whose gradient is Lipschitz continuous on bounded sets. We study the nonlinear dissipative dynamical system: [Formula: see text], plus Cauchy data, mainly in view of the unconstrained minimization of the function Φ. New results concerning the convergence of a solution to a critical point are given in various situations, including when Φ is convex (possibly with multiple minima) or is a Morse function (the critical point being then generically a local minimum); a counterexample shows that, without peculiar assumptions, a trajectory may not converge. By following the trajectories, we obtain a method for exploring local minima of Φ. A singular perturbation analysis links our results with those concerning gradient systems.