Heavy Ball with Friction Dynamical System

2019 ◽  
pp. 121-126
Author(s):  
Behzad Djafari-Rouhani ◽  
Hadi Khatibzadeh
Keyword(s):  
Dynamical System ◽  
Heavy Ball With Friction

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

2000 ◽  
Vol 02 (01) ◽  
pp. 1-34 ◽  
Author(s):  
H. ATTOUCH ◽  
X. GOUDOU ◽  
P. REDONT
Keyword(s):  
Dynamical System ◽  
Critical Point ◽  
Real Hilbert Space ◽  
Unconstrained Minimization ◽  
Cauchy Data ◽  
Local Minima ◽  
Lipschitz Continuous ◽  
Singular Perturbation Analysis ◽  
Dissipative Dynamical System ◽  
Heavy Ball With Friction

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.

Chaos and Creativity: The Psique as a Complex Dynamical System Creativiy and Arquetipal Psychology

2012 ◽  
Author(s):  
Magaly Villaobos ◽  
Yolanda Ng Lee ◽  
Maria Elena Gutierrez
Keyword(s):  
Dynamical System ◽  
Complex Dynamical System

On the Harmonic Oscillations for the Motion of a Dynamical System

2017 ◽  
Vol 13 (2) ◽  
pp. 4657-4670
Author(s):  
W. S. Amer
Keyword(s):  
Dynamical System ◽  
Numerical Solutions ◽  
Equation Of Motion ◽  
Parameter Method ◽  
Kutta Method ◽  
Small Parameter Method ◽  
Harmonic Oscillations ◽  
Pivot Point ◽  
Runge Kutta Method ◽  
High Consistency

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.

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