Complexity in spatio-temporal dynamics

1995 ◽  
Vol 125 (5) ◽  
pp. 959-974 ◽  
Author(s):  
D. L. Rod ◽  
B. D. Sleeman
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
Temporal Dynamics ◽  
Degree Of Freedom ◽  
Kovacic Algorithm ◽  
Spatio Temporal ◽  
Two Degree Of Freedom

Complex and chaotic structures in certain dynamical systems in biology arise as a consequence of noncomplete integrability of two-degree-of-freedom Hamiltonian systems. A study of this problem is made using Ziglin theory and implemented with the aid of the Kovacic algorithm.

scholarly journals Periodic solutions and their stability for some perturbed Hamiltonian systems

2020 ◽  
Vol 18 (01) ◽  
pp. 2150013
Author(s):  
Juan L. G. Guirao ◽  
Jaume Llibre ◽  
Juan A. Vera ◽  
Bruce A. Wade
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Degree Of Freedom ◽  
Averaging Theory ◽  
Differential Systems ◽  
Type Of Stability

We deal with non-autonomous Hamiltonian systems of one degree of freedom. For such differential systems, we compute analytically some of their periodic solutions, together with their type of stability. The tool for proving these results is the averaging theory of dynamical systems. We present some applications of these results.

Dynamics of Dissipative Systems with Hamiltonian Structures

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Xiaoming Zhang ◽  
Zhenbang Cao ◽  
Jianhua Xie ◽  
Denghui Li ◽  
Celso Grebogi
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian Structure ◽  
Degree Of Freedom ◽  
Dissipative Systems ◽  
Integral Invariant ◽  
Hamiltonian Structures ◽  
Cartan Type ◽  
Two Degree Of Freedom

In this work, we study a class of dissipative, nonsmooth [Formula: see text] degree-of-freedom dynamical systems. As the dissipation is assumed to be proportional to the momentum, the dynamics in such systems is conformally symplectic, allowing us to use some of the Hamiltonian structure. We initially show that there exists an integral invariant of the Poincaré–Cartan type in such systems. Then, we prove the existence of a generalized Liouville Formula for conformally symplectic systems with rigid constraints using the integral invariant. A two degree-of-freedom system is analyzed to support the relevance of our results.

Vibrations of Dynamical Systems With Quadratic Nonlinearities

1965 ◽  
Vol 32 (3) ◽  
pp. 576-582 ◽  
Author(s):  
P. R. Sethna
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Method ◽  
Degree Of Freedom ◽  
System Parameters ◽  
Quadratic Nonlinearities ◽  
The Stability ◽  
Physical Phenomena ◽  
Two Degree Of Freedom

General two-degree-of-freedom dynamical systems with weak quadratic nonlinearities are studied. With the aid of an asymptotic method of analysis a classification of these systems is made and the more interesting subclasses are studied in detail. The study includes an examination of the stability of the solutions. Depending on the values of the system parameters, several different physical phenomena are shown to occur. Among these is the phenomenon of amplitude-modulated motions with modulation periods that are much larger than the periods of the excitation forces.

Periodic Solutions for Coupled Van Der Pol Oscillators of Two-Degree-of-Freedom Solved by Homotopy Analysis Method

2009 ◽  
Author(s):  
Wei Zhang ◽  
Youhua Qian ◽  
Qian Wang
Keyword(s):  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Homotopy Analysis ◽  
Van Der Pol Oscillators ◽  
Two Degree Of Freedom

Innumerable engineering problems can be described by multi-degree-of-freedom (MDOF) nonlinear dynamical systems. The theoretical modelling of such systems is often governed by a set of coupled second-order differential equations. Albeit that it is extremely difficult to find their exact solutions, the research efforts are mainly concentrated on the approximate analytical solutions. The homotopy analysis method (HAM) is a useful analytic technique for solving nonlinear dynamical systems and the method is independent on the presence of small parameters in the governing equations. More importantly, unlike classical perturbation technique, it provides a simple way to ensure the convergence of solution series by means of an auxiliary parameter ħ. In this paper, the HAM is presented to establish the analytical approximate periodic solutions for two-degree-of-freedom coupled van der Pol oscillators. In addition, comparisons are conducted between the results obtained by the HAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the higher-order analytical solutions of the HAM agree well with the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses.

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