Linearly Stable Subharmonic Orbits in Strongly Monotone Time-Periodic Dynamical Systems

1992 ◽  
Vol 115 (3) ◽  
pp. 691 ◽  
Author(s):  
Peter Takac
Keyword(s):  
Dynamical Systems ◽  
Strongly Monotone ◽  
Time Periodic ◽  
Subharmonic Orbits

A construction of stable subharmonic orbits in monotone time-periodic dynamical systems

1993 ◽  
Vol 115 (3) ◽  
pp. 215-244 ◽  
Author(s):  
Peter Takáč
Keyword(s):  
Dynamical Systems ◽  
Time Periodic ◽  
Subharmonic Orbits

On the Analysis of Time-Periodic Nonlinear Hamiltonian Dynamical Systems

1995 ◽  
Author(s):  
Eric A. Butcher ◽  
S. C. Sinha
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Normal Form ◽  
Linear Part ◽  
Normal Form Theory ◽  
Mathieu’S Equation ◽  
Form Theory ◽  
Hamiltonian Dynamical Systems ◽  
Time Invariant ◽  
Time Periodic

Abstract In this paper, some analysis techniques for general time-periodic nonlinear Hamiltonian dynamical systems have been presented. Unlike the traditional perturbation or averaging methods, these techniques are applicable to systems whose Hamiltonians contain ‘strong’ parametric excitation terms. First, the well-known Liapunov-Floquet (L-F) transformation is utilized to convert the time-periodic dynamical system to a form in which the linear pan is time invariant. At this stage two viable alternatives are suggested. In the first approach, the resulting dynamical system is transformed to a Hamiltonian normal form through an application of permutation matrices. It is demonstrated that this approach is simple and straightforward as opposed to the traditional methods where a complicated set of algebraic manipulations are required. Since these operations yield Hamiltonians whose quadratic parts are integrable and time-invariant, further analysis can be carried out by the application of action-angle coordinate transformation and Hamiltonian perturbation theory. In the second approach, the resulting quasilinear time-periodic system (with a time-invariant linear part) is directly analyzed via time-dependent normal form theory. In many instances, the system can be analyzed via time-independent normal form theory or by the method of averaging. Examples of a nonlinear Mathieu’s equation and coupled nonlinear Mathieu’s equations are included and some preliminary results are presented.

scholarly journals Coherent structures and chaotic advection in three dimensions

2010 ◽  
Vol 654 ◽  
pp. 1-4 ◽  
Author(s):  
STEPHEN WIGGINS
Keyword(s):  
Dynamical Systems ◽  
Fluid Mechanics ◽  
Coherent Structures ◽  
Three Dimensional ◽  
Point Of View ◽  
Three Dimensions ◽  
Chaotic Advection ◽  
Periodic Flow ◽  
Two Dimensional ◽  
Time Periodic

In the 1980s the incorporation of ideas from dynamical systems theory into theoretical fluid mechanics, reinforced by elegant experiments, fundamentally changed the way in which we view and analyse Lagrangian transport. The majority of work along these lines was restricted to two-dimensional flows and the generalization of the dynamical systems point of view to fully three-dimensional flows has seen less progress. This situation may now change with the work of Pouransari et al. (J. Fluid Mech., this issue, vol. 654, 2010, pp. 5–34) who study transport in a three-dimensional time-periodic flow and show that completely new types of dynamical systems structures and consequently, coherent structures, form a geometrical template governing transport.

ON SOME NONAUTONOMOUS CHAOTIC SYSTEM ON THE PLANE

1999 ◽  
Vol 09 (09) ◽  
pp. 1853-1858 ◽  
Author(s):  
KLAUDIUSZ WÓJCIK
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Fixed Point Theorem ◽  
Chaotic System ◽  
Point Theorem ◽  
Chaotic Behavior ◽  
Topological Methods ◽  
Lefschetz Fixed Point Theorem ◽  
Nonautonomous Equations ◽  
Time Periodic

We prove the existence of the chaotic behavior in dynamical systems generated by some class of time periodic nonautonomous equations on the plane. We use topological methods based on the Lefschetz Fixed Point Theorem and the Ważewski Retract Theorem.

scholarly journals On the analysis of time-periodic nonlinear dynamical systems

Sadhana ◽  
1997 ◽  
Vol 22 (3) ◽  
pp. 411-434 ◽  
Author(s):  
S C Sinha
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Time Periodic

Semi-Discretization of Delayed Dynamical Systems

2001 ◽  
Author(s):  
Tamás Insperger ◽  
Gábor Stépán
Keyword(s):  
Dynamical Systems ◽  
Mathieu Equation ◽  
Special Kind ◽  
Delayed Systems ◽  
The Past ◽  
Linear Discrete System ◽  
Approximate System ◽  
Discretization Technique ◽  
The Stability ◽  
Time Periodic

Abstract An efficient numerical method is presented for the stability analysis of linear retarded dynamical systems. The method is based on a special kind of discretization technique with respect to the past effect only. The resulting approximate system is delayed and time-periodic in the same time, but still, it can be transformed analytically into a high dimensional linear discrete system. The method is especially efficient for time varying delayed systems, including the case when the time delay itself varies in time. The method is applied to determine the stability charts of the delayed Mathieu equation with damping.

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