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

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.

Coherent structures and chaotic advection in three dimensions

Journal of Fluid Mechanics ◽

10.1017/s0022112010002569 ◽

2010 ◽

Vol 654 ◽

pp. 1-4 ◽

Cited By ~ 24

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127499001322 ◽

1999 ◽

Vol 09 (09) ◽

pp. 1853-1858 ◽

Cited By ~ 6

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.

On the analysis of time-periodic nonlinear dynamical systems

Sadhana ◽

10.1007/bf02744481 ◽

1997 ◽

Vol 22 (3) ◽

pp. 411-434 ◽

Cited By ~ 4

Author(s):

S C Sinha

Keyword(s):

Dynamical Systems ◽

Nonlinear Dynamical Systems ◽

Nonlinear Dynamical ◽

Time Periodic

Semi-Discretization of Delayed Dynamical Systems

Volume 6B: 18th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc2001/vib-21446 ◽

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.

Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems

Volume 6: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

10.1115/detc2015-47486 ◽

2015 ◽

Author(s):

W. Grant Kirkland ◽

S. C. Sinha

Keyword(s):

Control System ◽

Dynamical Systems ◽

System Design ◽

Symbolic Computation ◽

Picard Iteration ◽

Control System Design ◽

Symbolic Form ◽

Varying Coefficients ◽

And Control ◽

Time Periodic

Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix Φ(t,α) associated with the linear part of the equation can be expressed in terms of the periodic Lyapunov-Floquét transformation matrix Q(t,α) and a time-invariant matrix R(α). Computation of Q(t,α) and R(α) in a symbolic form as a function of system parameters α is of paramount importance in stability, bifurcation analysis, and control system design. In the past, a methodology has been presented for computing Φ(t,α) in a symbolic form, however Q(t,α) and R(α) have never been calculated in a symbolic form. Since Q(t,α) and R(α) were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In this work a technique for symbolic computation of Q(t,α), and R(α) matrices is presented. First, Φ(t,α) is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method as suggested in the literature. Then R(α) is computed using the Gaussian quadrature integral formula. Finally Q(t,α) is computed using the matrix exponential summation method. Using Mathematica, this approach has successfully been applied to the well-known Mathieu equation and a four dimensional time-periodic system in order to demonstrate the applications of the proposed method to linear as well as nonlinear problems.

Dynamical systems analysis of fluid transport in time-periodic vortex ring flows

Physics of Fluids ◽

10.1063/1.2189867 ◽

2006 ◽

Vol 18 (4) ◽

pp. 047104 ◽

Cited By ~ 17

Author(s):

Karim Shariff ◽

Anthony Leonard ◽

Joel H. Ferziger

Keyword(s):

Dynamical Systems ◽

Vortex Ring ◽

Systems Analysis ◽

Fluid Transport ◽

Dynamical Systems Analysis ◽

Time Periodic

Global attractivity and stability in some monotone discrete dynamical systems

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700017032 ◽

1996 ◽

Vol 53 (2) ◽

pp. 305-324 ◽

Cited By ~ 38

Author(s):

Xiao-Qiang Zhao

Keyword(s):

Dynamical Systems ◽

Parabolic Equations ◽

Global Attractivity ◽

Discrete Dynamical Systems ◽

Nonnegative Solutions ◽

Globally Attractive ◽

Order Intervals ◽

Time Periodic ◽

Ordered Banach Spaces ◽

Asymptotic Behaviours

The existence of globally attractive order intervals for some strongly monotone discrete dynamical systems in ordered Banach spaces is first proved under some appropriate conditions. With the strict sublinearity assumption, threshold results on global asymptotic stability are then obtained. As applications, the global asymptotic behaviours of nonnegative solutions for time-periodic parabolic equations and cooperative systems of ordinary differential equations are discussed and some biological interpretations and concrete application examples are also given.