Parameterized Stable/Unstable Manifolds for Periodic Solutions of Implicitly Defined Dynamical Systems

2021 ◽  
Author(s):  
Jason Desmond Mireles James ◽  
Archana Neupane Timsina
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Unstable Manifolds

Finding Periodic Solutions of Forced Systems With Local Nonlinearities: A Mixed Shooting Harmonic Balance Method

2015 ◽  
Author(s):  
Frederic Schreyer ◽  
Remco Leine
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Shooting Method ◽  
Mechanical Systems ◽  
Harmonic Balance Method ◽  
Nonlinear Effects ◽  
Balance Method ◽  
Pros And Cons ◽  
Numerical Approaches

Several numerical approaches have been developed to capture nonlinear effects of dynamical systems. In this paper we present a mixed shooting-harmonic balance method to solve large mechanical systems with local nonlinearities efficiently. The Harmonic Balance Method as well as the shooting method have both their pros and cons. The proposed mixed shooting-HBM approach combines the efficiency of HBM and the accuracy of the shooting method and has therefore advantages of both.

ON NONLINEAR WAVE EQUATIONS WITH BREAKING LOOP-SOLUTIONS

2010 ◽  
Vol 20 (02) ◽  
pp. 519-537 ◽  
Author(s):  
JIBIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Nonlinear Wave ◽  
Traveling Wave ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Level Curves ◽  
Straight Line ◽  
Fixed Parameter ◽  
Unstable Manifolds

Using analytic methods from the dynamical systems theory, some new nonlinear wave equations are investigated, which have exact explicit parametric representations of breaking loop-solutions under some fixed parameter conditions. It is shown that these parametric representations are associated with some families of open level-curves of traveling wave systems corresponding to such nonlinear wave equations, each of which lies in an area bounded by a singular straight line and the stable and the unstable manifolds of a saddle point of such a system.

scholarly journals Unstable manifolds of analytic dynamical systems

1981 ◽  
Vol 21 (4) ◽  
pp. 763-785 ◽  
Author(s):  
Shigehiro Ushiki
Keyword(s):  
Dynamical Systems ◽  
Unstable Manifolds

A Wavelet Based Procedure to Study Vibrations and Stability

1999 ◽  
Author(s):  
S. Pernot ◽  
C. H. Lamarque
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Periodic Solutions ◽  
Time Varying ◽  
Differential Systems ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential

Abstract A Wavelet-Galerkin procedure is introduced in order to obtain periodic solutions of multidegrees-of-freedom dynamical systems with periodic time-varying coefficients. The procedure is then used to study the vibrations of parametrically excited mechanical systems. As problems of stability analysis of nonlinear systems are often reduced after linearization to problems involving linear differential systems with time-varying coefficients, we demonstrate the method provides efficient practical computations of Floquet exponents and consequently allows to give estimators for stability/instability levels. A few academic examples illustrate the relevance of the method.

Dynamical Systems with Simple Periodic Solutions

1986 ◽  
pp. 197-202
Author(s):  
N. Caranicolas ◽  
Th. Diplas
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions

scholarly journals Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

2010 ◽  
Vol 17 (1) ◽  
pp. 1-36 ◽  
Author(s):  
M. Branicki ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Vector Fields ◽  
Transient Flow ◽  
Time Interval ◽  
Stable And Unstable Manifolds ◽  
Lagrangian Transport ◽  
Transport Barriers ◽  
Unstable Manifolds ◽  
Hyperbolic Trajectories

Abstract. We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.

scholarly journals Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

2016 ◽  
Vol 260 (7) ◽  
pp. 6108-6129 ◽  
Author(s):  
Márcio R.A. Gouveia ◽  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Claudio Pessoa
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Normal Forms ◽  
Piecewise Smooth

scholarly journals Periodic solutions of a class of dynamical systems of second order

1991 ◽  
Vol 90 (2) ◽  
pp. 408-417 ◽  
Author(s):  
Zheng-chao Han
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Second Order
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