Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems

2012 ◽  
Vol 241 (22) ◽  
pp. 1962-1975 ◽  
Author(s):  
F. Battelli ◽  
M. Fečkan
Keyword(s):  
Homoclinic Orbits ◽  
Discontinuous Systems ◽  
Melnikov Functions

Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-de Vries equation

1994 ◽  
Vol 445 (1923) ◽  
pp. 1-21 ◽  
Keyword(s):  
Lyapunov Exponent ◽  
Vries Equation ◽  
Period Doubling ◽  
Homoclinic Orbits ◽  
Dynamical Behaviour ◽  
Chaotic Behaviour ◽  
Subharmonic Bifurcations ◽  
Melnikov Functions ◽  
De Vries ◽  
Period Doubling Bifurcations

The dynamical behaviour of a reduction of the forced (and damped) Korteweg-de Vries equation is studied numerically. Chaos arising from subharmonic instability and homoclinic crossings are observed. Both period-doubling bifurcations and the Melnikov sequence of subharmonic bifurcations are found and lead to chaotic behaviour, here characterised by a positive Lyapunov exponent. Supporting theoretical analysis includes the construction of periodic solutions and homoclinic orbits, and their behaviour under perturbation using Melnikov functions.

A FAST-MANIFOLD APPROACH TO MELNIKOV FUNCTIONS FOR SLOWLY VARYING OSCILLATORS

1996 ◽  
Vol 06 (08) ◽  
pp. 1575-1578 ◽  
Author(s):  
SHYH-LEH CHEN ◽  
STEVEN W. SHAW
Keyword(s):  
Present Method ◽  
Homoclinic Orbits ◽  
Melnikov Function ◽  
Three Dimensions ◽  
Distance Functions ◽  
New Approach ◽  
Alternative Derivation ◽  
Melnikov Functions ◽  
Melnikov Analysis ◽  
Insight Into

A new approach to obtaining the Melnikov function for homoclinic orbits in slowly varying oscillators is proposed. The present method applies the usual two-dimensional Melnikov analysis to the “fast” dynamics of the system which lie on an invariant manifold. It is shown that the resultant Melnikov function is the same as that obtained in the usual way involving distance functions in three dimensions [Wiggins and Holmes, 1987]. This alternative derivation provides some useful insight into the structure of the dynamical system.

Chaotic response of a Galerkin shell to constant load and periodic excitation

2005 ◽  
Vol 219 (1) ◽  
pp. 61-73
Author(s):  
L Dai ◽  
Q Han ◽  
M Dong
Keyword(s):  
Elastic Shell ◽  
Saddle Points ◽  
Time History ◽  
Homoclinic Orbits ◽  
Constant Load ◽  
Critical Conditions ◽  
Linear Elastic ◽  
Melnikov Functions ◽  
Non Linear ◽  
Lateral Excitation

This study intends to investigate the dynamic behaviour of a non-linear elastic shallow shell of large deflection subjected to constant boundary loading and harmonic lateral excitation. The general governing equation for the shell is established using the Galerkin Principle. Three types of dynamic equation of the shell are developed, corresponding to certain geometry and loading conditions. Melnikov functions are considered for each type. Non-linear responses of the shell to the loads are analysed theoretically. Centre points, saddle points, and homoclinic orbits are determined and analysed on the basis of the governing equations established. The critical conditions for chaos to occur are provided for the vibrations of the shell. Numerical analysis is also performed for the non-linear elastic shell. Chaotic and regular vibrations of the shell are analysed with presentations of time history plots, phase diagrams, and Poincaré maps.

scholarly journals A Melnikov Method for Homoclinic Orbits with Many Pulses

1998 ◽  
Vol 143 (2) ◽  
pp. 105-193 ◽  
Author(s):  
Roberto Camassa ◽  
Gregor Kovačič ◽  
Siu-Kei Tin
Keyword(s):  
Melnikov Method ◽  
Homoclinic Orbits

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

scholarly journals GEOMETRY OF HOMOCLINIC CONNECTIONS IN A PLANAR CIRCULAR RESTRICTED THREE-BODY PROBLEM

2007 ◽  
Vol 17 (04) ◽  
pp. 1151-1169 ◽  
Author(s):  
MARIAN GIDEA ◽  
JOSEP J. MASDEMONT
Keyword(s):  
Symbolic Dynamics ◽  
Body Problem ◽  
Invariant Manifolds ◽  
Libration Point ◽  
Homoclinic Orbits ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Lyapunov Orbits ◽  
Homoclinic Connections ◽  
Three Body

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1and the other collinear libration points L2, L3is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

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