What can we learn from homoclinic orbits in chaotic dynamics?

1983 ◽  
Vol 31 (3) ◽  
pp. 499-518 ◽  
Author(s):  
P. Gaspard ◽  
G. Nicolis
1992 ◽  
Vol 59 (1) ◽  
pp. 161-167 ◽  
Author(s):  
K. Yagasaki

A straight beam with fixed ends, forced with two frequencies is considered. By using Galerkin’s method, the equation of motion of the beam is reduced to a finite degree-of-freedom system. The resulting equation is transformed into a multi-frequency system and the averaging method is applied. It is shown, by using Melnikov’s method, that there exist transverse homoclinic orbits in the averaged system associated with the first-mode equation. This implies that chaotic motions may occur in the single-mode equation. Furthermore, the effect of higher modes and the implications of this result for the full beam motions are described.


2002 ◽  
Vol 12 (08) ◽  
pp. 1743-1754 ◽  
Author(s):  
VASSILIOS M. ROTHOS ◽  
CHRIS ANTONOPOULOS ◽  
LAMBROS DROSSOS

We study the chaotic dynamics of a near-integrable Hamiltonian Ablowitz–Ladik lattice, which is N + 2-dimensional if N is even (N + 1, if N is odd) and possesses, for all N, a circle of unstable equilibria at ε = 0, whose homoclinic orbits are shown to persist for ε ≠ 0 on whiskered tori. The persistence of homoclinic orbits is established through Mel'nikov conditions, directly from the Hamiltonian structure of the equations. Numerical experiments which combine space portraits and Lyapunov exponents are performed for the perturbed Ablowitz–Ladik lattice and large scale chaotic behavior is observed in the vicinity of the circle of unstable equilibria in the ε = 0 case. We conjecture that this large scale chaos is due to the occurrence of saddle-center type fixed points in a perturbed 1 d.o.f Hamiltonian to which the original system can be reduced for all N. As ε > 0 increases, the transient character of this chaotic behavior becomes apparent as the positive Lyapunov exponents steadily increase and the orbits escape to infinity.


Author(s):  
Stefano Lenci ◽  
Giuseppe Rega

We consider a four dimensional Hamiltonian system representing the reduced-order (two-mode) dynamics of a buckled beam. The system has a saddle-center equilibrium point, and we pay attention to the existence and detection of the stable-unstable nonlinear manifold and of homoclinic solutions, which are the sources of complex and chaotic dynamics observed in the system response. The system has also a coupling nonlinear parameter, which depends on the boundary conditions, and is zero, e.g., for the hinged-hinged beam and different from zero, e.g., for the fixed-fixed beam. The invariant manifold in the latter case is detected assuming that it can be represented as a graph over the plane spanned by the unstable (principal) variable and its velocity. We show by a series solution that the manifold exists but has a limited extension, not sufficient for the deployment of the homoclinic orbit. Thus, the homoclinic orbit is addressed directly, irrespective of its belonging to the invariant manifold. By means of the perturbation method it is shown that it exists only on some curves of the governing parameters space, which branch from a fundamental path. This shows that the homoclinic orbit is not generic. These results have been confirmed by numerical simulations and by a different analytical technique.


2018 ◽  
Vol 28 (11) ◽  
pp. 1850141 ◽  
Author(s):  
Tiantian Wu ◽  
Lei Wang ◽  
Xiao-Song Yang

The well-known Shil’nikov type theory provides an approach to proving the existence of chaotic invariant sets for some classes of smooth dynamical systems with homoclinic orbits or heteroclinic cycles. However, it cannot be applied to nonsmooth systems directly. Based on the similar ideas, this paper studies the existence of chaotic invariant sets for a class of two-zone four-dimensional piecewise affine systems with bifocal heteroclinic cycles that cross the switching manifold transversally at two points. It turns out that there exist countable infinite chaotic invariant sets in a neighborhood of the bifocal heteroclinic cycle under some eigenvalue conditions. Moreover, the horseshoes of the corresponding Poincaré map are topologically semi-conjugated to a full shift on four symbols.


1990 ◽  
Vol 57 (1) ◽  
pp. 209-217 ◽  
Author(s):  
Kazuyuki Yagasaki ◽  
Masaru Sakata ◽  
Koji Kimura

In this paper we study the dynamics of a weakly nonlinear single-degree-of-freedom system subjected to combined parametric and external excitation. The averaging method is used to establish the existence of invariant tori and analyze their stability. Furthermore, by applying the Melnikov technique to the average system it is shown that there exist transverse homoclinic orbits resulting in chaotic dynamics. Numerical simulation results are also given to demonstrate the theoretical results.


2017 ◽  
Vol 09 (04) ◽  
pp. 1750060 ◽  
Author(s):  
Y. Sun ◽  
W. Zhang ◽  
M. H. Yao

The multi-pulse homoclinic orbits and chaotic dynamics of an equivalent circular cylindrical shell for the circular mesh antenna are investigated in the case of 1:2 internal resonance in this paper for the first time. Applying the method of averaging, the four-dimensional averaged equation in the Cartesian form is obtained. The theory of normal form is used to reduce the averaged equation to a simpler form. Based on the simplified system, the energy phase method is employed to investigate the homoclinic bifurcations and the Shilnikov type multi-pulse chaotic dynamics. First, the energy difference function and the zeroes of the energy difference function are obtained. Then, the existence of the Shilnikov type multi-pulse orbits is determined. The homoclinic trees are depicted to describe the relationship among the layers diameter, the pulse numbers and the phase shift. Finally, we need to verify the condition which makes sure that any multi-pulse orbit departing from a slow sink comes back to the domain of attraction of one of the sinks. The results obtained here show the existence of the Shilnikov type multi-pulse chaotic motions of the circular mesh antenna. Numerical simulations are used to find multi-pulse chaotic motions. The results of the theoretical analysis are in qualitative agreement with the results obtained using numerical simulation.


2004 ◽  
Vol 14 (08) ◽  
pp. 2667-2687 ◽  
Author(s):  
SHYAN-SHIOU CHEN ◽  
CHIH-WEN SHIH

This presentation investigates the dynamics of discrete-time cellular neural networks (DT-CNN). In contrast to classical neural networks that are mostly gradient-like systems, DT-CNN possesses both complete stability and chaotic behaviors as different parameters are considered. An energy-like function which decreases along orbits of DT-CNN as well as the existence of a globally attracting set are derived. Complete stability can then be concluded, with further analysis on the sets on which the energy function is constant. The formations of saturated stationary patterns for DT-CNN are shown to be analogous to the ones in continuous-time CNN. Thus, DT-CNN shares similar properties with continuous-time CNN. By confirming the existence of snap-back repellers, hence transversal homoclinic orbits, we also conclude that DT-CNN with certain parameters exhibits chaotic dynamics, according to the theorem by Marotto.


2010 ◽  
Author(s):  
M. H. Yao ◽  
W. Zhang ◽  
D. X. Cao ◽  
Jane W. Z. Lu ◽  
Andrew Y. T. Leung ◽  
...  

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