Many Pulses Homoclinic Orbits and Chaotic Dynamics for Nonlinear Nonplanar Motion of a Cantilever Beam

2007 ◽  
pp. 267-276
Author(s):  
M. H. Yao ◽  
W. Zhang
Keyword(s):  
Cantilever Beam ◽  
Chaotic Dynamics ◽  
Homoclinic Orbits

What can we learn from homoclinic orbits in chaotic dynamics?

1983 ◽  
Vol 31 (3) ◽  
pp. 499-518 ◽  
Author(s):  
P. Gaspard ◽  
G. Nicolis
Keyword(s):  
Chaotic Dynamics ◽  
Homoclinic Orbits

Chaotic Dynamics of a Quasi-Periodically Forced Beam

1992 ◽  
Vol 59 (1) ◽  
pp. 161-167 ◽  
Author(s):  
K. Yagasaki
Keyword(s):  
Chaotic Dynamics ◽  
Averaging Method ◽  
Single Mode ◽  
Homoclinic Orbits ◽  
Equation Of Motion ◽  
Degree Of Freedom ◽  
Finite Degree ◽  
Chaotic Motions ◽  
Higher Modes ◽  
Straight Beam

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.

CHAOS IN A NEAR-INTEGRABLE HAMILTONIAN LATTICE

2002 ◽  
Vol 12 (08) ◽  
pp. 1743-1754 ◽  
Author(s):  
VASSILIOS M. ROTHOS ◽  
CHRIS ANTONOPOULOS ◽  
LAMBROS DROSSOS
Keyword(s):  
Lyapunov Exponents ◽  
Numerical Experiments ◽  
Chaotic Dynamics ◽  
Large Scale ◽  
Hamiltonian Structure ◽  
Chaotic Behavior ◽  
Original System ◽  
Homoclinic Orbits ◽  
Hamiltonian Lattice ◽  
Whiskered Tori

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.

Detecting Stable-Unstable Nonlinear Invariant Manifold and Homoclinic Orbits in Mechanical Systems

2008 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Invariant Manifold ◽  
Homoclinic Orbit ◽  
Chaotic Dynamics ◽  
Series Solution ◽  
Homoclinic Orbits ◽  
System Response ◽  
Reduced Order ◽  
Analytical Technique ◽  
Buckled Beam ◽  
Fixed Beam

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.

Chaotic Dynamics in Four-Dimensional Piecewise Affine Systems with Bifocal Heteroclinic Cycles

2018 ◽  
Vol 28 (11) ◽  
pp. 1850141 ◽  
Author(s):  
Tiantian Wu ◽  
Lei Wang ◽  
Xiao-Song Yang
Keyword(s):  
Chaotic Dynamics ◽  
Homoclinic Orbits ◽  
Invariant Sets ◽  
Heteroclinic Cycle ◽  
Piecewise Affine Systems ◽  
Heteroclinic Cycles ◽  
Affine Systems ◽  
Piecewise Affine ◽  
Nonsmooth Systems ◽  
Full Shift

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.

Dynamics of a Weakly Nonlinear System Subjected to Combined Parametric and External Excitation

1990 ◽  
Vol 57 (1) ◽  
pp. 209-217 ◽  
Author(s):  
Kazuyuki Yagasaki ◽  
Masaru Sakata ◽  
Koji Kimura
Keyword(s):  
Chaotic Dynamics ◽  
Averaging Method ◽  
Invariant Tori ◽  
Homoclinic Orbits ◽  
External Excitation ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Average System ◽  
Theoretical Results

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.

THEORIES OF MULTI-PULSE GLOBAL BIFURCATIONS FOR HIGH-DIMENSIONAL SYSTEMS AND APPLICATION TO CANTILEVER BEAM

2008 ◽  
Vol 22 (24) ◽  
pp. 4089-4141 ◽  
Author(s):  
W. ZHANG ◽  
M. H. YAO
Keyword(s):  
Nonlinear Systems ◽  
Cantilever Beam ◽  
Chaotic Dynamics ◽  
Adjoint Operator ◽  
Operator Method ◽  
Melnikov Method ◽  
Heteroclinic Orbits ◽  
High Dimensional ◽  
Phase Method ◽  
Global Bifurcations

The aim of this survey paper is to illustrate the perspectives on the theories of the single- and multi-pulse global bifurcations and chaotic dynamics of high-dimensional nonlinear systems and applications to several engineering problems in the past two decades. Two main methods for studying the Shilnikov type multi-pulse homoclinic and heteroclinic orbits in high-dimensional nonlinear systems, which are the energy-phase method and generalized Melnikov method, are briefly demonstrated in the theoretical frame. In addition, the theory of normal form and an improved adjoint operator method for high-dimensional nonlinear systems is also applied to describe a reducing procedure to high-dimensional nonlinear systems. The aforementioned methods are utilized to investigate the Shilnikov type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end. How to employ these methods to analyze the Shilnikov type multi-pulse homoclinic and heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example.

Multi-Pulse Chaotic Dynamics of Circular Mesh Antenna with 1:2 Internal Resonance

2017 ◽  
Vol 09 (04) ◽  
pp. 1750060 ◽  
Author(s):  
Y. Sun ◽  
W. Zhang ◽  
M. H. Yao
Keyword(s):  
Internal Resonance ◽  
Chaotic Dynamics ◽  
Circular Cylindrical Shell ◽  
Domain Of Attraction ◽  
Homoclinic Orbits ◽  
Energy Difference ◽  
Phase Method ◽  
Chaotic Motions ◽  
Difference Function ◽  
First Time

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.

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