Results 211 to 220 of about 10,534 (236)

Controlling homoclinic orbits

Theoretical and Computational Fluid Dynamics, 1989
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Bloch, A. M., Marsden, J. E.
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Homoclinic Orbits in the Complex Domain

International Journal of Bifurcation and Chaos, 1997
We consider the standard map, as a paradigm of area preserving map, when the variables are taken as complex. We study how to detect the complex homoclinic points, which cannot dissappear under a homoclinic tangency. This seems a promising tool to understand the stochastic zones of area preserving maps. The paper is mainly phenomenological and includes
Lazutkin, V. F., Simó, C.
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MULTIPLE HOMOCLINIC BIFURCATIONS FROM ORBIT-FLIP I: SUCCESSIVE HOMOCLINIC DOUBLINGS

International Journal of Bifurcation and Chaos, 1996
The purpose of this and forthcoming papers is to obtain a better understanding of complicated bifurcations for multiple homoclinic orbits. We shall take one particular type of codimension two homoclinic orbits called orbit-flip and study bifurcations to multiple homoclinic orbits appearing in a tubular neighborhood of the original orbit-flip. The main
Kokubu, Hiroshi, Komuro, Motomasa, Oka, Hiroe   +2 more
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N-Homoclinic bifurcations for homoclinic orbits changing their twisting

Journal of Dynamics and Differential Equations, 1996
The author considers two-parameter families of vector fields possessing a homoclinic orbit along a path in the parameter plane. These homoclinic orbits are homoclinic to a hyperbolic singularity that has a one-dimensional unstable manifold. The weakest stable and unstable eigenvalues of the linearized vector field at the singularity are supposed to be ...
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Conventional multipliers for homoclinic orbits

Nonlinearity, 1996
Summary: We introduce and describe conventional multipliers, a new characteristic of homoclinic orbits of saddle-node type periodic trajectories. We prove existence and smooth dependence of conventional multipliers on the initial point. We show that multipliers of periodic trajectories arising from the homoclinic ones as a result of the saddle-node ...
Afraimovich, Valentine, Liu, Weishi, Young, Todd R.   +2 more
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Differential Equations with Bifocal Homoclinic Orbits

International Journal of Bifurcation and Chaos, 1997
Global bifurcation theory can be used to understand complicated bifurcation phenomena in families of differential equations. There are many theoretical results relating to systems having a homoclinic orbit biasymptotic to a stationary point at some value of the parameters, and these results depend upon the eigenvalues of the Jacobian matrix of the ...
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Existence of optimal homoclinic orbits

2008 American Control Conference, 2008
The problem of optimal periodic control is considered from a geometric point of view. The objective is to determine the conditions under which a given optimal control problem admits a homoclinic orbit as an extremal solution. The analysis is performed on the Hamiltonian dynamical system obtained from the application of Pontryagin Maximum Principle ...
N. Hudon, K. Hoffner, M. Guay
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HOMOCLINIC ORBITS FOR 3−DIMENSIONAL SYSTEMS

SUT Journal of Mathematics, 1995
The bifurcation problem of a homoclinic loop for a three-dimensional system of ordinary differential equations is considered. Assuming that a system \[ \dot x= F(x, \mu)\qquad (F(0,\mu)= 0),\tag{1} \] where \(x\in \mathbb{R}^3\), \(\mu\in \mathbb{R}^m\) \((m\geq 3)\) is a parameter, \(F: \mathbb{R}^3\times \mathbb{R}^m\to \mathbb{R}^3\) is \(C^2\), has
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Homoclinic/heteroclinic recurrent orbits and horseshoe

Journal of Differential Equations
In this paper, the authors consider systems of ODEs \[ \dot z = g(z) + \mu h(t,z,\mu) \] with a small parameter \(\mu \in \mathbb{R}\). Assuming that for \(\mu=0\) the system has a solution \(\xi(t)\) that is homoclinic to a hyperbolic saddle point \(z_0\), as well as some other technical hypotheses, they show that for small non-zero \(|\mu|\) there is
Dong, Xiujuan, Li, Yong
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CONSTRUCTING HOMOCLINIC ORBITS AND CHAOTIC ATTRACTORS

International Journal of Bifurcation and Chaos, 1994
Homoclinic orbits and chaotic attractors are constructed progressively by singular perturbations. More specifically, lower dimensional slow subsystems and fast subsystems are constructed separately as building blocks. The former are then modulated onto the latter via homotopy.
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