Results 151 to 160 of about 481 (186)

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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Transversal Homoclinic Orbits in an Integrable System

American Journal of Mathematics, 1978
We construct a Hamiltonian system on TP' which admits a hyperbolic equilibrium point together with 2n transversal homoclinic orbits. However, the system is completely integrable, and hence X possesses no invariant subsystems topologically conjugate to the suspension of a Bernoulli shift.
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Shilnikov homoclinic orbit bifurcations in the Chua’s circuit

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2006
We analytically describe the complex scenario of homoclinic bifurcations in the Chua’s circuit. We obtain a general scaling law that gives the ratio between bifurcation parameters of different nearby homoclinic orbits. As an application of this theoretical approach, we estimate the number of higher order subsidiary homoclinic orbits that appear between
Medrano-T., R. O., Baptista, Murilo da Silva, Caldas, Ibere Luiz   +2 more
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Fuzzy homoclinic orbits and commuting fuzzifications

Fuzzy Sets and Systems, 2005
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Homoclinic orbits of a Hamiltonian system

Zeitschrift für angewandte Mathematik und Physik, 1999
The authors are interested in the existence of homoclinic orbits of the Hamiltonian system \(\dot x= JH_z(t,z)\) where \(z=(p,q)\in \mathbb{R}^N\times \mathbb{R}^N\), \(J\) is the standard symplectic matrix in \(\mathbb{R}^{2N}\), \(J= \left( \begin{smallmatrix} 0 &-\text{Id}\\ \text{Id} &0 \end{smallmatrix} \right)\), and \(H\in C(\mathbb{R}\times ...
Ding, Yanheng, Willem, Michel
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Periodic Orbits Near Homoclinic Orbits

1982
It is known that the orbit-structure of a dynamical system near a homoclinic orbit γ is extremely complicated. However, it is only recently that this complicated structure has begun to be understood. It has been shown (under some hypotheses) that, near γ there are infinitely many long periodic orbits. The flow, near γ, admits a singular Poincare map o:
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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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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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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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RIGOROUS VERIFICATION OF THE EXISTENCE OF TRANSVERSAL HOMOCLINIC ORBITS

International Journal of Bifurcation and Chaos, 2008
In this paper we present a computer-assisted technique that allows us to prove rigorously that a transversal homoclinic orbit of discrete dynamical systems with this method is simpler than previous works on this subject.
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