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A REMARK ON HETEROCLINIC BIFURCATIONS NEAR STEADY STATE/PITCHFORK BIFURCATIONS [PDF]

International Journal of Bifurcation and Chaos, 2004
We consider a bifurcation that occurs in some two-dimensional vector fields, namely a codimension-one bifurcation in which there is simultaneously the creation of a pair of equilibria via a steady state bifurcation and the destruction of a large amplitude periodic orbit.
Edgar Knobloch, Vivien Kirk
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Bifurcations of Heteroclinic Loop with Twisted Conditions

International Journal of Bifurcation and Chaos, 2017
The bifurcation problems of twisted heteroclinic loop with two hyperbolic critical points are studied for the case [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are the pair of principal eigenvalues of unperturbed system at the critical point [Formula: see text], [Formula: see text ...
Yuanyuan Liu, Dandan Xie, Suoling Yang, Suoling Yang, Yinlai Jin, Nana Zhang   +5 more
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Theory and Application of a Nongeneric Heteroclinic Loop Bifurcation

SIAM Journal on Applied Mathematics, 1999
Summary: Homoclinic and heteroclinic bifurcations from a heteroclinic loop are considered. The system under consideration has three parameters, two of which are not suitable for generic unfoldings. Analytical criteria in terms of derivatives to Melnikov's functions are given for nongeneric parameters.
Bo Deng, Mark J. Friedman, Shui-Nee Chow
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Homoclinic and heteroclinic bifurcations of Vector fields

Japan Journal of Applied Mathematics, 1988
The author studies a bifurcation of homoclinic and heteroclinic orbits in a k-parameter family of \((m+n)\)-dimensional ODES: \(\dot x=f(x)+g(x,\mu)\), \(x\in {\mathbb{R}}^{m+n}\), \(\mu \in {\mathbb{R}}^ k\) (k\(\geq 2)\), where f and g are smooth and \(g(x,0)=0\). Suppose that the system has three saddle equilibria \(0_ i(\mu)\), \(i=1,2,3\), and the
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Stationary bifurcation to limit cycles and heteroclinic cycles

Nonlinearity, 1991
The authors consider one-parametric vector fields which are equivariant under the action of the group \(\Gamma=\mathbb{Z}_ 4\cdot\mathbb{Z}^ 4_ 2\) (semi-direct product). It is supposed that \(\mathbb{R}^ 4\) is the absolutely irreducible space for \(\Gamma\). Thus the considered vector field is a perturbation of the field of the form \(\lambda x+Q(x)\)
James W. Swift, Michael Field
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BIFURCATIONS OF GENERIC HETEROCLINIC LOOP ACCOMPANIED BY TRANSCRITICAL BIFURCATION

International Journal of Bifurcation and Chaos, 2008
The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1and one hyperbolic saddle p2are investigated, where p1is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic ...
Deming Zhu, Fengjie Geng, Dan Liu
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Heteroclinic bifurcation of limit cycles in perturbed cubic Hamiltonian systems by higher-order analysis

Journal of Differential Equations, 2023
Wei-Jian Geng, Maoan Han, Yun Tian, Ai Ke   +3 more
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Stability of heteroclinic cycles in transverse bifurcations

Physica D: Nonlinear Phenomena, 2015
Abstract Heteroclinic cycles and networks exist robustly in dynamical systems with symmetry. They can be asymptotically stable, and gradually lose this stability through a variety of bifurcations, displaying different forms of non-asymptotic stability along the way.
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Periodic sinks and periodic saddle orbits induced by heteroclinic bifurcation in three-dimensional piecewise linear systems with two zones

Applied Mathematics and Computation, 2021
Lei Wang, Qingdu Li, Xiao-Song Yang
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Persistence of cycles and nonhyperbolic dynamics at heteroclinic bifurcations

Nonlinearity, 1995
In the paper arcs \((f_t)_{t\in I}\), \(I= [0, 1]\) of \(C^\infty\) diffeomorphisms \(f_t\) defined on an \(n\) \((n\geq 3)\)-dimensional manifold and bifurcating through the creation of heterodimensional connected cycles are considered. The arc \((f_t)_{t\in I}\) is said to create a heterodimensional cycle at \(t= b\) if there are periodic hyperbolic ...
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