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A competition between heteroclinic cycles
Nonlinearity, 1994This paper analyzes the dynamics of a particular family of ordinary differential equations in \(\mathbb{R}^ 4\) that possess a high degree of symmetry. Because of the symmetry there can be structurally stable configuration of 4 equilibria, \(p\), \(q\), \(r_ 1\) and \(r_ 2\) such that there are two heteroclinic cycles of the form \(C_ i = p \to q \to ...
Mary Silber, Vivien Kirk
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Homoclinic and Heteroclinic Bifurcations Close to a Twisted Heteroclinic Cycle
International Journal of Bifurcation and Chaos, 1998We study the interaction of a transcritical (or saddle-node) bifurcation with a codimension-0/codimension-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reaction–diffusion equation.
Mario A. Natiello, Martin Zimmermann
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Heteroclinic cycles in rings of coupled cells [PDF]
Physica D: Nonlinear Phenomena, 2000 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Pietro-Luciano Buono +2 more
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Izvestiya, Physics of the Solid Earth, 2020
Heteroclinic cycle is an invariant of a dynamical system comprised of steady states (or more general invariant subsets) and heteroclinic trajectories. The behavior of a dynamical system with a heteroclinic cycle is intermittent: a typical trajectory stays for a long time close to a steady state while the transitions between the states occur much faster.
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Heteroclinic cycle is an invariant of a dynamical system comprised of steady states (or more general invariant subsets) and heteroclinic trajectories. The behavior of a dynamical system with a heteroclinic cycle is intermittent: a typical trajectory stays for a long time close to a steady state while the transitions between the states occur much faster.
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Heteroclinic cycles in the repressilator model
Chaos, Solitons & Fractals, 2012Abstract A repressilator is a synthetic regulatory network that produces self-sustained oscillations. We analyze the evolution of the oscillatory solution in the repressilator model. We have established a connection between the evolution of the oscillatory solution and formation of a heteroclinic cycle at infinity.
Valentin Afraimovich +3 more
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Journal of Nonlinear Science, 1997
A heteroclinic cycle in a dynamical system is given by a finite cyclic sequence of trajectories each connecting two fixed points. Robustness means that a cycle nearby still exists after perturbing the system in a particular admissible way. Here, one allows for a stable state with high symmetry to lose its stability and some of its symmetry.
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A heteroclinic cycle in a dynamical system is given by a finite cyclic sequence of trajectories each connecting two fixed points. Robustness means that a cycle nearby still exists after perturbing the system in a particular admissible way. Here, one allows for a stable state with high symmetry to lose its stability and some of its symmetry.
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Stationary bifurcation to limit cycles and heteroclinic cycles
Nonlinearity, 1991The 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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Heteroclinic cycles in lattice dynamical systems
Science in China Series A: Mathematics, 1998zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Qian Min, Qin Wenxin, Qin Wenxin
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On the stability of double homoclinic and heteroclinic cycles
Nonlinear Analysis: Theory, Methods & Applications, 2003The authors study the stability of a polycycle \(L\) with two regular sides, \(L_1\) and \(L_2,\) and two hyperbolic corners, \(s_1\) and \(s_2\) (which eventually may coincide), of a \(C^5\) planar vector field \(X.\) It is not restrictive to assume that it is oriented clockwise and that the omega limit set of the side \(L_1\) is \(s_2.\) Denote by \(\
Shouchuan Hu, Xingbo Liu, Maoan Han
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Heteroclinic cycles and wreath product symmetries
Dynamics and Stability of Systems, 2000We consider the existence and stability of heteroclinic cycles arising by local bifurcation in dynamical systems with wreath product symmetry = Z 2 G, where Z 2 acts by - 1 on R and G is a transitive subgroup of the permutation group S N (thus G has degree N). The group acts absolutely irreducibly on R N .
Ana Paula S. Dias +2 more
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