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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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ON THE HOPF–PITCHFORK BIFURCATION IN THE CHUA'S EQUATION
International Journal of Bifurcation and Chaos, 2000We study some periodic and quasiperiodic behaviors exhibited by the Chua's equation with a cubic nonlinearity, near a Hopf–pitchfork bifurcation. We classify the types of this bifurcation in the nondegenerate cases, and point out the presence of a degenerate Hopf–pitchfork bifurcation. In this degenerate situation, analytical and numerical study shows
Manuel Merino +4 more
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Interactions of Hopf and Pitchfork Bifurcations
1980Non linear interactions between a Hopf bifurcation and a pitchfork-type stationary bifurcation can produce secondary bifurcations of periodic solutions, and tertiary bifurcations of periodic or aperiodic solutions lying on an invariant torus. A complete classification of the resulting bifurcation diagrams is presented, with emphasis on the cases which ...
William F. Langford, Gérard Iooss
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Time-reversible and equivariant pitchfork bifurcation
Physica D: Nonlinear Phenomena, 1998Abstract In this note we describe a new phenomenon in steady-state bifurcations of time-reversible and equivariant vector fields. When the eigenspace associated with a pair of imaginary eigenvalues that passes through zero onto the real line is separable from the nullspace associated with the fixed zeros of the spectrum, we prove that a pitchfork ...
Chjan C. Lim, I-Heng McComb
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Sensitivity of pitchfork bifurcation to stochastic perturbation
Dynamics and Stability of Systems, 1992For a stochastically perturbed two-dimensional system exhibiting a pitchfork bifurcation, asymptotic expansions for the Lyapunov exponent and the rotation number are obtained in the vicinity of the point of deterministic bifurcation. Using the result for the Lyapunov exponent, the shift in the bifurcation point due to the stochastic perturbation is ...
S. T. Ariaratnam, Wei-Chau Xie
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Hopf bifurcation at a degenerate stationary pitchfork
Nonlinear Analysis: Theory, Methods & Applications, 1986In this paper the branching of genuine periodic solutions for one parameter families of certain parabolic differential equations is investigated. The conditions given in the paper imply the existence of such solutions near a highly degenerate singularity, coming from the interaction of a simple zero eigenvalue and a pair of simple purely imaginary ...
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Applied Mathematics and Computation, 2021
Xindong Ma, Yue Yu, Lifeng Wang
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Xindong Ma, Yue Yu, Lifeng Wang
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Slow Periodic Crossing of a Pitchfork Bifurcation in an Oscillating System
Nonlinear Dynamics, 1997The nonlinear Mathieu equation is investigated as a case study of damped oscillating systems with slowly varying forcing function. The phase portrait of the system changes qualitatively with time with a pitchfork bifurcation point involved. A Poincaré map is constructed in order to describe the dynamics of the system as it repeatedly crosses the ...
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