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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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Interactions of Hopf and Pitchfork Bifurcations

1980
Non 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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additive noise destroys a pitchfork bifurcation

Journal of Dynamics and Differential Equations, 1998
In the deterministic pitchfork bifurcation the dynamical behavior of the system changes as the parameter crosses the bifurcation point. The stable fixed point loses its stability. Two new stable fixed points appear. The respective domains of attraction of those two fixed points split the state space into two macroscopically distinct regions.
Crauel, Hans, Flandoli, Franco
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ON THE HOPF–PITCHFORK BIFURCATION IN THE CHUA'S EQUATION

International Journal of Bifurcation and Chaos, 2000
We 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, Emilio Freire, E. Gamero, Alejandro J. Rodríguez-Luis, Antonio Algaba   +4 more
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Computing multiple pitchfork bifurcation points

Computing, 1997
A point (x*,λ*) is called apitchfork bifurcation point of multiplicityp≥1 of the nonlinear systemF(x, λ)=0,F:ℝn×ℝ1→ℝn, if rank∂ xF(x*, λ*)=n−1, and if the Ljapunov-Schmidt reduced equation has the normal formg(ξ, μ)=±ξ 2+ p±μξ=0. It is shown that such points satisfy a minimally extended systemG(y)=0,G:ℝ
Gerd Pönisch, Hubert Schwetlick, U. Schnabel   +2 more
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Time-reversible and equivariant pitchfork bifurcation

Physica D: Nonlinear Phenomena, 1998
Abstract 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, 1992
For 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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Spectral signature of the pitchfork bifurcation: Liouville equation approach

Physical Review E, 1995
The time evolution of probability densities of one-dimensional nonlinear vector fields is studied using a Liouville equation approach. It is shown that the Liouville operator admits a discrete spectrum of eigenvalues of decaying type if the vector field is far from bifurcation.
Gaspard, Pierre, Nicolis, Grégoire, Provata, Astero, Tasaki, Shuichi   +3 more
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Hopf bifurcation at a degenerate stationary pitchfork

Nonlinear Analysis: Theory, Methods & Applications, 1986
Soit E un espace de Hilbert reel. On considere l'existence de petites solutions periodiques pour une equation d'evolution sur E donnee par du/dt=G(u,λ), u∈E, λ∈R, avec G(0,λ)=0 ∀λ ...
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Pitchfork bifurcation for non-autonomous interval maps

Journal of Difference Equations and Applications, 2009
In this work, we investigate attracting periodic orbits for non-autonomous discrete dynamical systems with two maps using a new approach. We study some types of bifurcation in these systems. We show that the pitchfork bifurcation plays an important role in the creation of attracting orbits in families of alternating systems with negative Schwarzian ...
D'ANIELLO, Emma, OLIVEIRA H.
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