On the Normal Forms of Certain Parametrically Excited Systems

Zhang, W.Y.; Huseyin, K.; Ye, M.

doi:10.1006/jsvi.2001.4187

“…Equation (3.4) is thus the condition for the onset of 'static' instability of the plane wave. Whether this bifurcation is a pitchfork or transcritical one, and its subcritical or supercritical nature, may be readily determined by deriving an appropriate canonical system in the vicinity of (3.4) using any of a variety of normal form or perturbation methods [14,18,25]. One may also have the onset of dynamic instability ('flutter' in the language of Applied Mechanics) when a pair of eigenvalues of the Jacobian become purely imaginary.…”

Section: Stability Analysis For Individual Plane Wave Solutionsmentioning

confidence: 99%

Bifurcations of plane wave (CW) solutions in the complex cubic–quintic Ginzburg–Landau equation

Mancas

¹

,

Choudhury

²

2007

Mathematics and Computers in Simulation

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Singularity Theory is used to comprehensively investigate the bifurcations of the steady-states of the traveling wave ODEs of the cubic-quintic Ginzburg-Landau equation (CGLE). These correspond to plane waves of the PDE. In addition to the most general situation, we also derive the degeneracy conditions on the eight coefficients of the CGLE under which the equation for the steady states assumes each of the possible quartic (the quartic fold and an unnamed form), cubic (the pitchfork and the winged cusp), and quadratic (four possible cases) normal forms for singularities of codimension up to three. Since the actual governing equations are employed, all results are globally valid, and not just of local applicability. In each case, the recognition problem for the unfolded singularity is treated. The transition varieties, i.e. the hysteresis, isola, and double limit curves are presented for each normal form. For both the most general case, as well as for various combinations of coefficients relevant to the particular cases, the bifurcations curves are mapped out in the various regions of parameter space delimited by these varieties. The multiplicities and interactions of the plane wave solutions are then comprehensively deduced from the bifurcation plots in each regime, and include features such as regimes of hysteresis among co-existing states, domains featuring more than one interval of hysteresis, and isola behavior featuring dynamics unrelated to the primary solution branch in limited ranges of parameter space.

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“…Whether this bifurcation is a pitchfork or transcritical one, and its subcritical or supercritical nature, may be readily determined by deriving an appropriate canonical system in the vicinity of (3.4) using any of a variety of normal form or perturbation methods [21]- [23].…”

Section: Stability Analysis For Individual Plane Wave Solutionsmentioning

confidence: 99%

Bifurcations and competing coherent structures in the cubic-quintic Ginzburg–Landau equation I: Plane wave (CW) solutions

Mancas

¹

,

Choudhury

²

2006

Chaos, Solitons & Fractals

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Singularity Theory is used to comprehensively investigate the bifurcations of the steady-states of the traveling wave ODEs of the cubic-quintic Ginzburg-Landau equation (CGLE). These correspond to plane waves of the PDE. In addition to the most general situation, we also derive the degeneracy conditions on the eight coefficients of the CGLE under which the equation for the steady states assumes each of the possible quartic (the quartic fold and an unnamed form), cubic (the pitchfork and the winged cusp), and quadratic (four possible cases) normal forms for singularities of codimension up to three. Since the actual governing equations are employed, all results are globally valid, and not just of local applicability. In each case, the recognition problem for the unfolded singularity is treated. The transition varieties, i.e. the hysteresis, isola, and double limit curves are presented for each normal form. For both the most general case, as well as for various combinations of coefficients relevant to the particular cases, the bifurcations curves are mapped out in the various regions of parameter space delimited by these varieties. The multiplicities and interactions of the plane wave solutions are then comprehensively deduced from the bifurcation plots in each regime, and include features such as regimes of hysteresis among co-existing states, domains featuring more than one interval of hysteresis, and isola behavior featuring dynamics unrelated to the primary solution branch in limited ranges of parameter space.

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Hopf bifurcation control in a coupled nonlinear relative rotation dynamical system

Liu¹,

Liu²,

Yan³

et al. 2010

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On the Normal Forms of Certain Parametrically Excited Systems

Cited by 3 publications

References 16 publications

Bifurcations of plane wave (CW) solutions in the complex cubic–quintic Ginzburg–Landau equation

Bifurcations of plane wave (CW) solutions in the complex cubic–quintic Ginzburg–Landau equation

Bifurcations and competing coherent structures in the cubic-quintic Ginzburg–Landau equation I: Plane wave (CW) solutions

Hopf bifurcation control in a coupled nonlinear relative rotation dynamical system

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