Numerical Analysis and Control of Bifurcation Problems (Ii): Bifurcation in Infinite Dimensions

Doedel, Eusebius J.; Keller, Herbert B.; Kernevez, J. P.

doi:10.1142/s0218127491000555

Cited by 378 publications

(226 citation statements)

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“…given mean thicknesses we use continuation techniques 67,68,69 . We start with analytically known stationary periodic small-amplitude profiles, which correspond to the linear eigenfunctions for the critical wave number k c .…”

Section: Non-linear Behaviour a Stationary Solutions As Extrema mentioning

confidence: 99%

Morphology changes in the evolution of liquid two-layer films

Pototsky

Bestehorn

Merkt

et al. 2005

The Journal of Chemical Physics

163

202

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We consider a thin film consisting of two layers of immiscible liquids on a solid horizontal (heated) substrate. Both, the free liquid-liquid and the liquid-gas interface of such a bilayer liquid film may be unstable due to effective molecular interactions relevant for ultrathin layers below 100 nm thickness, or due to temperature-gradient caused Marangoni flows in the heated case. Using a long wave approximation we derive coupled evolution equations for the interface profiles for the general non-isothermal situation allowing for slip at the substrate. Linear and nonlinear analyses of the short-and long-time film evolution are performed for isothermal ultrathin layers taking into account destabilizing long-range and stabilizing short-range molecular interactions. It is shown that the initial instability can be of a varicose, zigzag or mixed type. However, in the nonlinear stage of the evolution the mode type and therefore the pattern morphology can change via switching between two different branches of stationary solutions or via coarsening along a single branch.

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Section: Non-linear Behaviour a Stationary Solutions As Extrema mentioning

confidence: 99%

Morphology changes in the evolution of liquid two-layer films

Pototsky

Bestehorn

Merkt

et al. 2005

The Journal of Chemical Physics

163

202

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“…Factoring out the group invariance (through so-called pinning conditions) is also an important component of many numerical methods for systems with symmetry. For instance, a similar templatebased phase condition is often used in the computation of periodic solutions of autonomous ordinary differential equations (ODEs) [4], and a related method was recently used in [19] for bifurcation analysis of systems where the governing equations are not explicitly known, but only a numerical timestepper is available.…”

Section: Introductionmentioning

confidence: 99%

Reduction and reconstruction for self-similar dynamical systems

et al. 2003

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We present a general method for analysing and numerically solving partial differential equations with self-similar solutions. The method employs ideas from symmetry reduction in geometric mechanics, and involves separating the dynamics on the shape space (which determines the overall shape of the solution) from those on the group space (which determines the size and scale of the solution). The method is computationally tractable as well, allowing one to compute self-similar solutions by evolving a dynamical system to a steady state, in a scaled reference frame where the self-similarity has been factored out. More generally, bifurcation techniques can be used to find self-similar solutions, and determine their behaviour as parameters in the equations are varied.The method is given for an arbitrary Lie group, providing equations for the dynamics on the reduced space, for reconstructing the full dynamics and for determining the resulting scaling laws for self-similar solutions. We illustrate the technique with a numerical example, computing self-similar solutions of the Burgers equation.Mathematics Subject Classification: 70G65, 76M60, 60G18, 34A26, 37J15, 65P99

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“…General methods [22][23][24][25] for the solution of nonlinear two-point boundary value problems tend to be geared toward ordinary differential equations, and can be prohibitively expensive for partial differential equations. Recently [26][27][28], Wilkening and Ambrose introduced an efficient method of computing timeperiodic solutions of nonlinear PDEs.…”

Section: The Adjoint Continuation Methods (Acm)mentioning

confidence: 99%

“…In general terms, the ACM operates by treating the task of finding the initial condition of a periodic orbit as an unconstrained minimization problem. One advantage of this approach over, say, the orthogonal collocation method implemented in AUTO [22,23] is that there are many fewer degrees of freedom to compute (as only the initial conditions are unknown). For example, in a typical simulation, we use 1024 grid points in space and 500 timesteps, each broken into 8 Runge-Kutta stages.…”

Section: Appendix a Acm Implementation Flowchartmentioning

confidence: 99%

Continuation of periodic solutions in the waveguide array mode-locked laser

Williams

Wilkening

Shlizerman

et al. 2011

Physica D: Nonlinear Phenomena

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We apply the adjoint continuation method to construct highly-accurate, periodic solutions that are observed to play a critical role in the multi-pulsing transition of mode-locked laser cavities. The method allows for the construction of solution branches and the identification of their bifurcation structure. Supplementing the adjoint continuation method with a computation of the Floquet multipliers allows for explicit determination of the stability of each branch. This method reveals that, when gain is increased, the multi-pulsing transition starts with a Hopf bifurcation, followed by a period-doubling bifurcation, and a saddle-node bifurcation for limit cycles. Finally, the system exhibits chaotic dynamics and transitions to the double-pulse solutions. Although this method is applied specifically to the waveguide array mode-locking model, the multi-pulsing transition is conjectured to be ubiquitous and these results agree with experimental and computational results from other models.

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Numerical Analysis and Control of Bifurcation Problems (Ii): Bifurcation in Infinite Dimensions

Cited by 378 publications

References 0 publications

Morphology changes in the evolution of liquid two-layer films

Morphology changes in the evolution of liquid two-layer films

Reduction and reconstruction for self-similar dynamical systems

Continuation of periodic solutions in the waveguide array mode-locked laser

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