By L.M. Pismen, Y. Pomeau
Spontaneous development formation in nonlinear dissipative platforms faraway from equilibrium happens in quite a few settings in nature and expertise, and has purposes starting from nonlinear optics via good and fluid mechanics, actual chemistry and chemical engineering to biology. This e-book explores the vanguard of present study, describing in-depth the analytical equipment that elucidate the advanced evolution of nonlinear dissipative platforms.
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Extra resources for Patterns and Interfaces in Dissipative Dynamics (Springer Series in Synergetics)
5. Stability of nonstationary and nonuniform states in the vicinity of a primary bifurcation is determined in a systematic way by weakly nonlinear analysis as described in Sects. 6. 1 Multiscale Expansion In the vicinity of a bifurcation manifold, the original system of equations can be reduced to a simple universal form that is characteristic to the particular type of bifurcation and retains the qualitative features of dynamic behavior of the underlying system in an adjacent parametric domain.
This will not happen when a mode with k = 0 is unstable, and the instability is absolute. 4 Instabilities of Periodic Orbits Periodic orbits or nonuniform stationary states emerging at a primary bifurcation may in turn lose stability at secondary bifurcation points. Stability of a T -periodic orbit u0 (t) = u0 (t + T ) to inﬁnitesimal perturbations is determined by linearizing the underlying equation in the vicinity of this solution. 1) and look for solutions in the form u(t) = u0 (t) + v(t), where v(t) is a small deviation from the periodic solution u0 (t).
At n ≤ 3, the resonance is strong and enters the lowest order amplitude equation; otherwise, it acts as a perturbation. Since, because of nonlinear frequency shift, resonances may appear in “Arnold tongues” beyond the bifurcation point, transition to chaos turns out to be a generic phenomenon under these conditions (Ruelle and Takens, 1971). 1 Global Dynamics: An Overview There are only two kinds of generic (codimension one) bifurcations in dynamical systems that can be detected by local linear analysis: a saddle-node bifurcation at zero eigenvalue that creates a pair of equilibria – a saddle and a node – and a Hopf bifurcation at imaginary eigenvalue creating a periodic orbit.
Patterns and Interfaces in Dissipative Dynamics (Springer Series in Synergetics) by L.M. Pismen, Y. Pomeau