Dynamics on Spectral Solutions of Forced Burgers Equation

Abstract

Burgers equation \(\frac{\partial u} {\partial t} + u\frac{\partial u} {\partial x} = \delta \frac{{\partial }^{2}u} {\partial {x}^{2}} + f\left (x\right )\) is one of the simplest partial nonlinear differential equation which can develop discontinuities, being the driven equation used to explore unidimensional “turbulence”. For low values of the viscosity coefficient δ, by discretization through spectral collocation methods, oscillations in Burgers equation can occur. For the Dirichlet problem and under a dynamic point of view, several bifurcations and stable attractors can be observed. Periodic orbits, nonperiodic and strange attractors may arise. Bistability can also be observed. Numerical simulations indicate that the loss of stability of the asymptotic solution of Burgers equation must occur by means of a supercritical Hopf bifurcation. Many nonlinear phenomena are modeled by spatiotemporal systems of infinite or very high dimension. Coupling and synchronization of spatially extended dynamical systems, periodic or chaotic, have many applications, including communications systems, chaos control, estimation of model parameters and model identifications. For the unidirectionally linear coupling, numerical studies show the presence of identical or generalized synchronization for different values of spacial points and different values of the viscosity coefficient δ. Also, nonlinear coupling by a convex linear combination of the drive and driven variables corresponding to the waves velocity, can achieve identical or generalized synchronization.