The exploration of solutions of the Ginzburg-Landau equation in the autowave approximation
RTU MIREA, Institute of Information Technologies, Department Applied Mathematics, Russia, 119454, Moscow, Vernadsky Avenue, 78, +7 (985) 255-01-18, firstname.lastname@example.org
The Ginzburg-Landau equation is used to model wave processes in self-oscillating environment. In our work, we explored the solution of the Ginzburg-Landau equation in the form: $w_t = w + (α + iC_1) w_xx- (1 + iC_2) w | w | ^ 2$ in the autowave approximation in order to explore the bifurcation diagram in the parameter space α and $C_0$, where $C_0$ - the propagation velocity of disturbances in the environment. To solve the problem, the Runge-Kutta method of 4 orders was used. The calculations were performed using a program implemented in the Jupiter Notebook environment (python3). The calculated accuracy of the representation of numbers is of the order of $10^-16$. The bifurcation of the birth of cycles with loss of stability of a fixed point of a system of ordinary differential equations in the autowave approximation and the bifurcation of the birth of a two-dimensional invariant torus with loss of stability of the limit cycle are established. It is shown that there is a cascade of bifurcations of the creation of tori of the doubled period in the outer cycle and a cascade of bifurcations of birth of the tori with periods in the outer cycle according to the Sharkovsky order. Finaly, we have built bifurcation diagram of solutions of the equation in the autowave approximation in the space of parameters α and $C_0$.