Pattern formation in a slowly flattening spherical cap: delayed bifurcation

This article describes a reduction of a non-autonomous Brusselator reaction–diffusion system of partial differential equations on a spherical cap with time-dependent curvature using the method of centre manifold reduction. Parameter values are chosen such that the change in curvature would cross critical values which would change the stability of the patternless solution in the constant domain case. The evolving domain functions and quasi-patternless solutions are derived as well as a method to obtain this non-autonomous normal form. The coefficients of such a normal form are computed and the reduction solutions are compared to numerical solutions.

DELAYED BIFURCATION IN FIRST-ORDER SINGULARLY PERTURBED PROBLEMS WITH A NONGENERIC TURNING POINT

In this paper, based on the method of upper and lower solutions, delayed bifurcation in first-order singularly perturbed problems with a nongeneric turning point is studied. The asymptotic behavior of the solutions is understood by constructing the upper and lower solutions with the desired dynamical properties. As an application of the obtained results, delayed phenomenon of degenerate Hopf bifurcation in a planar polynomial differential system with a slowly varying parameter is discussed in detail and the maximal delay is calculated. Numerical simulations are carried out to verify the theoretical results.

A Time-Dependent Bifurcation Model with Control and Pulsing Oscillations

A time-dependent bifurcation model and its control problem are studied. Firstly, the delayed bifurcating transition with memory effects due to time-dependent parameters are analysed. Secondly, a control problem with time-dependent parametric feedback in this bifurcation model is investigated. Finally, an important mechanism for pulsing oscillation is found as the result of the delayed bifurcation transition occurring when the bifurcation parameter varies periodically across the steady bifurcation value.

TIME-DEPENDENT BIFURCATION: A NEW METHOD AND APPLICATIONS

A new qualitative method is presented for studying the bifurcation problems of nonlinear systems with time-dependent parameters. The concept of stability on finite time intervals is introduced and some related stability theorems are established in order to analyze time-dependent bifurcation problems. As some applications of the new method, three different types of delayed bifurcation transitions and jump phenomena of time-dependent Duffing–van der Pol's equation are investigated qualitatively. The bifurcation transition values are predicted by constructing the V-functions or by the linear stability. The sensitivity of motion to the initial values and parameters is also discussed.