Bifurcations of asymptotically autonomous differential equations
This interesting paper is a contribution to a bifurcation theory for nonautonomous ordinary differential equations with asymptotically autonomous right hand side. The author's analysis is based on the assumption that an autonomous equation \[ \dot{x}=g(x,\alpha)\tag{1} \] with \(x\in{\mathbb R}\) and real bifurcation parameter \(\alpha\) undergoes a pitchfork, or a transcritical, or a saddle-node bifurcation. Moreover, consider a nonautonomous differential equation \[ \dot{x}=f(t,x,\alpha)\tag{2} \] and suppose that both the limits \(\lim_{t\to-\infty}D_x^if(t,x,\alpha)=D_x^ig(x,\alpha)\), \(i=0,1\), exist uniformly for \((x,\alpha)\) in a neighborhood of the bifurcation point of (1). Then the corresponding bifurcation results extend to the time-dependent equation (2) in the sense that corresponding past attractors instead of equilibria change. A related result is proved in the case of planar equations, i.e.\ for variables \(x\in{\mathbb R}^2\) and the situation where (1) undergoes a Hopf bifurcation. The nonautonomous bifurcation concepts used here are based on the concept of pullback attractors and repellers. They have been established in the earlier work of the author [Attractivity and bifurcation for nonautonomous dynamical systems. Lecture Notes in Mathematics 1907. Berlin: Springer (2007; Zbl 1131.37001)].
asymptotically autonomous differential equation
scientific article; zbMATH DE number 5529275