Versual deformation and local bifurcation analysis of time-periodic nonlinear systems
The authors study the parameter-dependent nonlinear time-periodic systems of the form \(\dot {x}(t,\alpha)\) \(=A(t,\alpha)x(t,\alpha)+f_2(x,t,\alpha)+\cdots +f_k(x,t, \alpha)+O(|x|^{k+1},\alpha,t)\). Here \(A(t+T,\alpha)=A(t,\alpha)\) is an \(n\times n\) state matrix, \(x\) and \(f_i\) are \(n\)-vectors, \(f_i\) contain monomials in \(x\) of order \(i\) and are \(T\)-periodic in time, and \(\alpha\) is an \(m\)-vector (\(m\leq n\)) of parameters. After simplification via Lyapunov-Floquet transformation, the authors use the time-periodic center manifold reduction to separate the critical states and reduce the dimension of the system. Then the center manifold equations are simplified via time-dependent normal form theory. The authors determine deformations of these time-invariant normal forms, and study the bifurcations in a neighborhood of the critical point.
parameter-dependent nonlinear time-periodic system
scientific article; zbMATH DE number 1455200