Normal form solutions of dynamical systems in the basin of attraction of their fixed points. (English) Zbl 0668.58031

Summary: The normal form theory of Poincaré, Siegel and Arnol’d is applied to an analytically solvable Lotka-Volterra system in the plane, and a periodically forced, dissipative Duffing’s equation with chaotic orbits in its 3-dimensional phase space. For the planar model, we determine exactly how the convergence region of normal forms about a nodal fixed point is limited by the presence of singularities of the solutions in the complex t-plane. Despite such limitations, however, we show, in the case of a periodically driven system, that normal forms can be used to obtain useful estimates of the basin of attraction of a stable fixed point of the Poincaré map, whose “boundary” is formed by the intersecting invariant manifolds of a second hyperbolic fixed point nearby.


37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37G99 Local and nonlocal bifurcation theory for dynamical systems
37C75 Stability theory for smooth dynamical systems
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