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

Bountis, Tassos; Tsarouhas, George; Herman, Russell

Physica D 33, No. 1-3, 34-50 (1988).

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.

MSC:

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

Keywords:

normal form theory; Lotka-Volterra system; dissipative Duffing’s equation; chaotic orbits; basin of attraction; fixed point

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