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**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.

### 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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\textit{T. Bountis} et al., Physica D 33, No. 1--3, 34--50 (1988; Zbl 0668.58031)

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### References:

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