Computation of Lyapunov-Type Numbers for Invariant Curves of Planar Maps

Numerical Approximation of Rough Invariant Curves of Planar Maps

“…We briefly summarize some known results; see, for example, [2,5]. The value λ 0 = 2.19 lies in an interval of phase locking as follows: For λ 1 < λ < λ 4 , where…”

“…However, the nonsmooth regions of Γ are small, namely, confined to seven spirals that are small in diameter, at least in the example considered below. The computations in [5] indicate that an attractivity factor ν = e −β is still numerically meaningful. In addition, the following observation is critical to understanding the performance of the suggested algorithm: The rough parts of Γ are precisely those parts that are dynamically most strongly attracting.…”

“…Under this assumption the Lyapunovtype numbers ν(p) and σ(p) are defined for all p ∈ Γ. See [7] for the definition of these numbers in the general context of overflowing invariant manifolds and see, for example, [5] for the specific case of invariant curves in the plane. The number ν = sup p ν(p) is assumed to be strictly less than 1, which expresses exponential attractivity of Γ.…”

“…See, for example, [9]. An extensive, computerassisted study of the fate of Γ λ , for increasing λ, has been given in [2]; see also [5] for the study of Lyapunov-type numbers related to the invariant curves Γ λ . It appears that Γ λ transforms itself from a smooth invariant curve (for 2 < λ < 2 + ε) to a strange attractor for λ ≈ 2.2701.…”

Lyapunov{type numbers for Poincaré maps

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In [3] we have addressed computational issues related to Lyapunovtype numbers for invariant curves of planar maps. These numbers are important for understanding persistence and breakdown of the invariant curves when the map is perturbed.

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