By Lakshmi Burra (auth.)

Using phase–plane research, findings from the idea of topological horseshoes and linked-twist maps, this e-book offers a singular technique to end up the lifestyles of chaotic dynamics. In dynamical platforms, advanced habit in a map may be indicated via exhibiting the lifestyles of a Smale-horseshoe-like constitution, both for the map itself or its iterates. This frequently calls for a few assumptions concerning the map, equivalent to a diffeomorphism and a few hyperbolicity stipulations. during this textual content, much less stringent definitions of a horseshoe were instructed which will reproduce a few geometrical positive aspects usual of the Smale horseshoe, whereas leaving out the hyperbolicity stipulations linked to it. This results in the examine of the so-called topological horseshoes. The presence of chaos-like dynamics in a vertically pushed planar pendulum, a pendulum of variable size, and in different extra basic similar equations is usually proved.

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**Sample text**

43 Fig. 4 An example of linked annuli determining the two regions P and Q for f (x) = sin(π x) (a) (b) Fig. 5 The two regions P and Q determined by the two systems of level lines are then oriented, by suitably choosing which are the [·]− -sets. a The upper region P . b The lower region Q A superimposition of the phase portraits of the two systems gives rise to linked annuli from which we construct the oriented rectangles. Such oriented rectangles are obtained by intersecting a region included between a pair of level lines of one system with a region between a pair of level lines of the other.

Firstly using the inverse of the homeomorphism h : G → R, we confine ourselves to the study of a compact set S ⊆ G having the property of meeting any path γ contained in G , with γ joining the left and the right sides of G . If, by contradiction, we assume that S does not contain any compact connected set C intersecting both the lower and the upper sides of G , by the Whyburn Lemma (the details of which we do not give here but which can be seen in [22]) we may find a decomposition of S into two disjoint compact subsets A and B such that A ∪ ([0, 1] × {0}) = ∅, A ∪ ([0, 1] × {1}) = ∅ B ∪ ([0, 1] × {1}) = ∅, B ∪ ([0, 1] × {0}) = ∅.

More formally, we note that, for any w ∈ J∞ , there exists a unique forward itinerary (wi )i∈N such that w0 = w and ψ(wi ) = wi+1 ∈ K , for every i ∈ N. Hence the function N which maps any w ∈ J into the one-sided sequence of points g1 : J∞ →, J∞ ∞ from the set J∞ sw := (wi )i∈N where wi := ψ i (w), ∀ i ∈ N, with the usual convention ψ 0 = I dJ∞ and ψ 1 = ψ, is well-defined. Since the sets K0 and K1 are disjoint, for every term wi of sw there exists a unique index si = si (wi ), with si ∈ {0, 1}, N → such that wi ∈ Ksi .