Dana C'Julio, Bernd Krauskopf, and Hinke M. Osinga
Abstract
The generalisation of Smale's horseshoe construction in spaces of at least dimension three can lead to a type of hyperbolic set called a blender. The characterising feature of a blender is that it has an invariant manifold that behaves as a geometric object of a higher dimension than expected from the dimension of the manifold itself. The presence of a blender can be used to prove the robust existence of a nontransverse heterodimensional cycle, leading to robust nonhyperbolic chaotic dynamics, also known as wild chaos. We consider a Hénon-like family of maps, which is one of the only known explicit examples that exhibits a blender generated by a three-dimensional horseshoe. We investigate how the orientation properties of the map affect the creation of a blender. To this end, we use advanced numerical techniques to compute the stable or unstable manifolds of the fixed points and period-two orbits of the map; these manifolds are a key element in the creation of a blender. We discover that the organisation of the manifolds, and hence, the blender, follow a consistent pattern as the parameters are varied, and we show in detail the order of events that bring about the generation or disappearance of blenders. This allows us to obtain an estimate for the lower bound of the parameter value at which a blender arises in this map. In particular, we contrast the results of the different orientation properties of the map, which is essential for understanding how blenders lose or gain their defining properties as parameters change.
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