Characterising the maximal invariant set of a map with (robust) heterodimensional cycles
Sam Doak, Bernd Krauskopf, and Hinke M. Osinga
Abstract
We present a case study of a three-dimensional diffeomorphism called the shear rotation map T. This one-parameter map has been proposed as the first example of an explicitly given discrete-time dynamical system that potentially has heterodimensional cycles. A heterodimensional cycle consists of two heteroclinic orbits between saddle periodic points of different index (number of unstable eigenvalues). In the (minimal) three-dimensional case considered here, one heteroclinic orbit lies in the non-transverse intersection of a one-dimensional stable and one-dimensional unstable manifold, and the other in the transverse intersection of the corresponding two-dimensional manifolds. Our goal is to characterize the bounded maximal invariant set Λ of T for a fixed parameter value. To this end, we design an efficient algorithm to compute millions of periodic orbits in Λ , and find that those with one-dimensional unstable manifolds are densely intermingled with those that have one-dimensional stable manifolds. Hence, Λ cannot be a hyperbolic set. We then focus on and compute to long arclengths the one-dimensional unstable and stable manifolds of the two fixed points of T and show that they have the so-called carpet property of `behaving like a surface'; this strongly suggests that T has robust heterodimensional cycles. Subsequently, we define suitably formulated boundary-value problems to find both a non-transverse heteroclinic orbit and a one-dimensional family of transverse return orbits between the two fixed points. This is the first heterodimensional cycle computed for an explicit diffeomorphism. Our case study of the properties of the invariant set Λ of T is representative in that it demonstrates generic properties of diffeomorphisms with robust heteterodimensional cycles — the key objects used to prove that nonhyperbolic dynamics can be persistent in open sets of diffeomorphisms.
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