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Register a new userChaos in the border-collision normal form: A computer-assisted proof using induced maps and invariant expanding cones
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In some maps the existence of an attractor with a positive Lyapunov exponent can be proved by constructing a trapping region in phase space and an invariant expanding cone in tangent space. If this approach fails it may be possible to adapt the strategy by considering an induced map (a first return map for a well-chosen subset of phase space). In this paper we show that such a construction can be applied to the two-dimensional border-collision normal form (a continuous piecewise-linear map) if a certain set of conditions are satisfied and develop an algorithm for checking these conditions. The algorithm requires relatively few computations, so it is a more efficient method than, for example, estimating the Lyapunov exponent from a single orbit in terms of speed, numerical accuracy, and rigor. The algorithm is used to prove the existence of an attractor with a positive Lyapunov exponent numerically in an area of parameter space where the map has strong rotational characteristics and the consideration of an induced map is critical for the proof of robust chaos.
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Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049--3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological attractor. We then construct an invariant expanding cone in tangent space to prove that tangent vectors expand and so no invariant set can have only negative Lyapunov exponents. Under additional assumptions we also characterise an attractor as the closure of the unstable manifold of a fixed point.
We show how the existence of three objects, $Omega_{rm trap}$, ${bf W}$, and $C$, for a continuous piecewise-linear map $f$ on $mathbb{R}^N$, implies that $f$ has a topological attractor with a positive Lyapunov exponent. First, $Omega_{rm trap} subset mathbb{R}^N$ is trapping region for $f$. Second, ${bf W}$ is a finite set of words that encodes the forward orbits of all points in $Omega_{rm trap}$. Finally, $C subset T mathbb{R}^N$ is an invariant expanding cone for derivatives of compositions of $f$ formed by the words in ${bf W}$. We develop an algorithm that identifies these objects for two-dimensional homeomorphisms comprised of two affine pieces. The main effort is in the explicit construction of $Omega_{rm trap}$ and $C$. Their existence is equated to a set of computable conditions in a general way. This results in a computer-assisted proof of chaos throughout a relatively large regime of parameter space. We also observe how the failure of $C$ to be expanding can coincide with a bifurcation of $f$. Lyapunov exponents are evaluated using one-sided directional derivatives so that forward orbits that intersect a switching manifold (where $f$ is not differentiable) can be included in the analysis.
The collision of a fixed point with a switching manifold (or border) in a piecewise-smooth map can create many different types of invariant sets. This paper explores two techniques that, combined, establish a chaotic attractor is created in a border-collision bifurcation in $mathbb{R}^d$ $(d ge 1)$. First, asymptotic stability of the fixed point at the bifurcation is characterised and shown to imply a local attractor is created. Second, a lower bound on the maximal Lyapunov exponent is obtained from the determinants of the one-sided Jacobian matrices associated with the fixed point. Special care is taken to accommodate points whose forward orbits intersect the switching manifold as such intersections can have a stabilising effect. The results are applied to the two-dimensional border-collision normal form focusing on parameter values for which the map is piecewise area-expanding.
The goal of this paper is to construct invariant dynamical objects for a (not necessarily invertible) smooth self map of a compact manifold. We prove a result that takes advantage of differences in rates of expansion in the terms of a sheaf cohomolog
ical long exact sequence to create unique lifts of finite dimensional invariant subspaces of one term of the sequence to invariant subspaces of the preceding term. This allows us to take invariant cohomological classes and under the right circumstances construct unique currents of a given type, including unique measures of a given type, that represent those classes and are invariant under pullback. A dynamically interesting self map may have a plethora of invariant measures, so the uniquess of the constructed currents is important. It means that if local growth is not too big compared to the growth rate of the cohomological class then the expanding cohomological class gives sufficient marching orders to the system to prohibit the formation of any other such invariant current of the same type (say from some local dynamical subsystem). Because we use subsheaves of the sheaf of currents we give conditions under which a subsheaf will have the same cohomology as the sheaf containing it. Using a smoothing argument this allows us to show that the sheaf cohomology of the currents under consideration can be canonically identified with the deRham cohomology groups. Our main theorem can be applied in both the smooth and holomorphic setting.
We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs-Markov-Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invariant probability measure which is absolutely continuous measure (with respect to Lebesgue) and for which all Lyapunov exponents are positive.
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