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Register a new userPiecewise-linear maps with heterogeneous chaos
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Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos), While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two more explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.
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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.
We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking and the quasi-periodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic ``sine-circle map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction.
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In this paper, we first show that any nonlinear monotonic increasing contracting maps with one discontinuous point on a unit interval which has an unique periodic point with period $n$ conjugates to a piecewise linear contracting map which has periodic point with same period. Second, we consider one parameter family of monotonic increasing contracting maps, and show that the family has the periodic structure called Arnold tongue for the parameter which is associated with the Farey series. This implies that there exist a parameter set with a positive Lebesgue measure such that the map has a periodic point with an arbitrary period. Moreover, the parameter set with period $(m+n)$ exists between the parameter set with period $m$ and $n$.
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