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Register a new userChaos on compact manifolds: Differentiable synchronizations beyond the Takens theorem
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This paper shows that a large class of fading memory state-space systems driven by discrete-time observations of dynamical systems defined on compact manifolds always yields continuously differentiable synchronizations. This general result provides a powerful tool for the representation, reconstruction, and forecasting of chaotic attractors. It also improves previous statements in the literature for differentiable generalized synchronizations, whose existence was so far guaranteed for a restricted family of systems and was detected using Holder exponent-based criteria.
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We prove that every $mathbb{Z}^{k}$-action $(X,mathbb{Z}^{k},T)$ of mean dimension less than $D/2$ admitting a factor $(Y,mathbb{Z}^{k},S)$ of Rokhlin dimension not greater than $L$ embeds in $(([0,1]^{(L+1)D})^{mathbb{Z}^{k}}times Y,sigmatimes S)$, where $Dinmathbb{N}$, $Linmathbb{N}cup{0}$ and $sigma$ is the shift on the Hilbert cube $([0,1]^{(L+1)D})^{mathbb{Z}^{k}}$; in particular, when $(Y,mathbb{Z}^{k},S)$ is an irrational $mathbb{Z}^{k}$-rotation on the $k$-torus, $(X,mathbb{Z}^{k},T)$ embeds in $(([0,1]^{2^kD+1})^{mathbb{Z}^k},sigma)$, which is compared to a previous result by the first named author, Lindenstrauss and Tsukamoto. Moreover, we give a complete and detailed proof of Takens embedding theorem with a continuous observable for $mathbb{Z}$-actions and deduce the analogous result for $mathbb{Z}^{k}$-actions. Lastly, we show that the Lindenstrauss--Tsukamoto conjecture for $mathbb{Z}$-actions holds generically, discuss an analogous conjecture for $mathbb{Z}^{k}$-actions appearing in a forthcoming paper by the first two authors and Tsukamoto and verify it for $mathbb{Z}^{k}$-actions on finite dimensional spaces.
Let G be a subgroup of finite index in SL(n,Z) for N > 4. Suppose G acts continuously on a manifold M, with fundamental group Z^n, preserving a measure that is positive on open sets. Further assume that the induced G action on H^1(M) is non-trivial. We show there exists a finite index subgroup G of G and a G equivariant continuous map from M to the n-torus that induces an isomorphism on fundamental groups. We prove more general results providing continuous quotients in cases where the fundamental group of M surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with G actions to which the theorems apply.
In this paper, we construct a homeomorphism on the unit closed disk to show that an invertible mapping on a compact metric space is Li-Yorke chaotic does not imply its inverse being Li-Yorke chaotic.
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.
In this paper, we study the bifurcate of limit cycles for Bogdanov-Takens system($dot{x}=y$, $dot{y}=-x+x^{2}$) under perturbations of piecewise smooth polynomials of degree $2$ and $n$ respectively. We bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles with switching curve $x=y^{2m}$($m$ is a positive integral) are $(39m+36)n+77m+21(mgeq 2)$ and $50n+52(m=1)$ (taking into account the multiplicity). The upper bounds number of limit cycles with switching lines $x=0$ and $y=0$ are 11 (taking into account the multiplicity) and it can be reached.
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