Browsing by Author "Lyu, Ming-Jiea"

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A simple Proof for persistence of snap-back repellers
(2009-04-15)
Li, Ming-Chia
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Lyu, Ming-Jiea
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應用數學系
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Department of Applied Mathematics
In this article, we show that if f has a snap-back repeller then any small C(1) perturbation of f has a snap-back repeller, and hence has Li-Yorke chaos and positive topological entropy, by simply using the implicit function theorem. We also give some examples. (C) 2008 Elsevier Inc. All rights reserved.
Covering relations and Lyapunov condition for topological conjugacy
(2016-01-02)
Li, Ming-Chia
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Lyu, Ming-Jiea
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應用數學系
;
Department of Applied Mathematics
In this paper, we study stability of non-autonomous discrete dynamical systems. For a two-sided non-autonomous systems with covering relations determined by a transition matrix A, we show that any small C-0 perturbed system has a sequence of compact invariant sets restricted to which the system is topologically semi-conjugate to sigma(A), the two-sided subshift of finite type induced by A. Together with Lyapunov condition of good rate, the semi-conjugacy will become conjugacy. Moreover, if the Lyapunov condition is strict and has perfect rate, then any small C-1 perturbed systems is topological conjugate to sigma(A). We also study topological chaos of one-sided systems and systems with limit functions. Lack of hyperbolicity and the time dependence of the rate prevent us from applying classical hyperbolic results or earlier works for autonomous systems: cone condition and Lyapunov function. Two examples are provided to demonstrate the existence of a non-trivial, non-hyperbolic invariant and essential of controlling rate.
Covering relations for coupled map networks
(2012-12-01)
Bunimovich, Leonid
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Li, Ming-Chia
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Lyu, Ming-Jiea
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應用數學系
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Department of Applied Mathematics
In this paper, we study coupled map networks over arbitrary finite graphs. An estimate from below for a topological entropy of a perturbed coupled map network is obtained via a topological entropy of an unperturbed network by making use of the covering relations for coupled map networks. The result is quite general; in particular, nonlinear coupling is allowed and no assumptions of hyperbolicity of the local dynamics are made. (C) 2012 Elsevier Inc. All rights reserved.
POSITIVE TOPOLOGICAL ENTROPY FOR MULTIDIMENSIONAL PERTURBATIONS OF TOPOLOGICALLY CROSSING HOMOCLINICITY
(2011-05-01)
Li, Ming-Chia
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Lyu, Ming-Jiea
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應用數學系
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Department of Applied Mathematics
In this paper, we consider a one-parameter family F(lambda) of continuous maps on R(m) or R(m) x R(k) with the singular map F(0) having one of the forms (i) F(0)(x) = f(x); (ii) F(0)(x,y) = (f(x), g(x)), where g : R(m) -> R(k) is continuous, and (iii) F(0)(x; y) = (f(x), g(x,y)), where g : R(m)xR(k) -> R(k) is continuous and locally trapping along the second variable y. We show that if f : R(m) -> R(m) is a C(1) diffeomorphism having a topologically crossing homoclinic point, then F(lambda) has positive topological entropy for all lambda close enough to 0.
TOPOLOGICAL CONJUGACY FOR LIPSCHITZ PERTURBATIONS OF NON-AUTONOMOUS SYSTEMS
(2016-09)
Li, Ming-Chia
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Lyu, Ming-Jiea
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應用數學系
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Department of Applied Mathematics
In this paper, topological conjugacy for two-sided non-hyperbolic and non-autonomous discrete dynamical systems is studied. It is shown that if the system has covering relations with weak Lyapunov condition determined by a transition matrix, there exists a sequence of compact invariant sets restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by the transition matrix. Moreover, if the systems have covering relations with exponential dichotomy and small Lipschitz perturbations, then there is a constructive verification proof of the weak Lyapunov condition, and so topological dynamics of these systems are fully understood by symbolic representations. In addition, the tolerance of Lipschitz perturbation can be characterised by the dichotomy tuple. Here, the weak Lyapunov condition is adapted from [12, 21, 15] and the exponential dichotomy is from [2].
Topological dynamics for multidimensional perturbations of maps with covering relations and Liapunov condition
(2011-01-15)
Li, Ming-Chia
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Lyu, Ming-Jiea
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應用數學系
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Department of Applied Mathematics
In this paper, we study topological dynamics of high-dimensional systems which are perturbed from a continuous map on R(m) x R(k) of the form (1(x), g(x, y)). Assume that f has covering relations determined by a transition matrix A. If g is locally trapping, we show that any small C(0) perturbed system has a compact positively invariant set restricted to which the system is topologically semi-conjugate to the one-sided subshift of finite type induced by A. In addition, if the covering relations satisfy a strong Liapunov condition and g is a contraction, we show that any small C(1) perturbed homeomorphism has a compact invariant set restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by A. Some other results about multidimensional perturbations of f are also obtained. The strong Liapunov condition for covering relations is adapted with modification from the cone condition in Zgliczynski (2009) [11]. Our results extend those in Juang et al. (2008) [1], Li et al. (2008) [2]. Li and Malkin (2006) [3]. Misiurewicz and Zgliczynski (2001) [4] by considering a larger class of maps f and their multidimensional perturbations, and by concluding conjugacy rather than entropy. Our results are applicable to both the logistic and Henon families. (C) 2010 Elsevier Inc. All rights reserved.
Topological entropy for multidimensional perturbations of snap-back repellers and one-dimensional maps
(2008-11-01)
Li, Ming-Chia
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Lyu, Ming-Jiea
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Zgliczynski, Piotr
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應用數學系
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Department of Applied Mathematics
We consider a one-parameter family of maps F(lambda) on R(m) x R(n) with the singular map F(0) having one of the two forms (i) F(0) (x, y) = (f (x), g(x)), where f : R(m) -> R(m) and g : R(m) -> R(n) are continuous, and (ii) F(0)( x, y) = (f (x), g(x, y)), where f : R(m) -> R(m) and g : R(m) x R(n) -> Rn are continuous and g is locally trapping along the second variable y. We show that if f is one-dimensional and has a positive topological entropy, or if f is high-dimensional and has a snap-back repeller, then F. has a positive topological entropy for all lambda close enough to 0.