Browsing by Author "Li, MC"

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Topological horseshoes for Arneodo-Coullet-Tresser maps
(2006)
Du, BS
;
Li, MC
;
Malkin, MI
;
應用數學系
;
Department of Applied Mathematics
In this paper, we study the family of Arneodo-Coullet-Tresser maps F(x, y, z) = (ax - b(y - z), bx + a(y - z), cx - dx(k) + ez) where a, b, c, d, e are real parameters with bd not equal 0 and k > 1 is an integer. We find regions of parameters near anti-integrable limits and near singularities for which there exist hyperbolic invariant sets such that the restriction of F to these sets is conjugate to the full shift on two or three symbols.
Topological horseshoes for perturbations of singular difference equations
(2006-04-01)
Li, MC
;
Malkin, M
;
應用數學系
;
Department of Applied Mathematics
In this paper, we Study solutions of difference equations Phi(lambda)(y(n), y(n+1),...,y(n+m)) = 0, n is an element of Z, of order in with parameter lambda, and consider the case when Phi(lambda) has a singular limit depending on a single variable as lambda -> lambda(0), i.e. Phi(lambda 0)(y(0),...,y(m)) = phi(y(N)), where N is an integer with 0 <= N <= m and phi is a function. We prove that if phi has k simple zeros then for lambda close enough to lambda(0), the difference equation has a k-horseshoe among its solutions, that is, the dynamics is conjugate to the full shift with k symbols. Moreover, we show that these horseshoes change continuously in the uniform topology as lambda varies. As applications of these results, we establish the horseshoe structure in families of generalized Henon-like maps and of Arneodo-Coullet-Tresser maps near their anti-integrable limits as well as in steady states for certain lattice models.