Browsing by Author "Malkin, Mikhail"

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Chaotic difference equations in two variables and their multidimensional perturbations
(2008-05-01)
Juang, Jonq
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Li, Ming-Chia
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Malkin, Mikhail
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應用數學系
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Department of Applied Mathematics
We consider difference equations Phi(lambda)(y(n), y(n+1), . . . , y(n+m)) = 0, n is an element of Z, of order m with parameter. close to that exceptional value lambda(0) for which the function Phi depends on two variables: Phi(lambda 0)(x(0), . . . , x(m)) = xi (x(N), x(N+L)) with 0 <= N, N + L <= m. It is also assumed that for the equation xi(x, y) = 0, there is a branch y = phi(x) with positive topological entropy h(top)(phi). Under these assumptions we prove that in the set of bi-infinite solutions of the difference equation with. in some neighbourhood of lambda(0), there is a closed ( in the product topology) invariant set to which the restriction of the shift map has topological entropy arbitrarily close to htop(phi)/vertical bar L vertical bar, and moreover, orbits of this invariant set depend continuously on. not only in the product topology but also in the uniform topology. We then apply this result to establish chaotic behaviour for Arneodo-Coullet-Tresser maps near degenerate ones, for quadratic volume preserving automorphisms of R(3) and for several lattice models including the generalized cellular neural networks (CNNs), the time discrete version of the CNNs and coupled Chua's circuit.
Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders
(2018-01-01)
Gonchenko, Sergey
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Li, Ming-Chia
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Malkin, Mikhail
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應用數學系
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Department of Applied Mathematics
Consider (m + 1)-dimensional, m 1, diffeomorphisms that have a saddle fixed point O with m-dimensional stable manifold W-s(O), one-dimensional unstable manifold W-u(O), and with the saddle value sigma different from 1. We assume that W-s(O) and W-u(O) are tangent at the points of some homoclinic orbit and we let the order of tangency be arbitrary. In the case when sigma < 1, we prove necessary and sufficient conditions of existence of topological horseshoes near homoclinic tangencies. In the case when sigma > 1, we also obtain the criterion of existence of horseshoes under the additional assumption that the homoclinic tangency is simple.
GENERALIZED HENON MAPS AND SMALE HORSESHOES OF NEW TYPES
(2008-10-01)
Gonchenko, Sergey
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Li, Ming-Chia
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Malkin, Mikhail
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應用數學系
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Department of Applied Mathematics
We study hyperbolic dynamics and bifurcations for generalized Henon maps in the form (x) over bar = y, (y) over bar = gamma y(1 - y) - bx + alpha xy (with b, alpha small and gamma > 4). Hyperbolic horseshoes with alternating orientation, called half-orientable horseshoes, are proved to represent the nonwandering set of the maps in certain parameter regions. We show that there are infinitely many classes of such horseshoes with respect to the local topological conjugacy. We also study transitions from the usual orientable and nonorientable horseshoes to half-orientable ones (and vice versa) as parameters vary.