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51.
Let F:Cn Cn be a holomorphic map, Fk be the kth iterate ofF, and p Cn be a periodic point of F of period k. That is,Fk(p) = p, but for any positive integer j with j < k, Fj(p) p. If p is hyperbolic, namely if DFk(p) has no eigenvalue ofmodulus 1, then it is well known that the dynamical behaviourof F is stable near the periodic orbit = {p, F(p),..., Fk1(p)}.But if is not hyperbolic, the dynamical behaviour of F near may be very complicated and unstable. In this case, a veryinteresting bifurcational phenomenon may occur even though may be the only periodic orbit in some neighbourhood of : forgiven M N\{1}, there may exist a Cr-arc {Ft: t [0,1]} (wherer N or r = ) in the space H(Cn) of holomorphic maps from Cninto Cn, such that F0 = F and, for t (0,1], Ft has an Mk-periodicorbit t with as t 0. Theperiod thus increases by a factor M under a Cr-small perturbation!If such an Ft does exist, then , as well as p, is said to beM-tupling bifurcational. This definition is independent of r. For the above F, there may exist a Cr-arc in H(Cn), with t [0,1], such that and, for t (0,1], has two distinct k-periodic orbits t,1 and t,2 with d(t,i, ) 0 as t 0 for i = 1,2. If such an does exist, then , as well as p, is said to be 1-tupling bifurcational. In recent decades, there have been many papers and remarkableresults which deal with period doubling bifurcations of periodicorbits of parametrized maps. L. Block and D. Hart pointed outthat period M-tupling bifurcations cannot occur for M >2 in the 1-dimensional case. There are examples showing thatfor any M N, period M-tupling bifurcations can occur in higher-dimensionalcases. An M-tupling bifurcational periodic orbit as defined here actsas a critical orbit which leads to period M-tupling bifurcationsin some parametrized maps. The main result of this paper isthe following. Theorem. Let k N and M N, and let F: C2 C2 be a holomorphicmap with k-periodic point p. Then p is M-tupling bifurcationalif and only if DFk(p) has a non-zero periodic point of periodM. 1991 Mathematics Subject Classification: 32H50, 58F14. 相似文献
52.
Several aspects of localized defects in the Frenkel-Kontorova, classicalXY chain and analogous models with a finite range of interactions are discussed from a general point of view. Precise definitions are given for defect phase shifts (charges) and for creation, pinning, and interaction energies. Corresponding definitions are also provided for interfaces (localized regions separating two phases). For the nearest-neighbor Frenkel-Kontorova model, the various defect energies are related to areas enclosed by contours joining heteroclinic points of the area-preserving map generated by the conditions of mechanical equilibrium. 相似文献
53.
The dynamics of bistable oscillators driven by periodic dichotomous noise is described. The stochastic differential equation governing the flow implies smooth trajectories between noise switching events. The dynamics of the two-branched map induced by this flow is a Markov process. Harmonic and quartic models of the bistable potential are studied in the overdamped limit. In the linear (harmonic) case the dynamics can be reduced to a stochastic one-dimensional map with two branches. The moments decay exponentially in this case, although the invariant measure may be multifractal. For strong damping, relaxation induces a cascade leading to a Cantor set and anomalous decay of the density in this case is modeled by a Markov chain. For the physically more realistic case of a quartic potential many additional features arise since the contraction factor is distance dependent. By tuning the barrier-height parameter in the quartic potential, noise-induced transition rates with the characteristics of intermittency are found. 相似文献
54.
CompletelyPositiveDefiniteMapsOverTopological-algebrasXuTianzhou(DepartmentofAppliedMathematics,BeijingInstituteofTechnofogy,... 相似文献
55.
56.
We study generalized equations of the following form: (render) where f is Fréchet differentiable in a neighborhood of a solution x* of (*) and g is Fréchet differentiable at x* and where F is a set-valued map acting in Banach spaces. We prove the existence of a sequence (xk) satisfying which is super-linearly convergent to a solution of (*). We also present other versions of this iterative procedure that have superlinear and quadratic convergence, respectively. 相似文献
0f(x)+g(x)+F(x),
57.
58.
Let be a nonempty closed convex subset of a real Banach space and be a Lipschitz pseudocontractive self-map of with . An iterative sequence is constructed for which as . If, in addition, is assumed to be bounded, this conclusion still holds without the requirement that Moreover, if, in addition, has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, then the sequence converges strongly to a fixed point of . Our iteration method is of independent interest.
59.
Zaqueu Coelho 《Transactions of the American Mathematical Society》2004,356(11):4427-4445
For a minimal circle homeomorphism we study convergence in law of rescaled hitting time point process of an interval of length 0$">. Although the point process in the natural time scale never converges in law, we study all possible limits under a subsequence. The new feature is the fact that, for rotation numbers of unbounded type, there is a sequence going to zero exhibiting coexistence of two non-trivial asymptotic limit point processes depending on the choice of time scales used when rescaling the point process. The phenomenon of loss of tightness of the first hitting time distribution is an indication of this coexistence behaviour. Moreover, tightness occurs if and only if the rotation number is of bounded type. Therefore tightness of time distributions is an intrinsic property of badly approximable irrational rotation numbers.
60.
Gamaliel?BléEmail author Víctor?Castellanos Manuel?J.?Falconi 《Bulletin of the Brazilian Mathematical Society》2003,34(2):333-345
In this paper we consider a system whose state
x changes to (x) if a perturbation occurs at the time
t, for
. Moreover, the state
x changes to the new state
(x) at time
t, for
. It is assumed that the
number of perturbations in an interval (0,
t) is a Poisson process. Here
and are measurable maps from a measure space
into itself. We give
conditions for the existence of a stationary distribution of the
system when the maps and commute, and we prove that any
stationary distribution is an invariant measure of these
maps. 相似文献