اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA flexible error estimate for the application of centre manifold theory
43
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In applications of centre manifold theory we need more flexible error estimates than that provided by, for example, the Approximation Theorem~3 by Carr (1981,1983). Here we extend the theory to cover the case where the order of approximation in parameters and that in dynamical variables may be completely different. This allows, for example, the effective evaluation of low-dimensional dynamical models at finite parameter values.
قيم البحث
اقرأ أيضاً
Given Holder continuous functions $f$ and $psi$ on a sub-shift of finite type $Sigma_A^{+}$ such that $psi$ is not cohomologous to a constant, the classical large deviation principle holds (cite{OP}, cite{Kif}, cite{Y}) with a rate function $I_psigeq
0$ such that $I_psi (p) = 0$ iff $p = int psi , d mu$, where $mu = mu_f$ is the equilibrium state of $f$. In this paper we derive a uniform estimate from below for $I_psi$ for $p$ outside an interval containing $tilde{psi} = int psi , dmu$, which depends only on the sub-shift, the function $f$, the norm $|psi|_infty$, the Holder constant of $psi$ and the integral $tilde{psi}$. Similar results can be derived in the same way e.g. for Axiom A diffeomorphisms on basic sets.
Large scale dynamical systems (e.g. many nonlinear coupled differential equations) can often be summarized in terms of only a few state variables (a few equations), a trait that reduces complexity and facilitates exploration of behavioral aspects of
otherwise intractable models. High model dimensionality and complexity makes symbolic, pen--and--paper model reduction tedious and impractical, a difficulty addressed by recently developed frameworks that computerize reduction. Symbolic work has the benefit, however, of identifying both reduced state variables and parameter combinations that matter most (effective parameters, inputs); whereas current computational reduction schemes leave the parameter reduction aspect mostly unaddressed. As the interest in mapping out and optimizing complex input--output relations keeps growing, it becomes clear that combating the curse of dimensionality also requires efficient schemes for input space exploration and reduction. Here, we explore systematic, data-driven parameter reduction by means of effective parameter identification, starting from current nonlinear manifold-learning techniques enabling state space reduction. Our approach aspires to extend the data-driven determination of effective state variables with the data-driven discovery of effective model parameters, and thus to accelerate the exploration of high-dimensional parameter spaces associated with complex models.
Let $M$ be a triangulable compact manifold. We prove that, among closed subgroups of $homeo_{0}(M)$ (the identity component of the group of homeomorphisms of $M$), the subgroup consisting of volume preserving elements is maximal.
In this paper, we first prove an abstract theorem on the existence of polynomial attractors and the concrete estimate of their attractive velocity for infinite-dimensional dynamical systems, then apply this theorem to a class of wave equations with n
onlocal weak damping and anti-damping in case that the nonlinear term~$f$~is of subcritical growth.
Local bifurcation theory typically deals with the response of a degenerate but isolated equilibrium state or periodic orbit of a dynamical system to perturbations controlled by one or more independent parameters, and characteristically uses tools fro
m singularity theory. There are many situations, however, in which the equilibrium state or periodic orbit is not isolated but belongs to a manifold $S$ of such states, typically as a result of continuous symmetries in the problem. In this case the bifurcation analysis requires a combination of local and global methods, and is most tractable in the case of normal nondegeneracy, that is when the degeneracy is only along $S$ itself and the system is nondegenerate in directions normal to $S$. In this paper we consider the consequences of relaxing normal nondegeneracy, which can generically occur within 1-parameter families of such systems. We pay particular attention to the simplest but important case where $dim S=1$ and where the normal degeneracy occurs with corank 1. Our main focus is on uniform degeneracy along $S$, although we also consider aspects of the branching structure for solutions when the degeneracy varies at different places on $S$. The tools are those of singularity theory adapted to global topology of $S$, which allow us to explain the bifurcation geometry in natural way. In particular, we extend and give a clear geometric setting for earlier analytical results of Hale and Taboas.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
