V. I. Arnol'd. Geometric Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, New York, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....approximations. Section 6 presents as simple exception to this rule the class of pseudo linear hybrid automata, for which linear phase portrait approximations can be generated automatically. Related work Phase portraits have been studied extensively in the literature on dynamical systems [HS74, Arn83] Typically, researchers concentrate on the continuous dynamics of a system and analyze extremely nontrivial properties, such as stability and convergence. Our work differs in two respects. First, we consider products of nondeterministic dynamical systems with discrete transition structures. ....

V. I. Arnol'd. Geometric Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, New York, 1983.

....persistence of coherent structures under perturbations of a dynamical system are fundamental issues in dynamical systems theory with implications in many fields of application. In the context of discrete, finite dimensional Hamiltonian systems, this issue is addressed by the celebrated KAM theorem [3], which guarantees the persistence of most invariant tori of the unperturbed dynamics under small Hamiltonian perturbations. For infinite dimensional systems, defined by Hamiltonian partial differential equations (PDEs) KAM type methods have recently been used to obtain results on the persistence ....

....(4.48) in a manner which makes explicit which terms determine the large time behavior of the amplitude and phase of A(t) 5. Dispersive Hamiltonian normal form To analyze the asymptotic behavior of A(t) or equivalently a(t) and j(t; x) as t 1 it is useful to use the idea of normal forms [3], 26] 52] from dynamical systems theory. We derive a perturbed normal form which makes the anticipated large time behavior of solutions transparent. Proposition 5.1. There exists a smooth near identity change of variables, A 7 A with the following properties: A = A h(A; t) h(A; t) ....

V.I. Arnol'd, Geometric Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York (1983)

....can replace standard analytic techniques for nonlinear behaviors. Indeed, our examples show that some understanding of differential inequalities is necessary for obtaining good approximations. Related work Phase portraits have been studied extensively in the literature on dynamical systems [HS74, Arn83] Typically, researchers concentrate on the complex dynamics of a system, and are able to prove extremely nontrivial properties, such as stability and convergence. Our work differs in two respects. First, we consider products of nondeterministic dynamical systems with discrete transition ....

V. I. Arnol'd. Geometric Methods in the Theory of Ordinary Differential Equations. Springer, 1983.

....flow is periodic or quasiperiodic on the two torus, depending on the rationality or irrationality of the rotation number x = y . The analysis of perturbations of such a flow is classical in dynamical systems theory. One may study the problem using a Poincar e return map that is a circle map [14]. The conclusion of the theory is that under the influence of small perturbations, the linear flow on the two torus tends to develop an attracting periodic orbit. This phenomenon is called mode locking. The associated phase diagram features the famous Arnol d tongues [14] In terms of the ....

....map that is a circle map [14] The conclusion of the theory is that under the influence of small perturbations, the linear flow on the two torus tends to develop an attracting periodic orbit. This phenomenon is called mode locking. The associated phase diagram features the famous Arnol d tongues [14]. In terms of the traveling waves in the planar system, mode locking of the torus flow means that the traveling wave tends to become a discrete traveling wave that moves on average in the direction of a lattice vector. We say that such a traveling wave is locked with the lattice. Notice that a ....

V.I. Arnol'd, Geometric methods in the theory of ordinary differential equations, GMW 250 (Springer, Berlin 1982).

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