Dynamics, Games and Science II: DYNA 2008, in Honor of by D. A. Rand (auth.), Mauricio Matos Peixoto, Alberto Adrego

By D. A. Rand (auth.), Mauricio Matos Peixoto, Alberto Adrego Pinto, David A. Rand (eds.)

Dynamics, video games and technology I and II are a range of surveys and study articles written by way of top researchers in arithmetic. the vast majority of the contributions are on dynamical platforms and video game conception, focusing both on primary and theoretical advancements or on functions to modeling in biology, ecomonics, engineering, funds and psychology.

The papers are in keeping with talks given on the foreign convention DYNA 2008, held in honor of Mauricio Peixoto and David Rand on the college of Braga, Portugal, on September 8-12, 2008.

The goal of those volumes is to give state of the art examine in those components to motivate graduate scholars and researchers in arithmetic and different fields to boost them further.

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Extra info for Dynamics, Games and Science II: DYNA 2008, in Honor of Maurício Peixoto and David Rand, University of Minho, Braga, Portugal, September 8-12, 2008

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Pt Manuel F. Silva Department of Electrotechnical Engineering, Institute of Engineering of Porto, Rua Dr. pt Marcelo Trindade dos Santos LNCC/MCT, Av. R. pt Michelle Wander Department of Natural Resources and Environmental Sciences, University of Illinois at Urbana-Champaign, 1301 W. N. A. Rand Abstract We present an approach to network control analysis that applies to some important time-dependent dynamical states for both autonomous and nonautonomous dynamical systems. In particular, the theory applies to periodic solutions of autonomous and periodically forced differential equations.

De Carvalho et al. there exist ˛ > 0 small enough and N 1 large enough such that, for every n N , the n-renormalized velocities yn W Œ0; 1 S1 Œ0; ˛n ! e; ÂI t/ D . 1/n 1 n;t 1 ! 1  qP n;t ! e; ÂI n `t ! 5) and ! 6). e; ÂI t/ are also dimensionless. 2. e; ÂI t/ D e sin `t 1 RC 0 ! q /. 1. 2. q /. e; ÂI t/ in the C 2 topology as n tends to infinity. e; ÂI t/. As stated in the following theorem, the pair formed by the asymptotic trajectories and the asymptotic velocities is the flow of a canonical Hamiltonian system.

3) are periodic. Thus, the trajectories q W Œ0;  S1 R ! E; I / for all 2 R, E 2 Œ0;  and 2 S1 . Furthermore, there exist ˛ > 0 small enough and N 1 large enough such that, for every n N , the n-renormalized trajectories xn W Œ0; 1 S1 Œ0; ˛n ! e; ÂI t/ D . 1/ n 1 n;t 1 Ä Â q n;t ! e; ÂI n `t ! q / and ! 6) Note that ! 3). n; t/-scaling parameter n;t . e; ÂI t/ are dimensionless. 1. e; ÂI t/ D e cos `t 1 e 2 C  ; RC 0 ! q /. The result below generalizes the main result of Carvalho et al. [8] in two ways.

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