Dynamics Games and Science II : DYNA 2008 in Honor of Maurício Peixoto and David Rand University of Minho Braga Portugal September 8 12 2008 Dynamics Games and Science I and II are a selection of surveys and research articles written by leading researchers in mathematics. The majority of the contributions are on dynamical systems and game theory focusing either on fundamental and theoretical developments or on applications to modeling in biology ecomonics engineering finances and psychology. The papers are based on talks given at the International Conference DYNA 2008 held in honor of Mauricio Peixoto and David Rand at the University of Braga Portugal on September 8 12 2008. The aim of these volumes is to present cutting edge research in these areas to encourage graduate students and researchers in mathematics and other fields to develop them further. 1 Network Control Analysis for Time Dependent Dynamical States 2 Renormalization and Focal Decomposition 3 Micro Foundation of the Social Change 4 Singularities Walrasian Economics and Economic Crisis 5 Probabilistic Methods in Dynamical Analysis: Population Growths Associated to Models Beta (p q) with Allee Effect 6 Power Indices Applied to Portuguese Parliament 7 A Methodological Contribution in the Theory of the Firm under Uncertainty 8 Explosion of Smoothness for Conjugacies Between Unimodal Maps 9 Multidimensional Rovella Like Attractors 10 Robust Heteroclinic Behaviour Synchronization and Ratcheting of Coupled Oscillators 11 An Economical Model for Dumping 12 Fractional Control of Dynamic Systems 13 A Dynamical Point of View of Quantum Information: Discrete Wigner Measures 14 Dynamics on Spectral Solutions of Forced Burgers Equation 15 Area Preserving Diffeomorphisms from the C1 Standpoint 16 A Theoretical Multidisciplinary View of Catastrophic Regime Change 17 General Relativistic Elasticity – Statics and Dynamics of Spherically Symmetric Metrics 18 Post Inflationary Scalar Field Phase Dynamics 19 An Application of the SIR Model to the Evolution of Epidemics in Portugal 20 Poissonian Tree Constructed from Independent Poisson Point Processes 21 Hamiltonian Systems on Polyhedra 22 Patents Moving to Bankrupcy Boundaries 23 Computation of Genus and Braid Index for Renormalizable Lorenz Links 24 Statistical Stability for Equilibrium States 25 Dynamic Games of Network Effects 26 Exit Times and Persistence of Solitons for a Stochastic Korteweg de Vries Equation 27 Optimization Approach to a Simulation Algorithm of a Mixer Settler System in the Transient State 28 Duality Theory Representation Formulas and Uniqueness Results for Viscosity Solutions of Hamilton Jacobi Equations 29 Microscopic Dynamics for the Porous Medium Equation 30 A Model for Wolf’s Sunspot Numbers 31 On Topological Classification of Morse Smale Diffeomorphisms 32 Isentropic Dynamics and Control in an Economic Model for Capital Accumulation 33 Hydrodynamic Limit of the Exclusion Process in Inhomogeneous Media 34 Application of Fractional Order Concepts in the Study of Electrical Potential 35 Economics of Bioenergy Crops for Electricity Generation: Implications for Land Use and Greenhouse Gases 36 An H Theorem for Chemically reacting Gases 37 Dynamics Systems Dynamical Systems and Interaction Graphs 38 The Dynamics of Scalar Fields in Cosmology 39 Resort Prices and Bankruptcy 40 On the Hannay Ozorio de Almeida Sum Formula 41 A Fourier Transform Method for Relaxation of Kinetic Equations 42 Bethe Ansatz Solution of the Finite Bernoulli Matching Model of Sequence Alignment 43 Fitness Function Evaluation Through Fractional Algorithms 44 An Exponential Observer for Systems on SE(3) with Implicit Outputs 45 Stability of Polydeuces Orbit 46 Peixoto Classification of 2 Dim Flows Revisited 47 Fractional Control of Legged Robots 48 Evolutionary Dynamics of the Spatial Prisoner’s Dilemma with Single and Multi Behaviors – a Multi Agent Application 49 Euclidean Jordan Algebras and Strongly Regular Graphs 50 Parameter Estimation in Stochastic Differential Equations 51 Stochastic Saddle Paths and Economic Theory.