# ȊwZ~i[ 2007

3 17 @Jean-Philippe Lessard (Rutgers University)
Braids, Chaos and Rigorous Numerics

In this talk, we prove that the stationary Swift-Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. The first step of the method relies on a computer assisted, rigorous, continuation method to prove the existence of a periodic orbit with certain geometric properties. The second step is topological: we use this periodic solution as a skeleton, through which we braid other solutions, thus forcing the existence of infinitely many braided periodic orbits. A semi-conjugacy to a subshift of finite type shows that the dynamics is chaotic.

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10 24 @с@aiswwȁj
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6 20 @HR@aiLwj
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5 23 @Pawel Pilarczykiswj
Automated Analysis of Dynamical Systems with the Use of Graph Algorithms

In this talk, I will introduce and explain some algorithmic methods for the automated analysis of dynamical systems. The discretization of the dynamics makes it possible to use fast graph algorithms to extract the recurrent and gradient-like dynamics, and to represent the system in terms of a Conley-Morse decomposition. Recently developed algorithms, including the automatic homology computation, provide additional information and make it possible to obtain mathematically meaningful, rigorous results. Moreover, if an n- parameter family of dynamical systems is considered, then using outer approximations of dynamics provides an algorithmic method to prove certain continuation results, as well as to detect possible bifurcations. A nonlinear Leslie population model will be used as a sample discrete dynamical system which illustrates the effectiveness of this approach.

4 11 @Pedro LimaiCentro de Matem\'atica e Aplica\c{c}\~oes, Instituto Superior T\'ecnicco, Lisboa, Portugalj
Numerical and Asymptotic Analysis of Nonlinear Boundary Value Problems with Bubble-type Solutions

In this talk we are concerned about singular boundary value problems arising in hydrodynamics and cosmology. In the case of spherical simmetry, the orginal partial differential equation may be reduced to a second order ordinary differential equation (ODE). This is the case, for example, of the formation of bubbles or droplets in a mixture gas-liquid. We are interested on solutions of the resulting ODE which are strictly increasing on the positive semi-axis and have finite limits at $0$ and $\infty$ (bubble-type solutions). Necessary and sufficient conditions for the existence of such solutions are obtained in the form of a restriction on the equation coefficients. The asymptotic behavior of certain solutions of this equation is analysed near the two singularities (when $r\rightarrow 0+$ and $r\rightarrow \infty$), where the considered boundary conditions define one-parameter families of solutions. Based on the analytic study, an efficient numerical method is proposed to compute approximately the needed solutions of the above problem. Some results of the numerical experiments are displayed and their physical interpretation is discussed.

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6 20 @HR@aiLwj
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5 23 @Pawel Pilarczykiswj
Automated Analysis of Dynamical Systems with the Use of Graph Algorithms

In this talk, I will introduce and explain some algorithmic methods for the automated analysis of dynamical systems. The discretization of the dynamics makes it possible to use fast graph algorithms to extract the recurrent and gradient-like dynamics, and to represent the system in terms of a Conley-Morse decomposition. Recently developed algorithms, including the automatic homology computation, provide additional information and make it possible to obtain mathematically meaningful, rigorous results. Moreover, if an n- parameter family of dynamical systems is considered, then using outer approximations of dynamics provides an algorithmic method to prove certain continuation results, as well as to detect possible bifurcations. A nonlinear Leslie population model will be used as a sample discrete dynamical system which illustrates the effectiveness of this approach.

4 11 @Pedro LimaiCentro de Matem\'atica e Aplica\c{c}\~oes, Instituto Superior T\'ecnicco, Lisboa, Portugalj
Numerical and Asymptotic Analysis of Nonlinear Boundary Value Problems with Bubble-type Solutions

In this talk we are concerned about singular boundary value problems arising in hydrodynamics and cosmology. In the case of spherical simmetry, the orginal partial differential equation may be reduced to a second order ordinary differential equation (ODE). This is the case, for example, of the formation of bubbles or droplets in a mixture gas-liquid. We are interested on solutions of the resulting ODE which are strictly increasing on the positive semi-axis and have finite limits at $0$ and $\infty$ (bubble-type solutions). Necessary and sufficient conditions for the existence of such solutions are obtained in the form of a restriction on the equation coefficients. The asymptotic behavior of certain solutions of this equation is analysed near the two singularities (when $r\rightarrow 0+$ and $r\rightarrow \infty$), where the considered boundary conditions define one-parameter families of solutions. Based on the analytic study, an efficient numerical method is proposed to compute approximately the needed solutions of the above problem. Some results of the numerical experiments are displayed and their physical interpretation is discussed.

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