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Adaptive networks are emerging a lot of fields of science, recently. In this paper, we consider about a mathematical model for adaptive network made by the plasmodium. The organism contains a tube network by means of which nutrients and signals circulate through the body. The tube network changes its shape to connect two exits through the shortest path when the organism is put in a maze and food is placed at two exits. Recently, a mathematical model for this adaptation process of the plasmodium has been proposed. Here we analyze it mathematically rigorously. In ring-shaped network and Wheatstone bridge-shaped network, we mainly show that the globally asymptotically stable equilibrium point of the model corresponds to the shortest path connecting two special points on the network in the case where the shortest path is determined uniquely. From the viewpoint of mathematical technique, especially in the case of Wheatstone bridge-shaped network, we show that there is a simple but novel device used here by which we prove the global asymptotic stability, even when Lyapunov function cannot be constructed.

We show the existence of global solution and the global attractor in L 2 (T) for the third order Lugiato-Lefever equation on T. Without damping and forcing terms, it has three conserved quantities, that is, the L 2 (T) norm, the momentum and the energy, but the leading term of the energy functional is not positive definite. So only the L 2 norm conservation is useful for the third order Lugiato-Lefever equation unlike the KdV and the cubic NLS equations. Therefore, it seems important and natural to construct the global attractor in L 2 (T). For the proof of the global attractor, we use the smoothing effect of cubic nonlinearity for the reduced equation.

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We study the stability of a stationary solution for the Lugiato-Lefever equation with the periodic boundary condition in one space dimension, which is a damped and driven nonlinear Schrödinger equation introduced to model the optical cavity. In this paper, we prove the Strichartz estimates for the linear damped Schrödinger equation with potential and external forcing and investigate the stability of certain stationary solutions under the initial perturbation within the framework of L 2 .

We study the usefulness of two most prominent publicly available rigorous ODE integrators: one provided by the CAPD group (capd.ii.uj.edu.pl) the other based on the COSY Infinity project (cosyinfinity.org). Both integrators are capable of handling entire sets of initial conditions and provide tight rigorous outer enclosures of the images under a time-T map. We conduct extensive benchmark computations using the well-known Lorenz system, and compare the computation time against the final accuracy achieved. We also discuss the effect of a few technical parameters, such as the order of the numerical integration method, the value of T , and the phase space resolution. We conclude that COSY may provide more precise results due to its ability of avoiding the variable dependency problem. However, the overall cost of computations conducted using CAPD is typically lower, especially when intervals of parameters are involved. Moreover,

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