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We provide a numerical validation method of blow-up solutions for finite dimensional vector fields admitting asymptotic quasi-homogeneity at infinity. Our methodology is based on quasi-homogeneous compactifications containing quasi-parabolic-type and directional-type compactifications. Divergent solutions including blow-up solutions then correspond to global trajectories of associated vector fields with appropriate time-variable transformation tending to equilibria on invariant manifolds representing infinity. We combine standard methodology of rigorous numerical integration of differential equations with Lyapunov function validations around equilibria corresponding to divergent directions, which yields rigorous upper and lower bounds of blow-up time as well as rigorous profile enclosures of blow-up solutions.

We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix., Comment: 24 pages. arXiv admin note: text overlap with arXiv:1505.00683

This paper aims at applications of Lyapunov functions as tools for analyzing concrete dynamical systems with computer assistance, even for non-gradient-like systems. We want to know concrete form of Lyapunov functions around invariant sets and their domains of definition for applying Lyapunov functions to various analysis of both continuous and discrete dynamical systems. Although there are several abstract results for the existence of Lyapunov functions, they cannot induce a systematic and concrete procedure of Lyapunov functions with explicit forms. In this paper, we present a numerical verification method which can validate Lyapunov functions with explicit forms and their explicit domains of definition, which can be applied to arbitrary dynamical systems with (hyperbolic) equilibria or fixed points. The proposed procedure provides us with a powerful validation tool for analyzing asymptotic behavior of dynamical systems.

This note states the correction of arguments in the proof of Theorem 3.2 in the original paper.

Our concerns here are blow-up solutions for ODEs with exponential nonlinearity from the viewpoint of dynamical systems and their numerical validations. As an example, the finite difference discretization of u t = u x x + e u m with the homogeneous Dirichlet boundary condition is considered. Our idea is based on compactification of phase spaces and time-scale desingularization as in previous works. In the present case, treatment of exponential nonlinearity is the main issue. Fortunately, under a kind of exponential homogeneity of vector field, we can treat the problem in the same way as polynomial vector fields. In particular, we can characterize and validate blow-up solutions with their blow-up times for differential equations with such exponential nonlinearity in the similar way to previous works. A series of technical treatments of exponential nonlinearity in blow-up problems is also shown with concrete validation examples.

This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method., Comment: Accepted version, to appear in Journal of Computational and Applied Mathematics