Rigorous numerics of blow-up solutions for ODEs with exponential nonlinearity

Abstract

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 ut = uxx + 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.

Construction of local Lyapunov functions around non-hyperbolic equilibria by verified numerics for two dimensional cases

“…Here Dq (2) (u) and Dq (3) (u) denote the derivatives of q (2) (u) and q (3) (u) with respect to u, respectively. 2.…”

“…Note that this process verifies that L(x) is indeed a local Lyapunov function within D L . The normal form theory simplifies D (2) (u) and D (3) (u) in (4) as possible by choosing appropriate q (2) (u) and q (3) (u). In our approach, we try to select q (2) (u) and q (3) (u) to control D (2) (u) and D (3) (u) in order that (dL/dt)(φ(t, u))| t=0 < 0 holds for any u ∈ B r \ {0} for a ball B r centered at 0 with some radius r > 0.…”

“…Since the matrix J has pure imaginary eigenvalues in this case, the real symmetric matrix Y cannot be chosen such that E (2) (u) < 0 in B r \ {0}. This means that E (3) (u) should be 0 for any u ∈ R n and E (4) (u) should be negative for any u ∈ B r \ {0}. We derive sufficient conditions for E (3) (u) = 0 and E (4) (u) < 0 in the next section.…”

“…This means that E (3) (u) should be 0 for any u ∈ R n and E (4) (u) should be negative for any u ∈ B r \ {0}. We derive sufficient conditions for E (3) (u) = 0 and E (4) (u) < 0 in the next section.…”

“…One can find a long list in [1]. Among them, the paper [2] by K.Matsue et al treats construction of local Lyapunov functions around hyperbolic equilibria, and there are some applications of their methods [3]. The present paper is one of their successors to give new methods to construct local Lyapunov functions around non-hyperbolic equilibria using verified computation.…”