Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs

Abstract

Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate time-scale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates.1. Apply compactifications of phase spaces and derivation of desingularized vector fields to (1.3).The infinity then corresponds to a hypersurface E called horizon.2. Specify an invariant set S on E for desingularized vector fields.3. Solve explicit solutions which converge to S. Transform the calculated solutions to those for the original system (1.3).Our main result is the following : if S is non-hyperbolic for desingularized vector field in the sense stated above and we solve trajectories on W c (S), then they correspond to divergent or blowup solutions with blow-up rates which are generally different from type-I. We can also say that the similar feature is revealed to other finite-time singularities such as finite-time extinction, compacton traveling waves or quenching.1 Precise meanings are shown in successive sections. 2The rest of the paper is organized as follows. In Section 2, we review the approach and result shown in [14,16] about characterization of blow-up solutions in terms of trajectories on stable manifolds of invariant sets at infinity for appropriately associated vector fields. These results gives characterization of type-I blow-ups. In Section 3, we discuss a methodology how the asymptotic behavior of blow-up solutions with different blow-up rates from type-I can be derived. The main issue there is a treatment of non-hyperbolic invariant sets at infinity. We thus review the type of equilibria, which is often discussed in singularities of (planar) vector fields (e.g., [7]). Several equivalences among dynamical systems are reviewed, which help us with treating asymptotic behavior of trajectories simply. Independently, we discuss the treatment for detecting other types of finite-time singularities such as finite-time extinctions and compactons. As indicated in preceding studies in partial differential equations (e.g., [12]), there are several common aspects among finite-time singularities including blow-ups, extinctions, compactons and quenching. Our present treatments reveals such common mechanisms of singularities from the viewpoint of dynamical systems. In Section 4, we provide various examples with calculating rigorous rates of finite-time singularities including blow-ups, extinctions, compactons, quenching and periodic blowups. We see that the asymptotic rates can be derived component-wise whether or not it is faster than that associated with nonlinearity of vector fields; namely type-I rates. Compactifications and type-I blow-upsFirst we collect our present sett...

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Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

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