Numerical validation of blow-up solutions with quasi-homogeneous compactifications

Matsue, Kaname; Takayasu, Atsushi

doi:10.1007/s00211-020-01125-z

Cited by 5 publications

(27 citation statements)

References 17 publications

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“…We also show that the present methodology enables us to provide an exact and explicit formula of the maximal existence time, equivalently the blow-up time, of solutions as a smooth or real-analytic function of initial points near blow-up, provided all our implementations work successfully. The present characterization of the maximal existence time provides us with a quantitative feature of blow-up solutions such as distributions of blow-up times depending on initial data which are not provided in preceding works [48,49,57]. The applicability of the present methodology is shown in successive sections.…”

Section: Organization Of the Papermentioning

confidence: 82%

“…The system possesses both stable and unstable equilibria at infinity. The parabolic-type compactification is applied and a saddle-type blow-up solution is firstly validated, while validations of blow-up solutions asymptotic to stable equilibria at infinity are already demonstrated in a preceding work [48]. The main aim of this section is to study a global nature of solution families in a neighborhood of saddle-type blow-up solutions.…”

Section: Organization Of the Papermentioning

confidence: 99%

“…A significant consequence of the preceding studies is that, a solution of (1.1) with bounded initial point is a blow-up solution, namely t max < ∞, if the image of the solution through a compactification mentioned above is on the stable manifold of a hyperbolic equilibrium on E for the desingularized vector field 2 . Simultaneously, the second and the third authors have developed a computer-assisted methodology for proving the existence of blow-up solutions for concretely given dynamical systems with rigorous bound of their blow-up times [48,49,57]. The basic idea is the combination of compactifications as well as time-scale desingularizations mentioned above with rigorous integrator of ODEs based on interval (and affine) arithmetic and topological characterizations of asymptotic behavior such as locally defined Lyapunov functions.…”

Section: Introductionmentioning

confidence: 99%

“…Keeping the success of computer-assisted proofs for various applications to dynamical systems (e.g. [13,16,48,49,57]) in mind, we believe that studying blow-up solutions with computer-assisted proofs provides rich insights into asymptotic behavior of solutions to differential equations as well as new research directions of global dynamics and finite-time singularities.…”

Section: Introductionmentioning

confidence: 99%

“…see [10,11,12]) to validating local stable manifolds of equilibria, which is the main difference from the preceding works. Once local stable manifolds of equilibria through the parameterization are validated, extension of these manifolds through flow trajectories can be also validated through the standard ODE integrations, as in preceding works [48,49,57]. There are two main contributions of applications of the parameterization method for validating blow-up solutions which have not been obtained in previous studies.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Lessard¹,

Matsue²,

Takayasu³

2021

Preprint

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In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial conditions are studied. Combining dynamical systems machinery (e.g. phase space compactifications, time-scale desingularizations of vector fields) with tools from computer-assisted proofs (e.g. rigorous integrators, parameterization method for invariant manifolds), these unstable blow-up solutions are obtained as trajectories on stable manifolds of hyperbolic (saddle) equilibria at infinity. In this process, important features are obtained: smooth dependence of blow-up times on initial points near blow-up, level set distribution of blow-up times, singular behavior of blow-up times on unstable blow-up solutions, organization of the phase space via separatrices (stable manifolds). In particular, we show that unstable blow-up solutions themselves, and solutions defined globally in time connected by those blow-up solutions can separate initial conditions into two regions where solution trajectories are either globally bounded or blow-up, no matter how the large initial points are.

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Section: Organization Of the Papermentioning

confidence: 82%

Section: Organization Of the Papermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Lessard¹,

Matsue²,

Takayasu³

2021

Preprint

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Rigorous numerics for nonlinear heat equations in the complex plane of time

et al. 2022

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In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explicit bounds for the inclusion of the solution of the Cauchy problem, which is rewritten as a zero-finding problem on a certain Banach space. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov–Perron method for calculating part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

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

2023

Self Cite

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In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery (e.g., compactifications, timescale desingularizations of vector fields) with tools from computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity. With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions, provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously. Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times and decomposition of the phase space playing a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior. Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed.

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Numerical validation of blow-up solutions with quasi-homogeneous compactifications

Cited by 5 publications

References 17 publications

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

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

Rigorous numerics for nonlinear heat equations in the complex plane of time

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

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