Boutroux asymptotic forms of solutions to Painlevé equations and power geometry

Bruno, A D; Goryuchkina, Irina

doi:10.1134/s1064562408050104

Cited by 21 publications

(36 citation statements)

References 3 publications

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“…The period T H (a) is one of the periods of the Weierstrass function ℘ associated to the cubic H (y 1 , y 2 ) = a (see e.g. [5], [1]). To compute it we can chose for instance…”

Section: Example: the Case Of The First Painlevé Equationmentioning

confidence: 99%

Doubly-Resonant Saddle-Nodes in $$\mathbb {C}^{3}$$ C 3 and the Fixed Singularity at Infinity in the Painlevé Equations: Formal Classification

Bittmann

2016

Qual. Theory Dyn. Syst.

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In this work we consider formal singular vector fields in C 3 with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlevé equations (Pj ), j ∈ {I, II, III, IV, V }, for generic values of the parameters. Under generic assumptions we give a complete formal classification for the action of formal diffeomorphisms (by changes of coordinates) fixing the origin and fibered in the independent variable x. We also identify all formal isotropies (self-conjugacies) of the normal forms. In the particular case where the flow preserves a transverse symplectic structure, e.g. for Painlevé equations, we prove that the normalizing map can be chosen to preserve the transverse symplectic form.

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“…The period T H (a) is one of the periods of the Weierstrass function ℘ associated to the cubic H (y 1 , y 2 ) = a (see e.g. [5], [1]). To compute it we can chose for instance…”

Section: Example: the Case Of The First Painlevé Equationmentioning

confidence: 99%

Doubly-Resonant Saddle-Nodes in $$\mathbb {C}^{3}$$ C 3 and the Fixed Singularity at Infinity in the Painlevé Equations: Formal Classification

Bittmann

2016

Qual. Theory Dyn. Syst.

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“…are 2π-periodic functions with respect to ϕ. Since (0, 0) is the equilibrium of system (1), we see that E = 0 is the fixed point of the first equation in (9): f (0, ϕ, t) ≡ 0 for all ϕ ∈ R and t > 0. Moreover, from (4) and ( 5) it follows that…”

mentioning

confidence: 85%

“…Note that such systems and perturbations arise in the study of various problems of mathematical physics. For example, phase synchronization models [33,24], autoresonance models [25,17], the Painlevé equations [15,9] and many other nonlinear non-autonomous systems [8,27] are reduced to systems of the form (1) with right-hand sides having power-law asymptotics with rational exponents. Note also that the powers k/q with q > 1 in the perturbations can be reduced to the integer exponents k by the transformation: τ = t 1/q .…”

mentioning

confidence: 99%

Bifurcations in asymptotically autonomous Hamiltonian systems under oscillatory perturbations

Sultanov¹

2021

DCDS

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The effect of decaying oscillatory perturbations on autonomous Hamiltonian systems in the plane with a stable equilibrium is investigated. It is assumed that perturbations preserve the equilibrium and satisfy a resonance condition. The behaviour of the perturbed trajectories in the vicinity of the equilibrium is investigated. Depending on the structure of the perturbations, various asymptotic regimes at infinity in time are possible. In particular, a phase locking and a phase drifting can occur in the systems. The paper investigates the bifurcations associated with a change of Lyapunov stability of the equilibrium in both regimes. The proposed stability analysis is based on a combination of the averaging method and the construction of Lyapunov functions.

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“…Note that the series in ( 6) are assumed to be asymptotic as t → ∞ uniformly for all (x, y) ∈ B r (see, for example, [29, §1]). Such decaying perturbations appear, for example, in the study of Painlevé equations [30,31], resonance and phase-locking phenomena [32,33] and in many other problems associated with nonlinear and non-autonomous systems [34][35][36]. It can easily be checked that the rational powers of the form k/q with q > 1 in ( 6) can be reduced to the integer exponents k by the change of the time variable θ = t 1/q in system (1).…”

Section: Problem Statementmentioning

confidence: 99%

Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

Sultanov¹

2021

Preprint

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The effect of multiplicative stochastic perturbations on Hamiltonian systems on the plane is investigated. It is assumed that perturbations fade with time and preserve a stable equilibrium of the limiting system. The paper investigates bifurcations associated with changes in the stability of the equilibrium and with the appearance of new stochastically stable states in the perturbed system. It is shown that depending on the structure and the parameters of the decaying perturbations the equilibrium can remain stable or become unstable. In some intermediate cases, a practical stability of the equilibrium with estimates for the length of the stability interval is justified. The proposed stability analysis is based on a combination of the averaging method and the construction of stochastic Lyapunov functions.

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Boutroux asymptotic forms of solutions to Painlevé equations and power geometry

Cited by 21 publications

References 3 publications

Doubly-Resonant Saddle-Nodes in $$\mathbb {C}^{3}$$ C 3 and the Fixed Singularity at Infinity in the Painlevé Equations: Formal Classification

Doubly-Resonant Saddle-Nodes in $$\mathbb {C}^{3}$$ C 3 and the Fixed Singularity at Infinity in the Painlevé Equations: Formal Classification

Bifurcations in asymptotically autonomous Hamiltonian systems under oscillatory perturbations

Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

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