Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equations

Cao, Dat; Hoang, Luan

doi:10.1007/s00028-020-00622-w

Cited by 13 publications

(36 citation statements)

References 19 publications

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“…Proof of Theorem 6. The proof is similar to that of Theorem 3 using the Lyapunov function (34) and the auxiliary function (36) with N = n + l.…”

Section: Stability Analysis In Case (14)mentioning

confidence: 86%

“…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%

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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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“…Proof of Theorem 6. The proof is similar to that of Theorem 3 using the Lyapunov function (34) and the auxiliary function (36) with N = n + l.…”

Section: Stability Analysis In Case (14)mentioning

confidence: 86%

Section: Problem Statementmentioning

confidence: 99%

Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

Sultanov¹

2021

Preprint

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“…This paper broadens the theory of the asymptotic expansions, as time tends to infinity, for solutions of systems of nonlinear ordinary differential equations (ODEs) that was studied in previous work [10]. The paper [10] itself was motivated by Foias-Saut's work [21] for the Navier-Stokes equations (NSE). In Foias and Saut's paper [21], the NSE are written in the functional form as an ODE u t + Au + B(u, u) = 0, (…”

Section: Introductionmentioning

confidence: 91%

“…The constructions of q k 's are not explicit and depend crucially on the analytic flows associated with the vector field F (y). On contrary, this paper and [10] do not require the analyticity and, hence, cannot use the constructions in [28]. The work [29], when viewed in the ODE context, requires A to be complex diagonalizable, which is more stringent than (b) above.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic expansions about infinity for solutions of nonlinear differential equations with coherently decaying forcing functions

Hoang¹

2021

Preprint

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This paper studies, in fine details, the long-time asymptotic behavior of decaying solutions of a general class of dissipative systems of nonlinear differential equations in complex Euclidean spaces. The forcing functions decay, as time tends to infinity, in a coherent way expressed by combinations of the exponential, power, logarithmic and iterated logarithmic functions. The decay may contain sinusoidal oscillations not only in time but also in the logarithm and iterated logarithm of time. It is proved that the decaying solutions admit corresponding asymptotic expansions, which can be constructed concretely. In the case of the real Euclidean spaces, the real-valued decaying solutions are proved to admit real-valued asymptotic expansions. Our results unite and extend the theory investigated in many previous works.

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Asymptotic Expansions for the Lagrangian Trajectories from Solutions of the Navier–Stokes Equations

Hoang

2020

Commun. Math. Phys.

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We study the three-dimensional Navier-Stokes equations in a periodic domain with the force decaying in time. Although the force has a certain coherent decay, as time tends to infinity, it can be too complicated for the previous theory of asymptotic expansions to be applicable. To deal with this issue, we systematically develop a new theory of asymptotic expansions containing the so-called subordinate variables which can be defined recursively. We apply it to obtain an asymptotic expansion for any Leray-Hopf weak solutions. The expansion, in fact, is constructed explicitly and the impact of the subordinate variables can be clearly specified. The complexifications of the Gevrey-Sobolev spaces, and of the Stokes and bilinear operators of the Navier-Stokes equations are utilized to facilitate such a construction.

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Asymptotic expansions with exponential, power, and logarithmic functions for non-autonomous nonlinear differential equations

Cited by 13 publications

References 19 publications

Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

Bifurcations in asymptotically autonomous Hamiltonian systems under multiplicative noise

Asymptotic expansions about infinity for solutions of nonlinear differential equations with coherently decaying forcing functions

Asymptotic Expansions for the Lagrangian Trajectories from Solutions of the Navier–Stokes Equations

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