On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents

Kieu, Chanh; Şengül, Taylan; Wang, Quan; Yan, Dongming

doi:10.1016/j.cnsns.2018.05.010

Cited by 18 publications

(12 citation statements)

References 20 publications

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“…Following this purpose, the dynamic transition theory shows that phase transitions are classified into three types: continuous, catastrophic, and random. Due to the classification of the phase transition is closely bound up with practical problems, the theory has been successfully applied in the study of a number of transition problems, including transitions of quasi‐geostrophic channel flows, instability and transitions of Rayleigh‐B

é

nard convection, dynamic transitions of Cahn‐Hilliard equation, and boundary layer separation, to name a few …”

Section: Introductionmentioning

confidence: 99%

“…Due to the classification of the phase transition is closely bound up with practical problems, the theory has been successfully applied in the study of a number of transition problems, including transitions of quasi-geostrophic channel flows, 9 instability and transitions of Rayleigh-Bénard convection, [12][13][14] dynamic transitions of Cahn-Hilliard equation, 15,16 and boundary layer separation, 17 to name a few. [18][19][20][21][22] Our present study relies on two methods. The first one is the continued-fraction method first arisen in Meshalkin and Sinai, 23 developed in Chen et al, 8 with which we try to verify the PES condition that is a necessary and sufficient condition for phase transition.…”

Section: Introductionmentioning

confidence: 99%

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On the stability and transition for the Navier‐Stokes‐alpha model

Wang

Yan

2019

Math Methods in App Sciences

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The main aim of this paper is to investigate the stability and transition of the Navier‐Stokes‐alpha model. By using the continued‐fraction method, combining with the dynamic transition theory, we show the existence of a Hopf bifurcation in this model as Reynolds number crosses a critical value. Upon deriving the explicit expression of a non‐dimensional number P called transition number, which is a function of the critical Reynolds number and the aspect ratio, we further analyze the transition associated with the Hopf bifurcation. More precisely, it is shown that the modeled flow exhibits either a continuous or catastrophic transition at the critical Reynolds number, whose specific type of the transition is determined by the sign of the real part of P at the critical Reynolds number, and the spatio‐temporal structure of the limit cycle bifurcated that corresponds to a wave that propagates slowly westward and is symmetric about the mid‐axis of the channel.

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é

nard convection, dynamic transitions of Cahn‐Hilliard equation, and boundary layer separation, to name a few …”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the stability and transition for the Navier‐Stokes‐alpha model

Wang

Yan

2019

Math Methods in App Sciences

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“…From Figure 5, the sufficient conditions (3.10)-(3.11) are satisfied for each d ∈ [6,13] and n ∈ [0.1, 1.5]. In addition, we know that the critical parameter value λ c is less than the critical parameter value Λ c for all d ∈ [11,13] and n ∈ [0.1, 1]. Namely, the assertion (1) of Theorem 3.1 is valid.…”

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confidence: 97%

Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food

Xing¹,

Pan²,

Luo³

2021

Communications on Pure &Amp; Applied Analysis

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The article aims to investigate the dynamic transitions of a toxinproducing phytoplankton zooplankton model with additional food in a 2Drectangular domain. The investigation is based on the dynamic transition theory for dissipative dynamical systems. Firstly, we verify the principle of exchange of stability by analysing the corresponding linear eigenvalue problem. Secondly, by using the technique of center manifold reduction, we determine the types of transitions. Our results imply that the model may bifurcate two new steady state solutions, which are either attractors or saddle points. In addition, the model may also bifurcate a new periodic solution as the control parameter passes critical value. Finally, some numerical results are given to illustrate our conclusions.

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“…Introduction. Quasi-geostrophic (QG) flow, a type of fluid motions that is very close to but not exactly in the geostrophic balance, plays an important role in the large-scale atmospheric and oceanic circulations [1,2,10,11,13,17,20,35]. Introduced by Charney in 1948 as an approach to filter high-frequency waves in the atmosphere, the QG formulation has led to the very first successful implementation of the numerical weather prediction [16].…”

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confidence: 99%

Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows

Pan¹,

Kieu²,

Wang³

2021

CPAA

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This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.

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On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents

Cited by 18 publications

References 20 publications

On the stability and transition for the Navier‐Stokes‐alpha model

On the stability and transition for the Navier‐Stokes‐alpha model

Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food

Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows

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