Interactions of \((m,n)\) and \((m+1,n)\) modes with real eigenvalues: a dynamic transition approach. (English) Zbl 1527.35053
Summary: In this work, we consider the multiplicity two dynamic transitions of a broad class of problems. The first main assumption is the existence of two critical eigenmodes of the linear operator which depend on at least two wave indices one of which are consecutive \(m,m+1\) and the other identical \(n\). The second main assumption is an orthogonality condition on the nonlinear interactions of the basis vectors of the phase space which is typical in many applications. Under this assumption we obtain a reduced system of ODE’s which describe the first dynamic transitions. We make a careful analysis of this reduced system to classify all possible transition behavior. We then apply our main theoretical findings to the 2D Rayleigh-Bénard convection with free-slip boundary conditions and show that this problem displays an \(S^1\) attractor bifurcation.
MSC:
| 35B36 | Pattern formations in context of PDEs |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
| 47J35 | Nonlinear evolution equations |
References:
| [1] | Temam, Roger, Infinite-dimensional dynamical systems in mechanics and physics. Applied mathematical sciences (1997), Springer: Springer New York · Zbl 0871.35001 |
| [2] | Chekroun, Mickaël D.; Liu, Honghu; McWilliams, James C.; Wang, Shouhong, Transitions in stochastic non-equilibrium systems: Efficient reduction and analysis. J Differential Equations, 145-204 (2023) · Zbl 1518.60055 |
| [3] | Ma, Tian; Wang, Shouhong, Phase transition dynamics (2019), Springer International Publishing: Springer International Publishing Cham · Zbl 1451.82003 |
| [4] | Ma, Tian; Wang, Shouhong |
| [5] | Şengül, Taylan, Dynamical transition theory of hexagonal pattern formations. Commun Nonlinear Sci Numer Simul (2020) · Zbl 1456.37103 |
| [6] | Şengül, Taylan; Tiryakioglu, Burhan, Dynamic transitions and bifurcations of 1D reaction-diffusion equations: The self-adjoint case. Math Methods Appl Sci, 5, 2871-2892 (2022) · Zbl 1529.35034 |
| [7] | Şengül, Taylan; Tiryakioglu, Burhan, Dynamic transitions and bifurcations of 1D reaction-diffusion equations: The non-self-adjoint case. J Math Anal Appl, 1 (2023) · Zbl 1511.35023 |
| [8] | Jia, Lan; Li, Liang, Stability and dynamic transition of vegetation model for flat arid terrains. Discrete Continuous Dyn Syst - B (2021) |
| [9] | Choi, Yuncherl; Ha, Taeyoung; Han, Jongmin; Kim, Sewoong; Lee, Doo Seok, Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discr Contin Dyn Syst, 9, 4255 (2021) · Zbl 1465.35043 |
| [10] | Muntari, Umar Faruk; Şengül, Taylan, Dynamic transitions and Turing patterns of the Brusselator model. Math Methods Appl Sci, 16, 9130-9151 (2022) · Zbl 1536.70026 |
| [11] | Wang, Quan; Yan, Dongming, On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete Continuous Dyn Syst - B, 7, 2607 (2020) · Zbl 1441.35063 |
| [12] | Liu, Honghu; Sengul, Taylan; Wang, Shouhong; Zhang, Pingwen, Dynamic transitions and pattern formations for a Cahn-Hilliard model with long-range repulsive interactions. Commun Math Sci, 5, 1289-1315 (2015) · Zbl 1332.35148 |
| [13] | Liu, Honghu; Sengul, Taylan; Wang, Shouhong, Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with Onsager mobility. J Math Phys, 2 (2012) · Zbl 1274.82039 |
| [14] | Ma, Tian; Wang, Shouhong, Dynamic bifurcation and stability in the Rayleigh-benard convection. Commun Math Sci, 2, 159-183 (2004) · Zbl 1133.76315 |
| [15] | Li, Junyan; Wu, Ruili, Dynamic transition analysis for activator-substrate system. J Nonlinear Math Phys (2023) · Zbl 1523.37094 |
| [16] | Han, Daozhi; Hernandez, Marco; Wang, Quan, Dynamical transitions of a low-dimensional model for Rayleigh-Bénard convection under a vertical magnetic field. Chaos Solitons Fractals, 370-380 (2018) · Zbl 1415.34073 |
| [17] | Li, Liang; Fan, Yanlong; Han, Daozhi; Wang, Quan, Dynamical transition and bifurcation of hydromagnetic convection in a rotating fluid layer. Commun Nonlinear Sci Numer Simul (2022) · Zbl 1506.35167 |
| [18] | Pan, Zhigang; Jia, Lan; Mao, Yiqiu; Wang, Quan, Transitions and bifurcations in couple stress fluid saturated porous media using a thermal non-equilibrium model. Appl Math Comput (2022) · Zbl 1510.35242 |
| [19] | Ma, Tian; Wang, Shouhong, Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Commun Pure Appl Anal, 4, 591 (2003) · Zbl 1210.37056 |
| [20] | Sengul, Taylan; Wang, Shouhong, Pattern formation in Rayleigh-Bénard convection. Commun Math Sci, 1, 315-343 (2013) · Zbl 1291.35235 |
| [21] | Sengul, Taylan; Shen, Jie; Wang, Shouhong, Pattern formations of 2D Rayleigh-Bénard convection with no-slip boundary conditions for the velocity at the critical length scales. Math Methods Appl Sci, 17, 3792-3806 (2015) · Zbl 1333.76036 |
| [22] | Han, Daozhi; Hernandez, Marco; Wang, Quan, Dynamic bifurcation and transition in the Rayleigh-Bénard convection with internal heating and varying gravity. Commun Math Sci, 1, 175-192 (2019) · Zbl 1415.37097 |
| [23] | Sengul, Taylan; Wang, Shouhong, Pattern formation and dynamic transition for magnetohydrodynamic convection. Commun Pure Appl Anal, 6, 2609-2639 (2014) · Zbl 1305.76130 |
| [24] | Chandrasekhar, Subrahmanyan |
| [25] | Pan, Zhigang; Wang, Quan; Sengul, Taylan, On the viscous instabilities and transitions of two-layer model with a layered topography. Commun Nonlinear Sci Numer Simul (2020) · Zbl 1444.76055 |
| [26] | Pan, Zhigang; Kieu, Chanh; Wang, Quan, Hopf bifurcations and transitions of two-dimensional quasi-geostrophic flows. Commun Pure Appl Anal, 4, 1385-1412 (2021) · Zbl 1470.37109 |
| [27] | Mao, Yiqiu; Chen, Zhimin; Kieu, Chanh; Wang, Quan, On the stability and bifurcation of the non-rotating Boussinesq equation with the Kolmogorov forcing at a low Péclet number. Commun Nonlinear Sci Numer Simul (2020) · Zbl 1444.76058 |
| [28] | Chekroun, Mickaël D.; Dijkstra, Henk; Şengül, Taylan; Wang, Shouhong, Transitions of zonal flows in a two-layer quasi-geostrophic ocean model. Nonlinear Dynam, 3, 1887-1904 (2022) · Zbl 1519.86001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
