Transition of patterns in the cell-chemotaxis system with proliferation source

Yari, Masoud

doi:10.1016/j.na.2015.01.017

Cited by 5 publications

(4 citation statements)

References 16 publications

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“…We first deal with the generic PES conditions (18) in Theorem 3 and (20) in Theorem 4 below. In the non-generic cases, the number of possible transition scenarios is much higher, and it is much more difficult to write a general transition theorem in that case.…”

Section: Pes Conditions For the Fourth-order Case With Periodicity Co...mentioning

confidence: 99%

“…The application of this methodology has given rise to a lot of results regarding the transitions of reaction-diffusion equations. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] Our main goal is to generalize these results which can cover most of the results obtained in these works. Namely, we want to obtain the relation between the linear and nonlinear terms and the effect of the boundaries which influence the transition phenomena in the most basic dissipative system possible, the 1D reaction-diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic transitions and bifurcations of 1D reaction–diffusion equations: The self‐adjoint case

Şengül

Tiryakioglu

2021

Math Methods in App Sciences

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This paper deals with the classification of transition phenomena in the most basic dissipative system possible, namely, the 1D reaction-diffusion equation.The emphasis is on the relation between the linear and nonlinear terms and the effect of the boundaries which influence the first transitions. We consider the cases where the linear part is self-adjoint with second-order and fourth-order derivatives which is the case which most often arises in applications. We assume that the nonlinear term depends on the unknown function and its first derivative which is basically the semilinear case for the second-order reaction-diffusion system. As for the boundary conditions, we consider the typical Dirichlet, Neumann, and periodic boundary settings. In all the cases, the equations admit a trivial steady state which loses stability at a critical parameter. We aim to classify all possible transitions and bifurcations that take place. Our analysis shows that these systems display all three types of transitions: continuous, jump and mixed.Moreover they exhibit transcritical, supercritical bifurcations with bifurcated states such as finitely many equilibria, circle of equilibria, and slowly rotating limit cycle. Many applications found in the literature are basically corollaries of our main results. We apply our results to classify the first transitions of the Chaffee-Infante equation, the Fisher-KPP equation, the Kuramoto-Sivashinsky equation, and the Swift-Hohenberg equation.

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Section: Pes Conditions For the Fourth-order Case With Periodicity Co...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamic transitions and bifurcations of 1D reaction–diffusion equations: The self‐adjoint case

Şengül

Tiryakioglu

2021

Math Methods in App Sciences

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“…The set of transition states which is often represented by a local attractor may lie near or away from the basic state. This theory has recently been successfully applied to many branches of nonlinear sciences, see [19,13,4,18] among others.…”

Section: Introductionmentioning

confidence: 99%

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

Özer

Şengül

2017

J. Math. Fluid Mech.

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Abstract. The main aim of the paper is to investigate the transitions of the thermohaline circulation in a spherical shell in a parameter regime which only allows transitions to multiple equilibria. We find that the first transition is either continuous (Type-I) or drastic (Type-II) depending on the sign of the transition number. The transition number depends on the system parameters and lc, which is the common degree of spherical harmonics of the first critical eigenmodes, and it can be written as a sum of terms describing the nonlinear interactions of various modes with the critical modes. We obtain the exact formulas of this transition number for lc = 1 and lc = 2 cases. Numerically, we find that the main contribution to the transition number is due to nonlinear interactions with modes having zero wave number and the contribution from the nonlinear interactions with higher frequency modes is negligible. In our numerical experiments we encountered both types of transition for Le < 1 but only continuous transition for Le > 1. In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2lc dimensional sphere and consists entirely of degenerate steady state solutions.

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Section: Introductionmentioning

confidence: 99%

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

Özer,

Şengül

2017

Preprint

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The main aim of the paper is to investigate the transitions of the thermohaline circulation in a spherical shell in a parameter regime which only allows transitions to multiple equilibria. We find that the first transition is either continuous (Type-I) or drastic (Type-II) depending on the sign of the transition number. The transition number depends on the system parameters and lc, which is the common degree of spherical harmonics of the first critical eigenmodes, and it can be written as a sum of terms describing the nonlinear interactions of various modes with the critical modes. We obtain the exact formulas of this transition number for lc = 1 and lc = 2 cases. Numerically, we find that the main contribution to the transition number is due to nonlinear interactions with modes having zero wave number and the contribution from the nonlinear interactions with higher frequency modes is negligible. In our numerical experiments we encountered both types of transition for Le < 1 but only continuous transition for Le > 1. In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2lc dimensional sphere and consists entirely of degenerate steady state solutions.

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Transition of patterns in the cell-chemotaxis system with proliferation source

Cited by 5 publications

References 16 publications

Dynamic transitions and bifurcations of 1D reaction–diffusion equations: The self‐adjoint case

Dynamic transitions and bifurcations of 1D reaction–diffusion equations: The self‐adjoint case

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

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