Secondary Hopf bifurcation caused by steady-state steady-state mode interaction

1995 ◽  
pp. 209-224
Author(s):  
Karin Gatermann ◽  
Bodo Werner
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Mode Interaction ◽  
Steady State Mode

Stability and Bifurcation in a Predator–Prey System with Prey-Taxis

2020 ◽  
Vol 30 (02) ◽  
pp. 2050022 ◽  
Author(s):  
Huanhuan Qiu ◽  
Shangjiang Guo ◽  
Shangzhi Li
Keyword(s):  
Steady State ◽  
Periodic Solutions ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Steady States ◽  
Mode Interaction ◽  
Small Range ◽  
Steady State Mode ◽  
Predator Prey ◽  
Prey System

In this paper, we consider a generalized predator–prey system with prey-taxis under Neumann boundary condition, that is, the predators can survive even in the absence of the prey species. It is proved that for an arbitrary spatial dimension, the corresponding initial boundary value problem possesses a unique global bounded classical solution when the prey-taxis is restricted to a small range. Moreover, the local stabilities of constant steady states (including trivial, semi-trivial and positive constant steady states) are investigated. A further study on the coexistence steady state implies that the prey-taxis term suppresses the global asymptotical stability and influences the steady-state/Hopf bifurcations (if they exist). Analyses of steady-state bifurcation, Hopf bifurcation, and even Hopf/steady-state mode interaction are carried out in detail by means of the Lyapunov–Schmidt procedure. In particular, we obtain stable or unstable steady states, time-periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found in numerical simulations.

THE 1:2 MODE INTERACTION IN RAYLEIGH–BÉNARD CONVECTION WITH WEAKLY BROKEN MIDPLANE SYMMETRY

2001 ◽  
Vol 11 (01) ◽  
pp. 27-41 ◽  
Author(s):  
ISABEL MERCADER ◽  
JOANA PRAT ◽  
EDGAR KNOBLOCH
Keyword(s):  
Steady State ◽  
Symmetry Breaking ◽  
Mode Interaction ◽  
Reflection Symmetry ◽  
Steady State Mode ◽  
Bénard Convection ◽  
Benard Convection ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The effects of weak breaking of the midplane reflection symmetry on the 1:2 steady state mode interaction in Rayleigh–Bénard convection are discussed in a PDE setting. Effects of this type arise from the inclusion of non-Boussinesq terms or due to small differences in the boundary conditions at the top and bottom of the convecting layer. The latter provides the simplest realization, and captures all qualitative effects of such symmetry breaking. The analysis is performed for two Prandtl numbers, σ=10 and σ=0.1, representing behavior typical of large and low Prandtl numbers, respectively.

A robust heteroclinic cycle in an steady-state mode interaction

Nonlinearity ◽  
1998 ◽  
Vol 11 (1) ◽  
pp. 89-104 ◽  
Author(s):  
P Hirschberg ◽  
E Knobloch
Keyword(s):  
Steady State ◽  
Mode Interaction ◽  
Heteroclinic Cycle ◽  
Steady State Mode
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