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Bifurcations and chaos in convection taking non-Fourier heat-flux. (English) Zbl 1375.34064

Summary: In this letter, we report the influences of thermal time-lag on the onset of convection, its bifurcations and chaos of a horizontal layer of Boussinesq fluid heated underneath taking non-Fourier Cattaneo-Christov hyperbolic model for heat propagation. A five-dimensional nonlinear system is obtained for a low-order Galerkin expansion, and it reduces to Lorenz system for Cattaneo number tending to zero. The linear stability agreed with existing results that depend on Cattaneo number \(C\). It also gives a threshold Cattaneo number, \(C_T\), above which only oscillatory solutions can persist. The oscillatory solutions branch terminates at the subcritical steady branch with a heteroclinic loop connecting a pair of saddle points for subcritical steady-state solutions. For subcritical onset of convection two stable solutions coexist, that is, hysteresis phenomenon occurs at this stage. The steady solution undergoes a Hopf bifurcation and is of subcritical type for small value of \(C\), while it becomes supercritical for moderate Cattaneo number. The system goes through period-doubling/noisy period-doubling transition to chaos depending on the control parameters. There after the system exhibits Shil’nikov chaos via homoclinic explosion. The complexity of spiral strange attractor is analyzed using fractal dimension and return map.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34F10 Bifurcation of solutions to ordinary differential equations involving randomness
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
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