Lower bounds for the number of limit cycles in a generalized Rayleigh-Liénard oscillator

Micropolar meets Newtonian in three dimensions. The Rayleigh--B{é}nard problem for large Prandtl numbers

We consider the Rayleigh--Bénard problem for the three--dimensional Boussinesq system for the micropolar fluid. We introduce the notion of the multivalued eventual semiflow and prove the existence of the two-space global attractor $\mathcal{A}^K$ corresponding to weak solutions, for every micropolar parameter $K\geq 0$ denoting the deviation of the considered system from the classical Rayleigh--Bénard problem for the Newtonian fluid. We prove that for every $K$ the attractor $\mathcal{A}^K$ is the smallest compact, attracting, and invariant set. Moreover, the semiflow restricted to this attrac…

含有趋光性微生物的非散射悬浮液中的热-生物对流

Thermal-bioconvection in a non-scattering suspension of phototactic microorganisms

本文通过线性稳定性分析和数值求解特征值问题，研究了从下方加热的趋光性微生物悬浮液中的热-生物对流，发现临界总光强和 Lewis 数会促进失稳，而高温下 Rayleigh-Bénard 对流将主导生物对流。

Systematic designing of bi-rhythmic and tri-rhythmic models in families of Van der Pol and Rayleigh oscillators

Van der Pol and Rayleigh oscillators are two traditional paradigms of nonlinear dynamics. They can be subsumed into a general form of Liénard--Levinson--Smith(LLS) system. Based on a recipe for finding out maximum number of limit cycles possible for a class of LLS oscillator, we propose here a scheme for systematic designing of generalised Rayleigh and Van der Pol families of oscillators with a desired number of multiple limit cycles. Numerical simulations are explicitly carried out for systematic search of the parameter space for bi-rhythmic and tri-rhythmic systems and their higher order var…

The limit cycles in a generalized Rayleigh-Liénard oscillator

We compute the cyclicity of open period annuli of the following generalized Rayleigh-Liénard equation $$\ddot{x}+ax+bx^3-(λ_1+λ_2 x^2+λ_3\dot{x}^2+λ_4 x^4+λ_5\dot{x}^4+λ_6 x^6)\dot{x}=0$$ and the equivalent planar system $X_λ$, where the coefficients of the perturbation $λ_j$ are independent small parameters and $a, b$ are fixed nonzero constants. Our main tool is the machinery of the so called higher-order Poincaré-Pontryagin-Melnikov functions (Melnikov functions $M_n$ for short), combined with the explicit computation of center conditions and the corresponding Bautin ideal. We consider firs…

Bifurcations and Amplitude Death from Distributed Delays in Coupled Landau-Stuart Oscillators and a Chaotic Parametrically Forced van der Pol-Rayleigh System

Distributed delays modeled by 'weak generic kernels' are introduced in the well-known coupled Landau-Stuart system, as well as a chaotic van der Pol-Rayleigh system with parametric forcing. The systems are close via the 'linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation which initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter $a=a_{Hopf}$ at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all …

光趋性生物对流在热浮力作用下的弱非线性分析

Weakly Nonlinear Analysis of Phototactic Bioconvection under Thermal Buoyancy

本研究通过弱非线性稳定性分析，探讨热浮力对光趋性生物对流的影响，发现热瑞利数接近临界值时对流由光趋性主导转为热驱动，并预测了超临界分岔行为。