Narrow Results

We study a nematic crystal model that appeared in [Liu et al., 2007], modeling stretching effects depending on the different shapes of the microscopic molecules of the material, under periodic boundary conditions. The aim of the present article is two-fold: to extend the results given in [Sun & Liu, 2009], to a model with more complete stretching terms and to obtain some stability and asymptotic stability properties for this model.

In this paper, we consider a class of flux controlled memristive circuits with a piecewise linear memristor (i.e. the characteristic curve of the memristor is given by a piecewise linear function). The mathematical model is described by a discontinuous planar piecewise smooth differential system, which is defined on three zones separated by two parallel straight lines $\left|x\right|=1$ (called as discontinuity lines in discontinuous differential systems). We first investigate the stability of equilibrium points and the existence and uniqueness of a crossing limit cycle for the memristor-based circuit under self-excited oscillation. We then analyze the existence of periodic orbits of forced nonlinear oscillation for the memristive circuit with an external exciting source. Finally, we give numerical simulations to show good matches between our theoretical and simulation results.

In this paper, we consider a simple equation which involves a parameter $k$, and its traveling wave system has a singular line.

Firstly, using the qualitative theory of differential equations and the bifurcation method for dynamical systems, we show the existence and bifurcations of peak-solitary waves and valley-solitary waves. Specially, we discover the following novel properties:

• (i)In the traveling wave system, there exist infinitely many periodic orbits intersecting at a point, or two points and passing through the singular line, and there is no singular point inside a homoclinic orbit.
• (ii)When $k<\frac{1}{2}$, in the equation there exist three types of bifurcations of valley-solitary waves including periodic wave, blow-up wave and double solitary wave.
• (iii)When $k\ge \frac{1}{2}$, in the equation there exist two types of bifurcations of valley-solitary wave including periodic wave and blow-up wave, but there is no double solitary wave bifurcation.

Secondly, we perform numerical simulations to visualize the above properties.

Finally, when $k<\frac{1}{8}$ and the constant wave speed equals $\frac{1}{2}(1\pm \sqrt{1-8k})$, we give exact expressions to the above phenomena.

For linear reaction–diffusion equations, a general geometric singular perturbation framework was developed, to study the impact of strong, spatially localized, smooth nonlinear impurities on the existence, stability, and bifurcation of localized structure, in the paper [Doelman et al., 2018]. The multiscale nature enables deriving algebraic conditions determining the existence of pinned single- and multi-pulses. Moreover, linearity enables treating the spectral stability issue for pinned pulses similarly to the problem of existence. In this paper, we move one step further to treat a special type of nonlinear reaction–diffusion equation with the same type of impurity. The additional nonlinear term generates richer and more complex dynamics. We derive algebraic conditions for determining the existence and stability of pinned pulses in terms of Legendre functions.