Wedevelop analytical andnumerical conditions to determinewhether limit cycle oscillations synchronize in diffusively coupled systems. We examine two classes of systems: reaction–diffusion PDEs with Neumann boundary conditions, and compartmental ODEs, where compartments are interconnected through diffusion terms with adjacent compartments. In both cases the uncoupled dynamics are governed by a n...

where g1(x) = εg11(x)+ε g12(x)+ε g13(x), g2(x) = εg21(x) + ε g22(x) + ε g23(x) and f(x) = εf1(x) + εf2(x) + ε f3(x) where g1i, g2i, f2i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when g1(x) = 0 we obtain the generalized Liénard polynomial differential systems. We provide an upper bound of the maximum number of limit cycles that the previous d...

We have fabricated NbSe3 structures with widths comparable to the Fukuyama-Lee-Rice phase-coherence length. For samples already in the two-dimensional pinning limit, we observe a crossover from twodimensional to one-dimensional collective pinning when the crystal width is less than 1.6 mm, corresponding to the phase-coherence length in this direction. Our results show that surface pinning is ne...

This paper treats limit cycles caused by friction. The goal has been to explain phenomena that have been observed experimentally in mechatronic systems. Experiments have shown that oscillations of qualitatively different types can be obtained simply by changing controller specifications. Stiction is important in some cases but not in others. Necessary conditions for limit cycle are given for th...

Dispersion curves to a oscillatory reaction-diffusion system with the selfconsistent flow have obtained by means of numerical calculations. The flow modulates the shape of dispersion curves and characteristics of traveling waves. The point of inflection which separates the dispersion curves into two branches corresponding to trigger and phase waves, moves according to the value of the advection...

In this paper we classify the phase portraits in the Poincaré disc of the centers of the generalized class of Kukles systems ẋ = −y, ẏ = x+ axy + bxy, symmetric with respect to the y-axis, and we study, using the averaging theory up to sixth order, the limit cycles which bifurcate from the periodic solutions of these centers when we perturb them inside the class of all polynomial differential s...

We consider the class of polynomial differential equations ẋ = λx + Pn(x, y), ẏ = μy + Qn(x, y) in R where Pn(x, y) and Qn(x, y) are homogeneous polynomials of degree n > 1 and λ 6= μ, i.e. the class of polynomial differential systems with a linear node with different eigenvalues and homogeneous nonlinearities. For this class of polynomial differential equations we study the existence and non–e...

In this paper, we show that perturbing a simple 3-d quadratic system with a center-type singular point can yield at least 10 small-amplitude limit cycles around a singular point. This result improves the 7 limit cycles obtained recently in a simple 3-d quadratic system around a Hopf singular point. Compared with Bautin’s result for quadratic planar vector fields, which can only have 3 small-amp...