It is proved in this paper that the maximum number of limit cycles of system { dx dt = y, dy dt = kx− (k + 1)x2 + x3 + (α + βx + γx2)y is equal to two in the finite plane, where k > 11+ √ 33 4 , 0 < | | 1, |α| + |β| + |γ| = 0. This is partial answer to the seventh question in [2], posed by Arnold.

The bifurcation of limit cycles by perturbing a planar system which has a continuous family of cycles, i.e. periodic orbits, has been an intensively studied phenomenon; see for instance [13, 16, 2] and references therein. The simplest planar system having a continuous family of cycles is the linear center, and a special family of its perturbations is given by the generalized polynomial Liénard ...

We study the stratum in the set of all quadratic differential systems ẋ = P2(x, y), ẏ = Q2(x, y) with a center, known as the codimension-four case Q4. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincaré-Pontryagin-Melnikov integral I. We show that the orbits of the unpertur...

Contraction theory provides an elegant way of analyzing the behaviors of systems subject to external inputs. Under sometimes easy to check hypotheses, systems can be shown to have the incremental stability property that all trajectories converge to a unique solution. This property is especially interesting when forcing functions are periodic (a globally attracting limit cycle results), as well ...

We consider the class of polynomial differential equations ẋ = Pn(x, y)+Pn+1(x, y) +Pn+2(x, y), ẏ = Qn(x, y)+Qn+1(x, y)+Qn+2(x, y), for n ≥ 1 and where Pi and Qi are homogeneous polynomials of degree i. These systems have a linearly zero singular point at the origin if n ≥ 2. Inside this class we identify a new subclass of Darboux integrable systems, and some of them having a degenerate center,...

A means was developed for extending the predictive capability of the Equivalent SingleTube Model (ESTM) to accurately predict the onset of a self-sustained oscillatory flow instability for a multitube condensing flow system. The model includes the effects of compressibility, subcooled liquid inertia, and thermal and flow distribution asymmetry. Previously, liquid inertia, a necessary mechanism ...

This paper deals with a synthesis of a nonautonomous system with a stable limit cycle. By extending Green’s method, by which arbitrary periodic solutions can be designed in autonomous systems, it is shown that a nonautonomous system whose transient trajectories converge to a prescribed limit cycle can be synthesized. Furthermore, we apply receding horizon control to determine the optimal time-d...

We study the length of the limit cycles of discrete monotone functions with symmetric connection graph. We construct a family of monotone functions such that the limit cycles are of maximum possible length, which is exponential in the number of variables. Furthermore, we prove for the class of monotone functions with more than two states and connection graph equal to a caterpillar that the leng...

We demonstrate the oscillatory decay of the survival probability of the stochastic dynamics dxε = a(xε) dt+ √ 2ε b(xε) dw, which is activated by small noise over the boundary of the domain of attraction D of a stable focus of the drift a(x). The boundary ∂D of the domain is an unstable limit cycle of a(x). The oscillations are explained by a singular perturbation expansion of the spectrum of th...