Narrow Results

This paper deals with two classes of polynomial differential systems that are generalized rigid systems, i.e. differential systems whose orbits rotate with a constant angular velocity. We give bounds for the maximum number of limit cycles of such polynomial differential systems provided by the averaging theory of first-order.

We provide an algorithm for studying invariant tori fulfilled by periodic orbits of a perturbed system which emerge from the set of periodic orbits of an unperturbed linear system in p : q resonance. We illustrate the algorithm with an application.

We study the bifurcation of limit cycles from the periodic orbits of a two-dimensional (resp. four-dimensional) linear center in ℝⁿ perturbed inside a class of discontinuous piecewise linear differential systems. Our main result shows that at most 1 (resp. 3) limit cycle can bifurcate up to first-order expansion of the displacement function with respect to the small parameter. This upper bound is reached. For proving these results, we use the averaging theory in a form where the differentiability of the system is not needed.

We study the number of limit cycles of the polynomial differential systems of the form

In this paper, we analyze the periodic structure of the anisotropic Kepler problem with the same type of perturbations using symplectic Delaunay coordinates and averaging theory. Moreover, sufficient conditions for existence and kind of stability of this class of perturbed Kepler problems are given.

In this paper, we investigate the number of limit cycles for a class of discontinuous planar differential systems with multiple sectors separated by many rays originating from the origin. In each sector, it is a smooth generalized Liénard polynomial differential system x′ = -y + g₁(x) + f₁(x)y and y′ = x + g₂(x) + f₂(x)y, where f_i(x) and g_i(x) for i = 1, 2 are polynomials of variable x with any given degree. By the averaging theory of first-order for discontinuous differential systems, we provide the criteria on the maximum number of medium amplitude limit cycles for the discontinuous generalized Liénard polynomial differential systems. The upper bound for the number of medium amplitude limit cycles can be attained by specific examples.

Recently some interest has appeared for the periodic FitzHugh–Nagumo differential systems. Here, we provide sufficient conditions for the existence of periodic solutions in such differential systems.

In this paper, we prove that at every energy level the anisotropic Kepler problem with small anisotropy has two periodic orbits which bifurcate from elliptic orbits of the Kepler problem with high eccentricity. Moreover we provide approximate analytic expressions for these periodic orbits. The tool for proving this result is the averaging theory.

We study the Hopf and the fold–Hopf bifurcations of the Rössler-type differential system

Lorenz studied the coupled Rosby waves and gravity waves using the differential system

We provide new results in studying a kind of stability of periodic orbits provided by the higher-order averaging theory. Then, we apply these results to determining the $k$-hyperbolicity of some periodic orbits of the Lorenz and Thomas differential systems.

In this paper, we study the periodic solutions bifurcating from a nonisolated zero–Hopf equilibrium in a polynomial differential system of degree two in ${ℝ}^3$. More specifically, we use recent results of averaging theory to improve the conditions for the existence of one or two periodic solutions bifurcating from such a zero–Hopf equilibrium. This new result is applied for studying the periodic solutions of differential systems in ${ℝ}^3$ having $n$-scroll chaotic attractors.

Recently there are several works studying the finance model

In this paper, we investigate the number of limit cycles for two classes of discontinuous Liénard polynomial perturbed differential systems. By the second-order averaging theorem of discontinuous differential equations, we provide several criteria on the lower upper bounds for the maximum number of limit cycles. The results show that the second-order averaging theorem of discontinuous differential equations can predict more limit cycles than the first-order one.

Consider the class of reversible quadratic systems

We study the zero-Hopf bifurcation of the Rössler differential system

For two families of planar piecewise smooth polynomial differential systems, whose unperturbed system has a degenerate center at the origin, we study the biggest lower bound for the maximum number of limit cycles bifurcating from the periodic orbits of the center. These results are extensions of the known ones on unperturbed nondegenerate $\mathrm{\Sigma }$-center, derived from a nonsmooth harmonic oscillator model, to degenerate $\mathrm{\Sigma }$-center. Our study involves some new computational treatments. The main tools are the generalized polar coordinate change and the generalized Lyapunov polar coordinate change together with an averaging theory for one-dimensional piecewise smooth differential equations. Finally, we present two Maple programs for computing the averaging functions and consequently the biggest lower bound on the maximum number of limit cycles of degenerate $(2k,2l)$-center under general polynomial perturbations of degree $n$.

The effect of retardation along second vector of angular velocity on a new attitude system of quad-rotor unmanned aerial vehicle (QUAV) is examined in this paper. Catastrophic and hovering conditions of rotor dynamics are verified through bifurcation analysis. It is analyzed that disturbed rotor dynamics during yaw maneuvering lead towards the existence of oscillations and hamper the efficiency of attitude system. We have categorically signified the situation where remote controller of attitude system cannot prevent flipping and consequently, shortens the life of QUAV. This type of situation arises in a strong wind friction zone where positive drag force of rotor dynamics is impeded by the magnitude of rotational moment of inertia. Moreover, subcritical and zero-Hopf bifurcations aroused due to retardational effect in attitude system are analyzed with the aid of normal form and averaging theory. Numerical simulations are also provided for validation of analytical results.

This paper deals with the limit cycle bifurcation from a reversible differential center of degree $2n+2$ due to small piecewise smooth homogeneous polynomial perturbations. By using the averaging theory for discontinuous systems and the complex method based on the Argument Principle, we obtain lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus around the center of the unperturbed system.

In this paper, the bounded invariant surfaces of a generalized Langford system are discussed. Firstly, by the first integrals of systems restricted in the Poincaré sections of a periodic orbit, the accurate expressions of a heteroclinic orbit, a family of invariant tori and a heteroclinic invariant ellipsoid are given near a periodic orbit. Then, applying the successor functions to compute the periods of periodic orbits for the systems in the Poincaré sections, we present the parameter conditions for the existence of periodic orbits with any periods on these invariant tori. Finally, using the averaging theory and the theory of the Poincaré bifurcation and by determining the monotonicity of the ratio of two Abelian integrals, we give the conditions respectively such that the system has a unique invariant torus and a unique heteroclinic invariant ellipsoid near a zero-Hopf equilibrium.