Piecewise systems are widely used to model real-world phenomena such as electrical circuits and neurons. Their rich dynamical behavior arises from two factors: the nonlinearity of the subsystems and the nonsmoothness induced by the switching curves. This article focuses on the latter factor. In particular, we consider an extended version of weak Hilbert's 16th problem for some planar piecewise Hamiltonian systems. In these systems, the subsystems have fixed C1 Hamiltonians, while the switching curves are perturbed in the family of C1 curves. These systems reveal the exclusive influence of the switching curves. To study the influence, we provide the formula of the first order Melnikov function corresponding to the family of k-crossing closed orbits and apply it to two problems. In the first application, we investigate a system with linear subsystems and an algebraic switching curve of order n, and prove that such systems have at least n & LeftFloor; 2n & RightFloor; limit cycles. This improves the results of Douglas D. Novaes published in Physica D. In the second application, we study a piecewise linear system with two subsystems and a hyperbola switching curve, and prove the existence of three crossing limit cycles that intersect the switching curve four times.

In this paper we obtain the global dynamics and phase portraits of quadratic and cubic quasi-homogeneous but non-homogeneous systems. We first prove that all planar quadratic and cubic quasi-homogeneous but non-homogeneous polynomial systems can be reduced to three homogeneous ones. Then for the homogeneous systems, we employ blow-up method, normal sector method, Poincaré compactification and other techniques to discuss their dynamics. Finally we characterize the global phase portraits of quadratic and cubic quasi-homogeneous but non-homogeneous polynomial systems.

This paper focuses on the first three order Melnikov functions of general planar piecewise Hamiltonian systems under the piecewise perturbations with a non-regular separation line. By using the first three order Melnikov functions, we obtain the exact upper bounds of the number of limit cycles bifurcated from two different piecewise linear near-Hamiltonian systems.

This paper provides a completed Melnikov analysis of any order for a class of perturbed polynomial differential systems, where the unperturbed system is a cubic center with a straight line of singular points. The emphasis on any order Melnikov function is crucial because of two major reasons: one is that higher order Melnikov functions are not in general expressible explicitly, the other one is the essential difficulty of computing elliptic integrals due to the complexity of many iterations. The exact upper bounds of the number of limit cycles bifurcated from the period annulus under different polynomial perturbations are obtained by analyzing the algebraic structures from explicit expressions of any order Melnikov functions. (c) 2024 Elsevier Inc. All rights reserved.

This paper focuses on the research of the algorithm for higher order Melnikov functions for piecewise Hamiltonian systems under piecewise perturbations in any order ϵ. Then the formula is applied to the any order ϵ linear perturbations of a general center-center type piecewise linear system. The maximum number of zeros for any order Melnikov functions is 2 and it can be reached. It is worth mentioning that the algorithm can also be applied to smooth case.

In this paper, the first fourth order Melnikov analyses are applied to find the limit cycles bifurcated from a cubic center under the perturbation of quadratic polynomials in ϵ up to the second order. The upper bounds of the number of limit cycles are given and can be reached.

In this paper, the perturbation of a class of cubic Hamiltonian systems is considered. Using Abelian integral, we prove that there exists a neighborhood of the center where the system has at most two limit cycles for arbitrary polynomial perturbation of degree three or four, and at most four limit cycles for arbitrary polynomial perturbation of degree five or six, respectively, which can be reached.

In this paper, we consider a class of cubic systems with polynomial perturbation of the degree at most \begin{document}$ n $\end{document}, and estimate the upper bound of the number of isolated zeros of its Abelian integral. Furthermore, we obtain the distributions of limit cycles bifurcated from a \begin{document}$ Z_4 $\end{document}-equivariant system with \begin{document}$ 5 $\end{document} centers.

In this paper, we focus on the number of nontrivial limit cycles in a kind of piecewise smooth generalized Abel equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Under the condition [Formula: see text], employing Melnikov functions of any order and using properties of Chebyshev systems, we prove that if [Formula: see text] is odd, then the maximum number of nontrivial limit cycles bifurcating from the periodic annulus of the unperturbed system is 6 and it is attainable, and if [Formula: see text] is even, then the maximum number is 3, and it can be attained too.

In this paper, we characterize the global dynamical behaviors of FitzHugh–Nagumo system $$\dot{x}=z$$ , $$\dot{y}=b(x-dy)$$ , $$\dot{z}=x(x-1)(x-a)+y+cz$$ which has invariant algebraic surfaces. As byproducts, we obtain some new dynamical phenomena related to the invariant surfaces at the infinity, which does not appear previously in the study of other models. In addition, since the system restricted to the invariant algebraic surfaces is not analytic, we adopt some new techniques to overcome this difficulty.

In this paper, we consider a perturbation of planar general piecewise Hamiltonian systems with nonregular separation line. A general explicit expression of the first and second order Melnikov functions is presented for such piecewise Hamiltonian systems. In addition, the maximum number of limit cycles for piecewise linear differential system is an important topic that many researchers are concerned with. Different from considering a piecewise linear perturbation of a linear center in the previous research, we apply our explicit expression to estimate the number of limit cycles bifurcated from a class of piecewise linear Hamiltonian systems. More specifically, the unperturbed system we study is piecewise smooth and constructed by a linear center and a constant differential system, which is difficult to be considered by a polar coordinate transformation. And we obtain the upper bounds to be 4 and 5 for the limit cycles bifurcated from the periodic orbits by using the first and second order Melnikov functions, respectively. Moreover the upper bounds are sharp.

By computing we obtain that P-1(0, 0, 1) is a zero-Hopf equilibrium point of nuclear spin generator system. We prove that there exist two families of nuclear spin generator system which has the zero-Hopf equilibrium point P-1(0, 0,1). Furthermore we prove that a limit cycle bifurcated from P-1(0, 0,1) by averaging method of one order and second order respectively.

Numerical simulations of the micron particles flow past a spheroidal droplet are carried out to determine the effect of droplet diameter in the range of 0.1-2 mm and the Reynolds number of 10-900 on capture efficiency, and to gain some insight into the inertial capture mechanism. The results suggest that the droplet inertial capture efficiency improves with the increase of Reynolds number and decreases with the increase of droplet diameter. For droplet diameter between 0.8-1 mm, there is a critical droplet diameter which changes the capture mechanism from inertial capture to interception and reduces the capture efficiency greatly. Micron particles trajectories are more likely to be affected by vortex-shedding which results in the occurrence of rear capture behavior when droplet diameter more than 1 mm. The rear capture behavior can be observed when the eddy turnover time and the particle relaxation time is in the same order magnitude.

Many natural phenomena can be modeled as discontinuous dynamical systems separated by a nonregular line. The number and distribution of limit cycles in discontinuous linear systems are important topics for research. In this paper, we focus on the limit cycles created by discontinuous planar piecewise linear systems separated by a nonregular line of center–center type, and prove that such systems have at most two limit cycles, which can be reached. Furthermore, the two limit cycles are nested and intersect the separation line at two points or four points, that is, either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points.

In this paper, we consider the general perturbations of piecewise Hamiltonian systems. A formula for the second order Melnikov functions is derived when the first order Melnikov functions vanish. As an application, we can improve an upper bound of the number of bifurcated limit cycles of a piecewise Hamiltonian system with quadratic polynomial perturbations.

We obtain the conditions for the existence of a center for a cubic planar differential system, which can be considered as a polynomial subfamily of the generalized Riccati system. We also investigate bifurcations of small limit cycles from the components of the center variety of the system.

In this paper, we consider the system x˙=y(1+x)2−ϵP(x,y), y˙=−x(1+x)2+ϵQ(x,y) where P(x,y) and Q(x,y) are arbitrary quadratic polynomials. We study the maximum number of limit cycles bifurcating from the periodic orbits by using the Melnikov function of any order. We prove that the upper bound for the number of limit cycles is 3 and reached.

We study arbitrary cubic perturbations of the symmetric 8-loop Hamiltonian, which are linear with respect to the small parameter. It is shown that when the first 4 coefficients in the expansion of the displacement functions corresponding to both period annuli inside the loop vanish, the system becomes integrable with the following three strata in the center manifold: Hamiltonian, reversible in y and reversible in x. In the latter case, the first integral is of Darboux type and we calculate it explicitly.Next we prove that the cyclicity of each of period annuli inside the loop is five and the total cyclicity of both is at most nine. For this, we use Abelian integrals method together with careful study the geometry of the separatrix solutions of related Riccati equations in connection to the second-order Melnikov functions.

In this paper, we characterize all the irreducible Darboux polynomials and polynomial first integrals of the FitzHugh-Nagumo system by the called characteristic curves method which involves the theory of weight homogeneous polynomials. It has been ever widely used to give a complete classification of Darboux polynomials of some systems. However, it does not work for the FitzHugh-Nagumo system. Thus, we introduce an “assistant” system and classify its invariant algebraic surfaces finally.