This paper investigates limit cycle bifurcations near a cuspidal loop L0 in a class of piecewisesmooth near-Hamiltonian systems. In earlier work, Wei (Nonlinear Analysis: Real World Applications, 2017) derived asymptotic expansions of the Melnikov functions near L0. However, due to computational complexity, the analysis of limit cycles was restricted to cusps of order at most two. In the present paper, we first show that the coefficients of terms of the same order in the expansions of the two Melnikov functions differ only by a constant multiple. Using this observation, we extend Wei's results and establish a general condition for the existence of limit cycles near L0 when the cusp has arbitrary order. Moreover, for a cubic piecewise Hamiltonian system with polynomial perturbations of degree n (n >= 1), we prove that, for suitable parameter values, the system admits at least 5n-4-3[ n ] limit cycles near L0. 2

This paper deals with the traveling wave solutions in the one-dimensional Euler-Poisson system equipped with the Boltzmann relation. By applying the method of dynamical systems to analyze the corresponding traveling wave system and find its phase portraits, we investigate the traveling wave solutions of the system without any restrictions. Under given parameter conditions, the existence of periodic wave solution families, compacton solution families, solitary wave solutions, kink wave solutions and anti-kink wave solutions is proved for the system with three variables. Our results complement the results in [Bae et al.; 2025] and demonstrate that the system cannot have peakon solutions.

For a kind of near-Hamiltonian systems of degree n subject to polynomial perturbations, we improve the existing results on the number of limit cycles near a homoclinic loop with a cusp. More significantly, the new method for calculating coefficients in the expansion of the first-order Melnikov function can be extended to many other cases.

For a cubic near-Hamiltonian system with a Hamiltonian function of the form H(x,y)=12y2−13x3+14x4, we obtain the expressions of the second-order and the third-order Melnikov functions. Hence, we study the expansions of them by using the expansions of the bases of corresponding Melnikov function. Furthermore, by using the two expansions, we study the number of limit cycles nearthe cuspidal loop L0 defined by the equation H(x,y)=0, and prove that there exist at least 10 limit cycles near the cuspidal loop, of which 6 (resp., 4) limit cycles are inside L0 and 4 (resp., 6) limit cycles are outside L0.

In this paper, we consider dynamics and exact solutions for the generalized Radhakrishnan-Kundu-Lakshmanan equation with four powers of nonlinearity. By investigating the solutions with the form q(x, t) = phi(x - vt)e(i(kappa x+omega t)), under given parameter conditions, the function phi(x - vt) satisfies a planar dynamical system depending on six parameters. For the parameter n = 1, 2, bifurcations of phase portraits of the dynamical system are studied. When n = 1, corresponding to all bounded solutions phi(xi) of dynamical system, 20 exact explicit parametric representations are derived. When n = 2, for the level curves having zero energy, exact explicit parametric representations with the form (psi(chi),xi(chi)) are given.

For two classes of piecewise smooth near-Hamiltonian systems, by studying some properties of the expansions of two Melnikov functions near a homoclinic loop, we give a simple relation between the coefficients of hj(j >= 0, j is an element of Z) appearing in the two expansions. Based on this, we further give a general condition for each of the two systems to have as many as possible limit cycles near the homoclinic loop. Hence, by using the above main results and some techniques we obtain a lower bound of the maximum number of limit cycles near a homoclinic loop for each of two concrete systems with polynomial perturbations of degree n(n >= 1).

In this paper, we first give the relation between the expansions of two Melnikov functions near a heteroclinic loop with two nilpotent cusps of order m_1 and m_2 ( m_i∈ℤ^+ ). Then, we derive a general condition for the existence of as many limit cycles as possible near the heteroclinic loop. Let ℋ(n, m) denote the maximum number of limit cycles for the Liénard equation ẍ+ε f(x)ẋ+g(x)=0 with f=n and g=m . We prove that ℋ(n, 7)≥ 2n-1 -2[ n+1/8] -[ n-1/8] for n≥ 1 , where the limit cycles are all near a heteroclinic loop with two nilpotent cusps. Notably, this result improves the existing lower bound of ℋ(n, 7) for n≥ 12 .

In this paper, we give a simple relation between the coefficients appearing in the expansions of n+2 (n∈Z+,n≥2) Melnikov functions near a compound cycle C(n), which can be used to simplify some computations. We further give some conditions for a general near-Hamiltonian system to have limit cycles as many as possible near C(n). Based on this, for a quintic Hamiltonian system with a compound cycle C(2) we prove that it can produce at least 72(n−2)+12(1+(−1)n) limit cycles near C(2) under polynomial perturbation of degree n(n≥2).

We study the bifurcation problem of limit cycles in near-Hamiltonian systems near a double homoclinic loop on the cylinder. We obtain a sufficient condition to find a lower bound of the maximal number of limit cycles near the loop by the coefficients of the expansions of the three Melnikov functions corresponding to the three families of periodic orbits near the double homoclinic loop. We also provide an application of our main results to a class of cylindrical systems.

In this paper, for a general near-Hamiltonian system we study the number and distributions of limit cycles near a double homoclinic loop. For a cubic Hamiltonian system with general polynomial perturbations, we obtain a lower bound of the maximum number of limit cycles near a double homoclinic loop.

In this paper, we consider the first-order Melnikov functions and limit cycle bifurcations of a near-Hamiltonian system near a cuspidal loop. By establishing relations between the coefficients in the expansions of the two Melnikov functions, we give a general method to obtain the number of limit cycles near the cuspidal loop. As an application, we consider a kind of Liénard systems and obtain a new estimation on the lower bound of the maximum number of limit cycles.

In this paper, we study the bifurcation problem of limit cycles near a general double homoclinic loop. We establish a general theory to obtain a lower bound of the maximal number of limit cycles near the double homoclinic loop. As an application, we prove that a near-Hamiltonian system of the form x˙=y(y2−1)+ε∑i=0naix2i+1,y˙=−x has at least [52n] limit cycles for 1≤n≤56. This number is maximal that we can find so far for the system.

In this paper, we study the relation between the coefficients in the expansions of two Melnikov functions near a heteroclinic loop with nilpotent cusps. Based on this relation, we give a condition of obtaining limit cycles near the heteroclinic loop. Further, we present a method to compute more coefficients in the expansions of two Melnikov functions near the heteroclinic loop. As an application, we consider a class of Liénard systems and study the number of limit cycles bifurcated from a heteroclinic loop and an elementary center.

In this paper, we study the expansion of the first order Melnikov function near a heteroclinic loop with two nilpotent cusps of general order. More precisely, the order of the two cusps is [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] [Formula: see text]. For general [Formula: see text] and [Formula: see text], we give the expansion of the first order Melnikov function and the formulas for the first few coefficients. We further give a general theorem on the number of limit cycles bifurcated from the heteroclinic loop. These results extend the existing results for [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text]. As an application, these results are applied to study the number of limit cycles near a heteroclinic loop with two cusps of different order.

In this paper, we give the different topological types of phase portrait for Liénard system $\dot {x}=y, \dot {y}=-g(x)$ in the case that $\deg g(x)=7$ and the system have six and seven singular points, respectively. For its perturbed system, the expansion of the Melnikov function near any of the above closed orbits, except that the closed orbit is a compound loop passing through a nilpotent cusp and two hyperbolic saddles or passing through three hyperbolic saddles, has been studied. In this paper, as one of main results, for a near-Hamiltonian system, we give the expansion of the Melnikov function near a compound loop with a nilpotent cusp and two hyperbolic saddles. Based on this, we present the conditions to obtain limit cycles.

For a centrally symmetric near-Hamiltonian system, we develop a method for computing all the coefficients in the expansions of three Melnikov functions near a double homoclinic loop. Moreover, we give a new estimation on the lower bound of H(2nˆ,5) for 11≤nˆ≤23, where H(2nˆ,5) is the maximal number of limit cycles for a kind of Liénard system, x˙=y,y˙=−g(x)+εf(x)y, with deg⁡g(x)=5 and deg⁡f(x)=2nˆ.

In this paper,we give all the different topological types of phase portrait for the unperturbed Liénard system (x) =y,(y) =-g(x) in the case that deg g(x) =7 and the system has 2,3,4 and 5 singular points,respectively.We then give the expansion of Melnikov function near a double heteroclinic loop with two nilpotent cusps and one hyperbolic saddle.We also give the conditions to obtain the limit cycles.

In this paper, the different topological types of phase portrait of the unperturbed Liénard system $$ {\dot{x}}=y,\ \ {\dot{y}}=-g(x)$$ are given, where $$\deg {g(x)}=6$$. We find that the expansion of the Melnikov function near any of closed orbits appeared in the above phase portraits, except a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle of order one, has been studied. In this paper, we give the expansion of the Melnikov function near this kind of heteroclinic loop. Further, we present the conditions to obtain limit cycles bifurcated from a compound loop with a hyperbolic saddle and a nilpotent saddle of order one, and apply it to study the number of limit cycles for a kind of Liénard system under perturbations.

In this paper, we mainly study the number of limit cycles for a quintic Lienard system under polynomial perturbations. In some cases, we give new estimations for the lower bound of the maximal number of limit cycles.

Homoclinic bifurcation is a difficult and important topic of bifurcation theory. As we know, a general theory for a homoclinic loop passing through a hyperbolic saddle was established by [Roussarie, 1986]. Then the method of stability-changing to find limit cycles near a double homoclinic loop passing through a hyperbolic saddle was given in [Han & Chen, 2000], and further developed by [Han et al., 2003; Han & Zhu, 2007]. For a homoclinic loop passing through a nilpotent saddle there are essentially two different cases, which we distinguish by cuspidal type and smooth type, respectively. For the cuspidal type a general theory was recently established in [Zang et al., 2008]. In this paper, we consider limit cycle bifurcation near a double homoclinic loop passing through a nilpotent saddle by studying the analytical property of the first order Melnikov functions for general near-Hamiltonian systems and obtain the conditions for the perturbed system to have 8, 10 or 12 limit cycles in a neighborhood of the loop with seven different distributions. In particular, for the homoclinic loop of smooth type, a general theory is obtained as a consequence. We finally consider some polynomial systems and find a lower bound of the maximal number of limit cycles as an application of our main results.