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In this article, the integrability and linearizability of a class of cubic Z 2 -equivariant systems x ˙ = - 1 2 x - a 21 y + 1 2 x 3 + a 21 x 2 y + a 12 x y 2 + a 03 y 3 , y ˙ = (- q - b 21) y + b 21 x 2 y + b 12 x y 2 + b 03 y 3 , are studied. For any positive integer q, we obtain the first three saddle quantities of the above systems by theoretical analysis. Moreover, for any positive integer q, we derive the necessary and sufficient conditions for the linearizability of the above systems under some assumptions. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the number of sliding limit cycles of regularized piecewise linear visible–invisible twofolds using the notion of slow divergence integral. We focus on limit cycles produced by canard cycles located in the half-plane with an invisible fold point. We prove that the integral has at most 1 zero counting multiplicity (when it is not identically zero). This will imply that the canard cycles can produce at most 2 limit cycles. Moreover, we detect regions in the parameter space with 2 limit cycles. [ABSTRACT FROM AUTHOR]

We study the maximum number of limit cycles that bifurcate from the periodic annulus of the isochronous cubic centre of discontinuous and continuous piecewise differential systems with three zones formed by the discontinuity set Σ = { (x , y) ∈ R 2 : (y = 0) ∨ (x = 0 ∧ y ≥ 0) } . More precisely, we consider the following perturbed systems x ˙ = - y + x 2 y + ϵ p i (x , y) , y ˙ = x + x y 2 + ϵ q i (x , y) , i = 1 , 2 , 3 , where p i and q i are polynomials of degree m. Using the averaging theory of first order, we prove that for m = 1 , 2 , 3 , at most 3, 9 and 15 limit cycles bifurcate from the periodic annulus of the isochronous cubic centre in the discontinuous case, respectively. While for the continuous case, it can appear 1, 5 and 6 limit cycles from the periodic orbits of these centres, respectively. Furthermore, we extend our study when p i and q i are homogeneous polynomials with 1 ≤ m ≤ 3 , obtaining respectively for m = 1 , 2 , 3 at most one, seven and at least twelve limit cycles, which bifurcate from the periodic orbits of the isochronous cubic centre. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the maximum number of limit cycles that can bifurcate from the periodic annulus of the linear center for discontinuous piecewise quadratic polynomial differential systems with four zones separated by two nonlinear curves y = ± x 2 . By analyzing the first order averaged function, we prove that at most 7 crossing limit cycles can produce from periodic annulus of the linear center, and the upper bound is reached. In addition, under particular conditions, we obtain that at least 8 crossing limit cycles can bifurcate from periodic annulus in this class of systems by using the second order averaged function. Our results show that multiple switching curves increase the number of crossing limit cycles in comparison with the case where there is only a single switching curve. [ABSTRACT FROM AUTHOR]

In this paper, we revisit a Leslie type predator–prey system with strong Allee effect on prey and simplified Holling type IV functional response proposed. Some more complex dynamical properties have been discovered. First, it is proved that the system exhibits a degenerate cusp of codimension 4 and a degenerate focus, saddle or elliptic of codimension 3 for different parameter values analytically. Further, various possible high codimensional bifurcation analyses are performed. It is shown that the system undergoes degenerate focus, saddle or elliptic type Bogdanov–Takens bifurcation of codimension 3 and degenerate cusp type Bogdanov–Takens bifurcation of codimension 4 as the parameters vary. Moreover, the existence and spatial location of the limit cycles are explored by calculating Hopf bifurcation and homoclinic bifurcation surfaces. Numerical simulations are carried out, and it is observed that three limit cycles coexist. [ABSTRACT FROM AUTHOR]

It is known that for smooth differential systems in the plane R 2 the Melnikov and the averaging methods for studying the limit cycles produce the same results. Here we prove that this is not the case for nonsmooth differential systems in the plane. More precisely, we prove that the linear system x ˙ = y , y ˙ = - x , can produce at most 5 crossing limit cycles using the averaging theory of first order and also produce at most 5 crossing limit cycles using the averaging theory of second order, when it is perturbed by discontinuous piecewise polynomials of two pieces separated by the cubic curve y = x 3 , and having in each piece a quadratic polynomial differential system. While using the Melnikov theory up to second order these discontinuous piecewise differential systems already produce 7 crossing limit cycles having in each piece a linear polynomial differential system. Note that the class of the linear polynomial differential systems is contained into the class of quadratic polynomial differential systems. [ABSTRACT FROM AUTHOR]

The present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to present the behavior generated by piecewise smooth vector fields tangent to foliations. We consider two smooth foliations F 1 and F 2 that are coupled to produce a foliation composed by nested topological tori. Moreover, a piecewise smooth vector field composed of periodic orbits and tangent to these tori is considered. We perturb this piecewise vector field (and, consequently, the foliations) and the birth of either finite or infinitely many limit cycles for the 3D piecewise smooth vector field is observed. [ABSTRACT FROM AUTHOR]

There are various types in generalized Liénard systems. In this paper, we treat a Liénard-type system in their paper. Though the existence of homoclinic orbits was discussed in their paper, our aim is to present a tool for the unique existence of limit cycles of the system with one parameter. It is an improvement of the criterion in Hayashi (Adv Dyn Syst Appl 14:179–187, 2019). As applications, the examples in the above paper of Hayashi or Villari and Zanolin (Dyn Syst Appl 25:321–334, 2016) are investigated under the more small parameter for the uniqueness of the limit cycle. [ABSTRACT FROM AUTHOR]

In this paper, we consider the family of generalized Abel equations of the form x ′ = A (t) x m + B (t) x n , where A, B are trigonometric polynomials and m , n ∈ N . We characterize the existence of non-trivial limit cycles in this family, in terms of the trigonometric monomials. [ABSTRACT FROM AUTHOR]

A general family of planar piecewise linear ODEs with two zones both having a real focus and separated by a straight line is considered. By analyzing the number of zero points of a new function related to the intersection points of the trajectories of the linear subsystems with the separation line, complete results on the existence and number of limit cycles are obtained. In particular, complete parameter regions for the existence of 1–2 limit cycles are provided with two concrete examples, which will be helpful in studying some kinds of discontinuity-induced bifurcations (i.e., DIBs). Based on the main results, it is obtained that the family of planar piecewise linear ODEs with focus–focus dynamics separated by a straight line can have 3 limit cycles if and only if one subsystem has a real focus and the other one has a virtual focus. Moreover, it is showed that five is the least value of the number of parameters needed in canonical forms of such systems. [ABSTRACT FROM AUTHOR]

In this paper, we study the limit cycle bifurcation by perturbing the period annuluses of two perturbed hyper-elliptic Hamiltonian systems of degree seven. The period annuluses are bounded by heteroclinic loops, inside or outside of which there exist two nilpotent cusps. The bifurcation function is Abelian integral which is the first-order approximation of the Poincaré map. The sharp bounds of the number of limit cycles bifurcated from the periodic annuluses are obtained by Chebyshev criterion and asymptotic analysis. [ABSTRACT FROM AUTHOR]

In this paper we devote to study the piecewise smooth equation of the form: dx dt = S (t , x) = S 1 (t , x) = a 1 (t) x m + b (t) , if x ≥ 0 , S 2 (t , x) = a 2 (t) x m + b (t) , if x < 0 , where (t , x) ∈ [ 0 , 2 π ] × R , m ∈ Z + and a 1 (t) , a 2 (t) , b (t) are 2 π -periodic smooth functions. A solution of the equation satisfying x (0) = x (2 π) is called a periodic solution. Moreover, such solution is called a limit cycle if and only if it is isolated. We obtain that the maximum number of limit cycles for this equation is 1 (resp. 2) if (- 1) m a 1 (t) · a 2 (t) < 0 (resp. (- 1) m a 1 (t) · a 2 (t) > 0 ). In this study we pay more attention to the examples in which the equation has limit cycle(s) crossing the separation straight line x = 0 . In the end, we apply this result on a kind of piecewise smooth planar system which has a separation curve x 2 + y 2 = 1 . [ABSTRACT FROM AUTHOR]

In this paper we prove that the quartic Liénard equation with linear damping { x ˙ = y , y ˙ = - (a 0 + x) y - (b 0 + b 1 x + b 2 x 2 + b 3 x 3 + x 4) } can have at most two limit cycles, for the parameters kept in a small neighborhood of the origin (a 0 , b 0 , b 1 , b 2 , b 3) = (0 , 0 , 0 , 0 , 0) . Near the origin in the parameter space, the Liénard equation is of singular type and we use singular perturbation theory and the family blow up. To study the limit cycles globally in the phase space we need a suitable Poincaré–Lyapunov compactification. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of limit cycles in the perturbed Hamiltonian system dH=εF1+ε2F2+ε3F3 with Fi, the vector valued homogeneous polynomials of degree i and 4-i for i=1,2,3, and small positive parameter ε. The Hamiltonian function has the form H=y2/2+U(x), where U is a univariate polynomial of degree four without symmetry. We compute higher order Melnikov functions until we obtain reversible perturbations. Then we find the upper bounds for the number of limit cycles that can bifurcate from the periodic orbits of dH=0. [ABSTRACT FROM AUTHOR]

In this work we study degenerate fold-Hopf bifurcation of codimension three in a differential system of the form x˙=-x-y-z,y˙=x+ay+bxy,z˙=bx-cz.We prove that the classical theory of the fold-Hopf bifurcation cannot be applied for studying the system’s behavior. However, using tools from averaging theory we prove the existence of periodic orbits generated by this bifurcation. [ABSTRACT FROM AUTHOR]

We prove a linear in $$\deg \omega $$ upper bound on the number of real zeros of the Abelian integral $$I(t)=\int _{\delta (t)}\omega $$ , where $$\delta (t)\subset {\mathbb R}^2$$ is the real oval $$x^2y(1-x-y)=t$$ and $$\omega $$ is a one-form with polynomial coefficients. [ABSTRACT FROM AUTHOR]

In this work we give sufficient conditions for the existence of a Dulac function for an arbitrary differential system. This Dulac function allows to discard the existence of limit cycles in its domain of definition if this domain is a simply connected region. If the domain of definition is $$\ell $$ -multiple connected then the Dulac function can estimate the number of limit cycles inside the domain. [ABSTRACT FROM AUTHOR]

In this paper, a class of polynomial Liénard systems is considered. Some conditions of the existence, non-existence and uniqueness of limit cycles are obtained by using Filippov transformations and Zhang's theorem. We obtain that the above system has at most one limit cycle surrounding the origin if a a < 0 or b = 0. And, one example is given to illustrate that the system can have three limit cycles. [ABSTRACT FROM AUTHOR]

We study a three dimensional Lotka-Volterra systems. In the paper Bobieński and Żołądek (J Ergod Theory Dyn Syst 25:759-791, ) four cases of center (i.e. an invariant surface supporting a center) were found. In this paper, we study a codimension 2 component LV and its versal deformation. We prove that at most one invariant torus may appear. This invariant torus corresponds to the limit cycle bifurcating in the amplitude system. [ABSTRACT FROM AUTHOR]

We prove that perturbing the two periodic annuli of the quadratic polynomial reversible Lotka-Volterra differential system ẋ = − y + x − y, ẏ = x(1 + 2 y), inside the class of all quadratic polynomial differential systems we can obtain the following configurations of limit cycles (0, 0), (1, 0), (2, 0), (1, 1) and (1, 2). [ABSTRACT FROM AUTHOR]

We prove that there are one-parameter families of planar differential equations for which the center problem has a trivial solution and on the other hand the cyclicity of the weak focus is arbitrarily high. We illustrate this phenomenon in several examples for which this cyclicity is computed. [ABSTRACT FROM AUTHOR]

We study the number of limit cycles of T-periodic Chini equations and some generalized Abel equations and apply the results obtained to illustrate the existence of isochronous centers in three-dimensional autonomous differential systems. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of uniqueness of limit cycles for particular second order differential equations and we give a generalization of the classical Massera’s Theorem. [ABSTRACT FROM AUTHOR]

In this paper we study the infinitesimal Hilbert’s 16th problem on $${\mathbb{R}}^{2}$$ and $${\mathbb{C}}^{2}$$, describing some techniques used for its research in the last few years. We make an alternative proof of a theorem of Gavrilov and Ilyashenko which allows the study of higher order Poincaré–Pontryagin functions. We apply these techniques to the research of limit cycles of systems of Liénard type obtained by special perturbations of the harmonic oscillator. For this class of systems, we obtain an upper bound, depending on the degree of the perturbation, for the number of limit cycles that can be generated from cycles of the harmonic oscillator. Explicit examples are given. [ABSTRACT FROM AUTHOR]

In this paper we study the bifurcations of a class of polycycles, called lips, occurring in generic three-parameter smooth families of vector fields on a Möbius band. The lips consists of a set of polycycles formed by two saddle-nodes, one attracting and the other repelling, connected by the hyperbolic separatrices of the saddle-nodes and by orbits interior to both nodal sectors. We determine, under certain genericity hypotheses, the maximum number of limits cycles that may bifurcate from a graphic belonging to the lips and we describe its bifurcation diagram. [ABSTRACT FROM AUTHOR]

We investigate the bifurcation of small-amplitude limit cycles in generalized Liénard equations. We use the simplicity of the Liénard family, to illustrate the advantages of the approach based on Bautin ideals. Essentially, this Bautin ideal is generated by the so-called Lyapunov quantities, that are computed for generalized Liénard equations and used to detect the presence of a Hopf-Takens bifurcation. Furthermore, the cyclicity is computed exactly. [ABSTRACT FROM AUTHOR]

In this paper, we study the distribution and number of limit cycles for quadratic systems with two foci. It is proved that a quadratic system with two foci has at most one limit cycles around one of the two foci, and hence the limit cycles of the quadratic system with two foci must be (0, 1)-distribution or (1, i)-distribution ( i=0,1,2,...) [ABSTRACT FROM AUTHOR]

We show that the classical procedure of succesive bifurcations at the origin can be applied also at infinity, both for polynomial and non-polynomial systems. [ABSTRACT FROM AUTHOR]

Using first integrals we introduce an alternative tool to the Poincaré maps in order to study the limit cycles of piecewise linear differential systems. It is also shown that an usual relaxation oscillation phenomenon occurrs in a two-parameter family of planar piecewise linear vector fields. [ABSTRACT FROM AUTHOR]

The pseudo-normal form is presented as an alternative to the Birkhoff normal form for a planar vector field with the origin as an equilibrium point. Its convergence is proved for non-zero critical exponents λ,−λ, and some consequences for the center-focus, problem are presented. [ABSTRACT FROM AUTHOR]

In this paper we prove that several elementary graphics surrounding a focus or center in quadratic systems have finite cyclicity. This paper represents an additional step in the large program to prove the existence of a uniform bound for the number of limit cycles of a quadratic vector field which we can call the finiteness part of Hilbert’s 16th problem for quadratic vector fields. It nearly finishes the part of the program concerned with elementary graphics. In [3] this problem was reduced to the proof that 121 graphics have finite cyclicity. The graphics considered here are the hemicycles ( H ), ( H ) and ( H ) together with ( I ), ( I , ( I ) in the notation of [3] (Figure 1). [ABSTRACT FROM AUTHOR]