1. Introduction

In the last century, Hilbert presented a list of problems that almost all of them are solved. One problem that remains open consists in determining the maximal number $\mathcal {H}(n)$ of limit cycles, and their relative positions, of planar polynomial vector fields of degree $n$. This problem is known as the second part of the 16th Hilbert's problem. In the year of 1977, Arnol'd in [Reference Arnol'd4] proposed a weakened version, focused on the study of the number of limit cycles bifurcating from the period annulus of Hamiltonian systems.

In this work, we are interested in another local version, that consists in providing the maximum number $\mathcal {M}(n)$ of small amplitude limit cycles bifurcating from an elementary centre or an elementary focus, clearly $\mathcal {M}(n)\leq \mathcal {H}(n).$ In other words, $\mathcal {M}(n)$ is an upper bound of the (local) cyclicity of such equilibrium points. For more details, see [Reference Roussarie49]. For $n=2$, Bautin proved that $\mathcal {M}(2) = 3,$ see [Reference Bautin5]. For $n=3$, the family of cubic systems without quadratic terms was studied in [Reference Blows and Lloyd7, Reference Sibirskiĭ51]. The proof of $\mathcal {M}_h(3)=5$ can be found in [Reference Żoła̧dek54]. Żoła̧dek in [Reference Żoła̧dek55] shows the first evidence that $\mathcal {M}(3)\ge 11.$ However, this problem has been recently revisited by himself in [Reference Żoła̧dek57]. The first proof of this fact was obtained some years later by Christopher in [Reference Christopher14], studying first-order perturbations of another cubic centre also provided by Żoła̧dek in [Reference Żoła̧dek56].

In 2012, Giné conjectures that $\mathcal {M}(n)= n^{2}+3n-7$, see [Reference Giné25, Reference Giné26]. This suggests a higher value for $\mathcal {M}(n)$ for polynomial vector fields of low degree. Gouveia and Torregrosa in [Reference Gouveia and Torregrosa30] show that $\mathcal {M}(5)\geq 33$, $\mathcal {M}(7) \geq 61,$ $\mathcal {M}(8) \geq 76,$ and $\mathcal {M}(9) \geq 88$. The first evidence that this conjecture fails is given in [Reference Yu and Tian53]. Recently, a complete proof that $\mathcal {M}(3) \geq 12$ is provided by Giné, Gouveia, and Torregrosa in [Reference Giné, Gouveia and Torregrosa27]. Moreover, in the same paper, they show that $\mathcal {M}(4) \geq 21.$ For $n=6$, the best lower bound was given in [Reference Liang and Torregrosa42], proving that $\mathcal {M}(6)\geq 40$. The first main result of this paper updates this value.

In this work, we are also interested in piecewise polynomial vector fields. Andronov, in [Reference Andronov, Vitt and Khaikin2] was the first to study such class of systems. In the last years, they have been widely studied, since many problems of engineering, physics, and biology can be modelled by them, see [Reference Acary, Bonnefon and Brogliato1, Reference di Bernardo, Budd, Champneys and Kowalczyk6]. One of the most studied situations in the plane is given by two vector fields defined in two half-planes separated by a straight line. As in the case of the classical qualitative theory of polynomial systems, the study of the number and location of the isolated periodic orbits, also called limit cycles, have received special attention, see for example [Reference Buzzi, Pessoa and Torregrosa11, Reference Coombes17, Reference Giannakopoulos and Pliete24, Reference Han and Zhang34, Reference Leine and van Campen41, Reference Llibre and Ponce43]. In particular, it can be seen as an extension of the 16th-Hilbert problem for planar piecewise polynomial vector fields, see [Reference Ilyashenko38].

Here, we study limit cycles bifurcating from the origin in the class of piecewise differential equations of the form

\[ \begin{aligned} \left\{\begin{array}{@{}lcr} ( \dot x, \dot y) & = & (P^{+}(x,y,\lambda),Q^{+}(x,y,\lambda)), \text{ when } y\ge 0,\\ (\dot x, \dot y) & = & (P^{-}(x,y,\lambda),Q^{-}(x,y,\lambda)),\text{ when } y<0, \end{array} \right. \end{aligned} \]

where $P^{\pm }(x,\,y,\,\lambda )$ and $Q^{\pm }(x,\,y,\,\lambda )$ are polynomials in $x$ and $y$. The straight line $\Sigma =\{y=0\}$ divides the plane in two half-planes, $\Sigma ^{\pm }=\{(x,\,y) :\pm y>0\},$ and the trajectories on $\Sigma$ are defined following the Filippov convention, see [Reference Filippov20]. Here we will consider only limit cycles of crossing type, that is, when both vector fields point out in the same direction on the separation line $\Sigma.$ Then, we denote by $\mathcal {H}^{c}_{p}(n)$ the number of limit cycles of crossing type for piecewise polynomial vector fields of degree $n.$ Similarly, $\mathcal {M}_{p}^{c}(n)$ counts the ones of small amplitude bifurcating from a monodromic equilibrium point. The upper index $c$ means for crossing limit cycles and the subscript $p$ for piecewise class. Clearly, $\mathcal {M}(n) \leq \mathcal {H}(n)$ and $\mathcal {M}_{p}^{c}(n) \leq \mathcal {H}_{p}^{c}(n).$

For $n=1$, Huan and Yang in [Reference Huan and Yang37] show a numerical evidence that $\mathcal {H}^{c}_{p}(1) \geq 3.$ Llibre and Ponce in [Reference Llibre and Ponce43] provide an analytical proof of this property. Using the averaging technique, this lower bound was reobtained by Buzzi, Pessoa, and Torregrosa in [Reference Buzzi, Pessoa and Torregrosa11]. Freire, Ponce, and Torres also obtained the same number in [Reference Freire, Ponce and Torres21]. In this last work, the three limit cycles are explained by studying the full return map, two appear near the origin and the other one far from it. But the three limit cycles appear nested and surrounding one sliding segment. In fact, the two limit cycles of small amplitude appearing from an equilibrium point provide the lower bound $\mathcal {M}^{c}_{p}(1) \geq 2.$ This value can be proved with the results in [Reference Freire, Ponce and Torres21] and we will recall it in the next section. Recently in [Reference Freire, Ponce, Torregrosa and Torres23], the three limit cycles have been obtained near infinity in a Hopf type bifurcation.

For $n=2$, using averaging theory of fifth-order, and perturbing the linear centre, Llibre and Tang in [Reference Llibre and Tang44] proved that $\mathcal {H}^{c}_{p}(2) \geq 8.$ Recently, da Cruz, Novaes, and Torregrosa in [Reference da Cruz, Novaes and Torregrosa18] improve this lower bound and, using local developments of the difference map, Gouveia and Torregrosa in [Reference Gouveia and Torregrosa29] provide the first high lower bounds for piecewise polynomial classes of degrees three, four and five: $\mathcal {M}_{p}^{c}(3)\geq 26$, $\mathcal {M}_{p}^{c}(4)\geq 40$, and $\mathcal {M}_{p}^{c}(5)\geq 58.$ Here we update two of these lower bounds using averaging theory in our second main result.

In this paper, we are interesting also in differential equations containing a privileged $\varepsilon$ that acts as a small perturbation of a periodic behaviour. Lagrange, in his study about the three-body problem, formulated the idea of averaging. During some years, this method was used in many fields without people bothering about proofs validity. In 1928, Fatou in [Reference Fatou19] gave the first analytic proof. Nowadays, in planar differential systems, the averaging theory is also used for proving the existence of limit cycles which appear, after perturbation, from a fully periodic neighbourhood. A similar mechanism, when the differential equation is nonautonomous, used for the same purpose is the Melnikov method. Moreover, it is an excellent tool for studying global bifurcations that occur near homoclinic or heteroclinic loops in one-parameter families.

Let us consider the perturbation of a Hamiltonian system

(1)\begin{equation} \left\{\begin{aligned} x' & ={-}H_{y}+\varepsilon \,P(x,y,\varepsilon,\lambda),\\ y' & =H_{x}+\varepsilon \, Q(x,y,\varepsilon,\lambda). \end{aligned}\right. \end{equation}

Then, the first Melnikov function writes as

(2)\begin{equation} M(h)=\displaystyle\int_{\Gamma_h} Q(x,y,0,\lambda)dx-P(x,y,0,\lambda)dy, \end{equation}

where $\Gamma _h=\{H(x,\,y)=h^{2}\}$ are closed ovals. This first-order analysis is based on the Implicit Function Theorem. In fact, for $\varepsilon$ small enough, the simple zeros of $M(h)$ correspond to limit cycles of the perturbed system (1). That is, if $h^{*}$ satisfies that $M(h^{*})=0$ and $M'(h^{*})\ne 0$ then, there exists an hyperbolic limit cycle of (1) that goes to $\Gamma _{h^{*}}$ when $h$ goes to $h^{*}.$

This result can be generalized also for the perturbation of centres, $(\dot x,\,\dot y)=(P_c,\,Q_c)$, having an inverse integrating factor $V(x,\,y)$. Then after a time rescaling the perturbed system

(3)\begin{equation} \left\{\begin{aligned} \dot x & =P_c(x,y)+\varepsilon \, P(x,y,\varepsilon,\lambda),\\ \dot y & =Q_c(x,y)+\varepsilon \, Q(x,y,\varepsilon,\lambda), \end{aligned}\right. \end{equation}

can be written as

(4)\begin{equation} \left\{\begin{aligned} x' & ={-}H_{y}+\varepsilon \, \dfrac{P(x,y,\varepsilon,\lambda)}{V(x,y)},\\ y' & =H_{x}+\varepsilon \, \dfrac{Q(x,y,\varepsilon,\lambda)}{V(x,y)}. \end{aligned}\right. \end{equation}

So, the corresponding generalized first Melnikov function of (4), or also of (3), is

(5)\begin{equation} M(h)=\displaystyle\int_{\Gamma_h} \displaystyle\frac{Q(x,y,0,\lambda)dx-P(x,y,0,\lambda)dy}{V(x,y)}. \end{equation}

In this paper, we only consider the above Melnikov function for the study of perturbations of periodic orbits near planar autonomous differential systems. In this case, there are some works explaining the equivalence between Melnikov studies and the averaging theory, see [Reference Buică9, Reference Han, Romanovski and Zhang35]. For more details in this theory, we refer the reader to [Reference Buică and Llibre10, Reference Han and Yu33, Reference Sanders and Verhulst50, Reference Verhulst52].

This bifurcation mechanism can be used also in piecewise differential equations, see [Reference Llibre, Novaes and Teixeira45–Reference Llibre, Mereu and Novaes47]. Although this problem can be treated in a more general way, in this work, we will consider a simplified version of it. We will study that only the perturbation in (3) is defined by piecewise functions. This perturbation will be defined in two zones separated by a straight line, that is, the differential equation to consider can be written as

(6)\begin{equation} (\dot x,\dot y)=\left\{\begin{array}{@{}l} (P_c(x,y),Q_c(x,y))+\varepsilon\, (P^{+}(x,y,\varepsilon,\lambda),Q^{+}(x,y,\varepsilon,\lambda)), \text{ if } y\geq 0,\\ (P_c(x,y),Q_c(x,y))+\varepsilon\, (P^{-}(x,y,\varepsilon,\lambda),Q^{-}(x,y,\varepsilon,\lambda)), \text{ if } y<0. \end{array} \right. \end{equation}

Therefore, the piecewise generalized first Melnikov function of (6) is

(7)\begin{equation} M_{p}(h)=\frac{1}{H_x^+(x_0,0)}M^{+}(h)+\frac{1}{H_x^-(x_0,0)}M^{-}(h), \end{equation}

where

\[ M^{{\pm}}(h)=\displaystyle\int_{\Gamma_h^{{\pm}}} \displaystyle\frac{Q^{{\pm}}(x,y,0,\lambda)dx-P^{{\pm}}(x,y,0,\lambda)dy}{V(x,y)} \]

with $\Gamma _h^{\pm }=\Gamma _h \cap \{\pm y>0\}(x_0,0)\Gamma_hy=0x_0>0.\Gamma _h^{\pm }=\Gamma _h \cap \{\pm y>0\}(x_0,0)\Gamma_hy=0x_0>0.$ $\Gamma _h^{\pm }=\Gamma _h \cap \{\pm y>0\}(x_0,0)\Gamma_hy=0x_0\,{>}\,0.\Gamma _h^{\pm }=\Gamma _h \cap \{\pm y>0\}(x_0,0)\Gamma_hy=0x_0\,{>}\,0.$ $\Gamma _h^{\pm } =\Gamma _h \cap \{\pm y>0\}(x_0,0)\Gamma_hy=0x_0>0.$ Clearly, in both piecewise regions, we have the same centre having $V(x,\,y)$ as an inverse integrating factor. The main results of this work could be extended to perturbing piecewise centres but this is not the aim of this paper.

From the mechanism described above, the functions (2), (5), and (7) provide the number of limit cycles bifurcating from the period annulus up to first-order analysis. Another tool for obtaining limit cycles bifurcating from the origin comes from the study of the Taylor development of the return map, the so-called Lyapunov constants. In this context, the perturbation terms of the non-degenerate linear centre, $(\dot x,\,\dot y)=(-y,\,x)$, usually have degree at least two because in this case, the Lyapunov constants are polynomials in the perturbation parameters, [Reference Cima, Gasull, Mañosa and Mañosas15]. Another small limit cycle bifurcates from the origin using the classical Hopf bifurcation when the linear coefficients are added in the perturbation terms and the trace parameter change sign adequately. This bifurcation phenomenon is also known as the degenerated Hopf bifurcation. More details on it can be seen in [Reference Andronov, Leontovich, Gordon and Maĭer3]. In piecewise differential equations, a similar bifurcation can also be introduced, but adding two limit cycles instead of only one. The linear terms are responsible for one of them and the other is due to the existence of what is known as the sliding segment that appears when the constant terms are added. The latter is known as the pseudo-Hopf bifurcation and they are summarized in the next result.

Proposition 1.3 Gouveia and Torregrosa [Reference Gouveia and Torregrosa28]

Consider the perturbed system

\[ \left\{\begin{array}{@{}l} \dot x ={-}(1+c^{2})y+\displaystyle\sum\limits_{k+\ell=2}^{\infty} a^{+}_{k\ell} x^{k}y^{\ell}, \\ \dot y = x+2cy+\displaystyle\sum\limits_{k+\ell=2}^{\infty} b^{+}_{k\ell} x^{k}y^{\ell}, \end{array}\right.\quad \left\{\begin{array}{@{}l} \dot x ={-}y+\sum\limits_{k+\ell=2}^{\infty} a^{-}_{k\ell} x^{k}y^{\ell}, \\ \dot y = d+x+\sum\limits_{k+\ell=2}^{\infty} b^{-}_{k\ell} x^{k}y^{\ell}, \end{array} \right. \]

for $y\ge 0$ and $y<0$, respectively. If $a^{+}_{11}-a^{-}_{11}+2b^{+}_{02}-2b^{-}_{02}+b^{+}_{20}-b^{-}_{20}\ne 0$ then there exist $c$ and $d$ small enough such that two crossing limit cycles of small amplitude bifurcate from the origin.

We notice that when $c=d=0,$ the expression $a^{+}_{11}-a^{-}_{11}+2b^{+}_{02}-2b^{-}_{02}+b^{+}_{20}-b^{-}_{20}$ defines the stability of the origin. So, we could define it as the first Lyapunov constant for the piecewise perturbation of the linear centre. More details on the pseudo-Hopf bifurcation are collected in [Reference Castillo, Llibre and Verduzco12, Reference Freire, Ponce and Torres21, Reference Freire, Ponce and Torres22]. In [Reference Christopher14, Reference Giné, Gouveia and Torregrosa27, Reference Gouveia and Torregrosa29, Reference Gouveia and Torregrosa30], we can see that a good and simple tool for studying the number of limit cycles of small amplitude bifurcating from the origin is the first-order development of the Lyapunov constants. We notice that, in this case, the Jacobian matrix of the vector field at the origin must be of non-degenerate monodromic type.

Then, a natural question arises. Are there any relation between both mechanisms? Chicone and Jacobs in 1991 provided a positive answer for quadratic families of vector fields, see [Reference Chicone and Jacobs13]. But, their proof is also valid for polynomial vector fields of any degree. Hence, we can say that the next result comes from their original work. These ideas appear also in the works of Han and Yu, see [Reference Han and Yu33, Reference Han, Sheng and Zhang36], where the Melnikov function in the Hopf bifurcation is studied.

Theorem 1.4 Chicone and Jacobs [Reference Chicone and Jacobs13]

Let $(\dot x,\,\dot y)=(P_c(x,\,y),\,Q_c(x,\,y))$ be a polynomial vector field of degree $n,$ with a non degenerated centre at the origin. Consider the perturbed systems, in the class of polynomial vector fields of degree $n,$

(8)\begin{equation} (\dot x,\dot y)=\left(P_c(x,y)+\displaystyle\displaystyle\sum_{k+l=2}^{n} a_{kl}x^{k}y^{l},Q_c(x,y)+\displaystyle\sum_{k+l=2}^{n} b_{kl}x^{k}y^{l}\right) \end{equation}

and

(9)\begin{equation} (\dot x,\dot y)=\left(P_c(x,y)+\varepsilon\displaystyle\displaystyle\sum_{k+l=2}^{n} a_{kl}x^{k}y^{l} +O(\varepsilon^{2}),Q_c(x,y)+\varepsilon\displaystyle\sum_{k+l=2}^{n} b_{kl}x^{k}y^{l}+O(\varepsilon^{2})\right). \end{equation}

If we denote by $L_k^{[1]}$ the first-order expansion, with respect to the parameters $a_{kl},\, b_{kl}$, for the Taylor series of the Lyapunov constant $L_k$ associated with (8) then, for $\rho$ small, the first Melnikov function of (9) is

(10)\begin{equation} M(\rho)=\displaystyle\sum_{k=1}^{N}{L}_{k}^{[1]}\left(1+\sum_{j=1}^{\infty}\alpha_{kj0}\rho^{j}\right)\rho^{2k+2}, \end{equation}

with the Bautin ideal $\langle L_1,\,\ldots,\,L_N,\,\ldots \rangle =\langle L_1,\,\ldots,\,L_N \rangle.$

We remark that the perturbations in the above result are taken without linear terms. This restriction is due to the fact that the elements in the Bautin ideal should be polynomials. A direct consequence is the first part of the next corollary. For the second part, the classical Hopf bifurcation is necessarily to be used. As usual, $O(\varepsilon ^{2})$ denotes the terms of degree at least $2$ in the perturbative parameter $\varepsilon$.

Recently, see [Reference Giné, Gouveia and Torregrosa27, Reference Gouveia and Torregrosa29, Reference Gouveia and Torregrosa30], a detailed study of the return map of a differential equation near a non-degenerated monodromic point located at the origin has been done. The authors focus their attention not only on first-order Taylor developments, with respect to parameters, of the coefficients of the return map with respect to the initial condition. These papers are a continuation of the work given by Christopher in [Reference Christopher14]. The degenerated Hopf bifurcation technique applies when $L_k^{[1]}=u_k$ for $k=1,\,\ldots,\,N,$ and all $u_k$ are new independent parameters. Then, the existence of $N$ limit cycles of small amplitude bifurcating from the origin is guaranteed. This approach fails when the rank does not increase in each step. That is when at least one $L_k^{[1]}$ is a linear combination of the previous ones. The main advantage of Theorem 1.4 is that we can compute easier the expressions of $L_k^{[1]}$ than the coefficients of the series expansion of $M.$ In [Reference Han and Yang32], the study of the multiplicity of zeros is also taken into account and moreover it provides upper bounds for the number of limit cycles up to this perturbation order.

This paper is structured as follows. In § 2, for completeness, we prove Theorem 1.1 recovering the original proof for quadratic vector fields obtained in [Reference Chicone and Jacobs13]. As a natural application of it, we also prove Corollary 1.5. Then, we can get limit cycles of small amplitude bifurcating from a centre equilibrium point. Finally, we extend Theorem 1.4 and Corollary 1.5 to the class of piecewise polynomial vector fields. In § 3, we show, with a simple example of a polynomial vector field of degree $6$, that the maximal computed rank of the linear developments does not coincide with the subscript of the corresponding Lyapunov constant. Then, the result using only linear parts given by Christopher in [Reference Christopher14] does not apply. But using Theorem 1.4, we provide more limit cycles of small amplitude. In § 4, we prove our first main result, Theorem 1.1, and we study the local cyclicity problem for some vector fields of degrees $7$, $8$, and $10,$ proving that $\mathcal {M}(7)\geq 60$, $\mathcal {M}(8)\geq 70$, and $\mathcal {M}(10)\geq 97.$ Finally, § 5 is devoted to piecewise polynomial vector fields of degrees $3,\,4,$ and $5$, proving our second main result Theorem 1.2. We also provide a new proof of $\mathcal {M}_p^{c}(3)\ge 26,$ different from the one published in [Reference Gouveia and Torregrosa29], using now only first-order developments.