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Register a new userLimit cycles by perturbing quadratic isochronous centers inside piecewise smooth polynomial differential systems
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In the present paper, we study the number of zeros of the first order Melnikov function for piecewise smooth polynomial differential system, to estimate the number of limit cycles bifurcated from the period annulus of quadratic isochronous centers, when they are perturbed inside the class of all piecewise smooth polynomial differential systems of degree $n$ with the straight line of discontinuity $x=0$. An explicit and fairly accurate upper bound for the number of zeros of the first order Melnikov functions with respect to quadratic isochronous centers $S_1, S_2$ and $S_3$ is provided. For quadratic isochronous center $S_4$, we give a rough estimate for the number of zeros of the first order Melnikov function due to its complexity. Furthermore, we improve the upper bound associated with $S_4$, from $14n+11$ in cite{LLLZ}, $12n-1$ in cite{SZ} to $[(5n-5)/2]$, when it is perturbed inside all smooth polynomial differential systems of degree $n$. Besides, some evidence on the equivalence of the first order Melnikov function and the first order Averaged function for piecewise smooth polynomial differential systems is found.
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In this article we study the existence of limit cycles in families of piecewise smooth differential equations having the unit circle as discontinuity region. We consider families presenting singularities of center or saddle type, visible or invisible, as well as the case without singularities. We establish an upper bound for the number of limit cycles and give examples showing that the maximum number of limit cycles can be reached. We also discuss the existence of homoclinic cycles for such differential equations in the saddle-center case.
We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or cubic polynomial differential systems separated by a straight line. We compute the maximum number of limit cycles of these discontinuous piecewise polynomial perturbations of the linear center, which can be obtained by using the averaging theory of order $n$ for $n=1,2,3,4,5$. Of course these limit cycles bifurcate from the periodic orbits of the linear center. As it was expected, using the averaging theory of the same order, the results show that the discontinuous quadratic and cubic polynomial perturbations of the linear center have more limit cycles than the ones found for continuous and discontinuous linear perturbations. Moreover we provide sufficient and necessary conditions for the existence of a center or a focus at infinity if the discontinuous piecewise perturbations of the linear center are general quadratic polynomials or cubic quasi--homogenous polynomials.
In this paper, we extend the slow divergence-integral from slow-fast systems, due to De Maesschalck, Dumortier and Roussarie, to smooth systems that limit onto piecewise smooth ones as $epsilonrightarrow 0$. In slow-fast systems, the slow divergence-integral is an integral of the divergence along a canard cycle with respect to the slow time and it has proven very useful in obtaining good lower and upper bounds of limit cycles in planar polynomial systems. In this paper, our slow divergence-integral is based upon integration along a generalized canard cycle for a piecewise smooth two-fold bifurcation (of type visible-invisible called $VI_3$). We use this framework to show that the number of limit cycles in regularized piecewise smooth polynomial systems is unbounded.
In this paper we research global dynamics and bifurcations of planar piecewise smooth quadratic quasi--homogeneous but non-homogeneous polynomial differential systems. We present sufficient and necessary conditions for the existence of a center in piecewise smooth quadratic quasi--homogeneous systems. Moreover, the center is global and non-isochronous if it exists, which cannot appear in smooth quadratic quasi-homogeneous systems. Then the global structures of piecewise smooth quadratic quasi--homogeneous but non-homogeneous systems are studied. Finally we investigate limit cycle bifurcations of the piecewise smooth quadratic quasi-homogeneous center and give the maximal number of limit cycles bifurcating from the periodic orbits of the center by applying the Melnikov method for piecewise smooth near-Hamiltonian systems.
In this paper, we give a direct method to study the isochronous centers on center manifolds of three dimensional polynomial differential systems. Firstly, the isochronous constants of the three dimensional system are defined and its recursive formulas are given. The conditions of the isochronous center are determined by the computation of isochronous constants in which it doesnt need compute center manifolds of three dimensional systems. Then the isochronous center conditions of two specific systems are discussed as the applications of our method. The method is an extension and development of the formal series method for the fine focus of planar differential systems and also readily done with using computer algebra system such as Mathematica or Maple.
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