Higher order Melnikov function for a quartic hamiltonian with cuspidal loop
Copyright policy )Higher order Melnikov function for a quartic hamiltonian with cuspidal loop
The authors consider the polynomial perturbations \[ X_{\varepsilon}=X_H+ \varepsilon f(x,y,\varepsilon)\frac{\partial}{\partial x}+ \varepsilon g(x,y,\varepsilon)\frac{\partial}{\partial y}, \] where \(f(x,y,\varepsilon)\) and \(g(x,y,\varepsilon)\) are polynomials in \(x,y\) with coefficients depending analytically on the small parameter \(\varepsilon\), \(n=\max \{\deg f(x,y,\varepsilon),\deg g(x,y,\varepsilon)\}\), of the polynomial Hamiltonian vector field \(X_H=H_y\frac{\partial}{\partial x}-H_x\frac{\partial}{\partial y}\), where the Hamiltonian \(H(x,y)=\frac{1}{2}y^2+U(x)\) has one center and one cuspidal loop and \(\deg U(x)=4\). Here, an upper bound for the number of zeros of the \(k\)th-order Melnikov function \(M_k(h)\) for arbitrary polynomials \(f(x,y,\varepsilon)\) and \(g(x,y,\varepsilon)\) is found in terms of \(k\) and \(n\). This estimate allows one to determine the number of limit cycles of \(X_{\varepsilon}\) emerging from periodic orbits of the unperturbed Hamiltonian system \(X_H\).
polynomial Hamiltonian vector field, limit cycle, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), Abelian integrals, Bifurcations of limit cycles and periodic orbits in dynamical systems, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, \(k\)th-order Melnikov function, cuspidal loop, weakened Hilbert's 16th problem
polynomial Hamiltonian vector field, limit cycle, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), Abelian integrals, Bifurcations of limit cycles and periodic orbits in dynamical systems, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, \(k\)th-order Melnikov function, cuspidal loop, weakened Hilbert's 16th problem
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