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On the control of parameters of distributions of limit cycles for a \(Z_2\)-equivariant perturbed planar Hamiltonian polynomial vector field. (English) Zbl 1077.34037

The authors study the Hopf and saddle-connection bifurcations of limit cycles in the system \[ \dot{x}=H_y,\qquad \dot{y}=-H_x+\varepsilon Q(x,y),\tag \(X_\varepsilon\) \] where \[ H=\frac12(x^2+y^2)-\frac58x^4-\frac{5}{32}y^4+\frac16x^6+\frac{1}{96}y^6, \] \(\varepsilon\) is a small parameter and \[ Q=y(\lambda+\mu x^2+ry^2+kx^4+nx^2y^2+my^4). \] The unperturbed system \(X_0\) possesses a rotational symmetry of order 2 and has a complicated phase portrait, containing 13 centres, 12 saddles and 19 different period annuli.
Using numerical and symbolic calculations, the authors show that for small \(\varepsilon\) at least 23 limit cycles bifurcate simultaneously near the centers and separatrix connections of \(X_0\). They also point out that some of the results on the same system obtained earlier by G. Chen, Y. Wu and X. Yang [J. Aust. Math. Soc. 73, 37–53 (2002; Zbl 1017.34028)] are incorrect (one can hardly check up this because the results in both papers are based mainly on heavy computer calculations).

MSC:

34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations

Citations:

Zbl 1017.34028
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References:

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