On the stability of double homoclinic and heteroclinic cycles.
The authors study the stability of a polycycle \(L\) with two regular sides, \(L_1\) and \(L_2,\) and two hyperbolic corners, \(s_1\) and \(s_2\) (which eventually may coincide), of a \(C^5\) planar vector field \(X.\) It is not restrictive to assume that it is oriented clockwise and that the omega limit set of the side \(L_1\) is \(s_2.\) Denote by \(\lambda_i<0<\mu_i\) the eigenvalues of \(DX(s_i)\) and by \(r_i=-\lambda_i/\mu_i\) the hyperbolicity ratio of \(s_i.\) It is well known that \(L\) is stable (resp. unstable) if \(r_1r_2>1\) (resp. \(r_1r_2<1\)). It is also known that, when \(r_1=r_2=1\), the stability of \(L\) is given by \(\sigma:=\sigma_1+\sigma_2,\) with \(\sigma_i=\int_{-\infty}^\infty \text{div\,} X((x_i(t),y_i(t)))\,dt\) and \(L_i=\{(x_i(t),y_i(t)),\,t\in\mathbb{R}\}.\) The authors consider the case \(r_1=r_2=1\) and \(\sigma=0.\) Concretely, they prove that, under the above hypotheses, the stability of \(L\) is given by the sign of \(R_1+R_2\mu_1e^{\sigma_1}/\mu_2,\) where the value \(R_i\) is the so-called first saddle value of \(X\) at the weak saddle \(s_i.\) These saddle values are obtained from the terms of degree smaller or equal than 3 in the Taylor expansion of \(X\) at the saddle points. This result is applied to obtain the stability of the polycycle \(H=0\) of two systems of the form \(\dot x= H_y+aH_xH,\) \(\dot y= -H_x-H_yH.\) The first one with \(H=(y^2-x^2)/2+x^4/4\) and the second one with \(H=x(y^2+x-1).\)
scientific article; zbMATH DE number 1903499
scientific article