Poincaré components of quadratic systems on periodic annular domains bounded by hyperbolas and equatorial arcs and their applications. (Chinese) Zbl 0938.34021
The authors study the Poincaré bifurcation:
\[
\dot x= xy+ \mu(\ell_1 x^2+ m_1 xy+ n_1y^2+ p_1x+ q_1y+ f_1),\tag{1}
\]
\[ \dot y= \textstyle{{1\over 2}}-{1\over 2} x^2+ 2y^2+ \mu(\ell_2x^2+ m_2 xy+ n_2y^2+ p_2x+ q_2y+ f_2), \] of the integrable system: \[ \dot x= xy,\quad \dot y= \textstyle{{1\over 2}}- {1\over 2} x^2+ 2y^2.\tag{2} \] System (2) has two centers \(A(-1,0)\) and \(B(1,0)\), a straight line integral \(x= 0\) and two infinite separatrix cycles formed by the hyperbola \(2x^2- 4y^2= 1\) and two arcs of the equator.
They prove that for suitably chosen coefficients \(\ell_i,m_i,\dots, f_i\) and \(0<|\mu|\ll 1\), system (1) has an approximate system which can have either two limit cycles, or one semistable limit cycle, or one limit cycle and one infinite separatrix cycle in the right half-plane \(x>0\), around the focus \(B(1,0)\).
\[ \dot y= \textstyle{{1\over 2}}-{1\over 2} x^2+ 2y^2+ \mu(\ell_2x^2+ m_2 xy+ n_2y^2+ p_2x+ q_2y+ f_2), \] of the integrable system: \[ \dot x= xy,\quad \dot y= \textstyle{{1\over 2}}- {1\over 2} x^2+ 2y^2.\tag{2} \] System (2) has two centers \(A(-1,0)\) and \(B(1,0)\), a straight line integral \(x= 0\) and two infinite separatrix cycles formed by the hyperbola \(2x^2- 4y^2= 1\) and two arcs of the equator.
They prove that for suitably chosen coefficients \(\ell_i,m_i,\dots, f_i\) and \(0<|\mu|\ll 1\), system (1) has an approximate system which can have either two limit cycles, or one semistable limit cycle, or one limit cycle and one infinite separatrix cycle in the right half-plane \(x>0\), around the focus \(B(1,0)\).
Reviewer: Ye Yanqian (Nanjing)
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 |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 37J20 | Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems |
