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Normal form of a planar autonomous system and critical points of periods of closed orbits. (Chinese) Zbl 0744.34041

Assume \(O(0,0)\) is a generalized center of the complex analytic system \[ {dZ\over dT}=z+\sum^ \infty_{\alpha+\beta \geq 2}a_{\alpha\beta}z^ \alpha w^ \beta,\qquad {dw\over dT}=-w-\sum^ \infty_{\alpha+\beta\geq z}b_{\alpha\beta}W^ \alpha z^ \beta,(1) \] i.e., the critical point quantities (see [1]: Acta Acad. Sin. A3, 245-255 (1989)] of all order at \(O\) are zero. (1) can be transformed by the canonical transformation \[ z=\varphi+\sum^ \infty_{k=2}\sum_{\alpha+\beta=k}A_{\alpha\beta}\varphi^ \alpha \psi^ \beta,\qquad w=\psi+\sum^ \infty_{k=2}\sum_{\alpha+\beta=k}B_{\alpha\beta}\psi^ \alpha\varphi^ \beta(2) \] into the normal form \[ d\varphi/dT=\varphi\sum^ \infty_{k=0}p_{2k+1}\varphi^ k\psi^ k,\qquad d\psi/dT=-\psi\sum^ \infty_{k=0}p_{2k+1}\varphi^ k\psi^ k,(3) \] where \(p_ 1=1\) and \(A_{k+1,k}=B_{k+1,k}=0\). If \[ p_ 3=p_ 5=\dots=p_{2k+1}=0,\qquad p_{2(k+1)+1}\neq 0,(4) \] \(O\) is called a generalized center of order \(k\); if \(p_{2k+1}=0\) for all \(k\in Z^ +\), \(O\) is an isodonous center. When (1) is derived from (5) \(dx/dt=y-F(x,y)\), \(dy/dt=x+G(x,y)\) by the linear transformation \(z=x+iy\), \(w=x-iy\), \(idt=dT\), \(idt=dT\), \(i=\sqrt{-1}\), so that \(a_{\alpha\beta}=b_{\alpha\beta}\), then (5) and (2) are called adjoint systems, and (3) becomes (6) \(du/dt=-v\sum^ \infty_{k=0}p_{2k+1}(u^ 2+v^ 2)^ k\), \(dv/dt=u\sum^ \infty_{k=0}p_{2k+1}(u^ 2+v^ 2)^ k\), or in polar coordinates: (7) \(d\gamma/dt=0\), \(d\theta/dt=\sum^ \infty_{k=0}p_{2k+1}\gamma^{2k}\), for which we see the period of the periodic orbit \(\gamma=C\) is \(Pe(\gamma)=2\pi[1+Q_ 3\gamma^ 2+Q_ 3\gamma^ 4+\cdots]\), where \(Q_ i\)’s are polynomials of the \(p_{2k+1}\)’s. Under (4) we have \(P_ e(\gamma)=2\pi[1- p_{2(k+1)+1}\gamma^{2(k+1)}+\hbox{h.o.t.}].\) In this paper, formulae for the calculation of \(P_{2k+1}\)are given. Moreover, from known conditions [1] for \(O\) to be a generalized center of (8) \(dz/dT=z+a_{30}z^ 3+a_{21}z^ 2w+a_{12}zw^ 2+a_{03}w^ 3\), \(dw/dT=-w-b_{30}w^ 3-b_{21}w^ 2z-b_{12}wz^ 2-b_{03}z^ 3\) all conditions are obtained for \(O\) to be a generalized isoclonous center of (8). The number of critical points of the periods of closed orbits of the system adjoint to (8): (9) \(dx/dt=-y-c_ 1x^ 3-c_ 2x^ 2y-c_ 3xy^ 2-c_ 4y^ 3\), \(dy/dt=x+d_ 1x^ 3+d_ 2x^ 2y+d_ 3xy^ 2+d_ 4y^ 3\), i.e., the number of limit cycles that can be generated when (9) is perturbed is studied. It is proved that the number is 2 when \(O\) is a weak generalized center but not isodonous.

MSC:

34C25 Periodic solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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