MATHEMATICS

Yu. I. NEIMARK, L. P. SHILNIKOV

ON ONE CASE OF THE BIRTH OF PERIODIC MOTIONS

(Presented by Academician A. Yu. Ishlinskii, 28 VII 1964)

Suppose that, for the dynamical system under consideration, depending on a parameter \(\mu\), when \(\mu=0\) there is a phase trajectory \(\Gamma_0\) issuing from a simple saddle equilibrium state \(O\) and returning to it. We shall assume that the following conditions are satisfied:

1) In some domain \(G\) containing the equilibrium state \(O\), the equations of motion are written in the form:

\[ \dot{x}_i=\lambda_i(\mu)x_i \qquad (i=1,2,\ldots,n), \tag{1} \]

where \(\operatorname{Re}\lambda_m \le \cdots \le \operatorname{Re}\lambda_2 \le \operatorname{Re}\lambda_1<0<\lambda_n<\operatorname{Re}\lambda_{n-1}\le \cdots \le \operatorname{Re}\lambda_{m+1}\), and \(\lambda_i(\mu)\) are sufficiently smooth functions of the parameter \(\mu\).

2) The phase trajectory \(\Gamma_0\), as \(t\to-\infty\), enters the saddle \(O\), tangent to the leading axis \(Ox_n\).

3) In the domain \(G\) the trajectory \(\Gamma_0\) can be intersected by sufficiently smooth cross-section surfaces \(S\) \((S(x_1,\ldots,x_m)=0)\) and \(\bar S\) \((x_n=d,\ d>0)\) in such a way that the point mapping, generated by the phase trajectories of the system, of the surface \(\bar S\) into \(S\) in neighborhoods of the points of intersection of \(\bar S\) and \(S\) with \(\Gamma_0\) exists and depends sufficiently smoothly on the variables \(x_1,x_2,\ldots,x_{n-1}\) and the parameter \(\mu\) for all \(|\mu|<\mu_0\), and \(\partial S/\partial x_1\ne 0\).

The problem of the present work is to clarify the conditions under which, when the parameter \(\mu\) is varied from zero, a periodic motion is born from the phase trajectory \(\Gamma_0\). The results obtained are formulated in Theorems 1—4*.

1. Denote by \(M^0(x_1^0,\ldots,x_m^0,0,\ldots,0)\) and \(\bar M^0(0,\ldots,0,\bar x_{m+1}^0,\ldots,\bar x_n^0)\) the points of intersection of \(\Gamma_0\) with the cross-section surfaces \(S\) and \(\bar S\). Let \(M(x_1,\ldots,x_n)\) be a point on the surface \(S\), sufficiently close to \(M^0\). The point mapping of the surface \(S\) into \(\bar S\), according to (1), can be written in the form

\[ \bar x_j=x_j e^{\lambda_j\bar t} =x_j\left(\frac{x_n}{d}\right)^{-\lambda_j/\lambda_n} \qquad (j=1,2,\ldots,n-1), \tag{2} \]

where \(\bar t\) is the root of the equation \(d=x_n e^{\lambda_n\bar t}\). We denote the mapping (2) by \(T_0\). Obviously, \(T_0\) is defined for sufficiently small \(x_n>0\). By assumption, the point \(\bar M\) on the surface \(\bar S\), lying in some \(\varepsilon\)-neighborhood of the point \(\bar M^0\), is transformed into the point \(\tilde M\) of the surface \(S\) so that

\[ \tilde x_j=f_j(\bar x_1,\ldots,\bar x_{n-1},\mu) \qquad (j=1,\ldots,n), \tag{3} \]

where \(f_j\) are sufficiently smooth functions of their variables and \(S(\tilde x_1,\ldots,\tilde x_m)=0\). We denote the mapping (3) of the surface \(\bar S\) into \(S\) by \(T_1\). Note that, for \(\mu=0\), \(M^0=T_1\bar M^0\).

Consider the mapping \(T=T_1T_0\)

\[ \tilde x_j = f_j\left( x_1\left(\frac{x_n}{d}\right)^{\nu_1}, \ldots, x_{n-1}\left(\frac{x_n}{d}\right)^{-\nu_{n-1}}, \mu \right), \tag{4} \]

\[ \text{* For a second-order system, the conditions for the birth of a limit cycle from a separatrix loop were found in the works of A. A. Andronov and E. A. Leontovich }(^{1,2}).\ \text{The case of arbitrary }n\text{ and }m=n-1\text{ was considered in a work of L. P. Shilnikov }(^{3}). \]

where \(v_j(\mu)=-\lambda_j\lambda_n^{-1}\) for \(j=1,2,\ldots,m\) and \(v_j(\mu)=+\lambda_j\lambda_n^{-1}\) for \(j=m+1,\ldots,n-1\) (\(\operatorname{Re} v_j>0\) for all \(1\le j\le n-1\)) and \(S(x_1,\ldots,x_m)=0\). The mapping \(T\) is defined for all points \(M\) for which \(\rho(TM,\overline M^{0})<\varepsilon\). Introduce new variables \(\xi_1,\ldots,\xi_n\) according to the formulas

\[ x_i=x_i^0+\xi_i,\qquad x_j\left(\frac{x_n}{d}\right)^{-v_j}=\bar x_j^0+\xi_j,\qquad x_n=\xi_n, \tag{5} \]

\[ i=1,\ldots,m;\quad j=m+1,\ldots,n-1. \]

In the new variables \(\xi_1,\ldots,\xi_n\), the coordinates of the fixed points
\(M^*(\xi_1^*,\ldots,\xi_n^*)\) of the transformation \(T\) will satisfy the system

\[ \xi_i^* = f_i\left( (x_1^0+\xi_1^*)\left(\frac{\xi_n^*}{d}\right)^{v_1}, \ldots,\bar x_{n-1}^0+\xi_{n-1}^*;\mu \right)-x_i^0,\qquad i=1,\ldots,m; \]

\[ (\bar x_j^0+\xi_j^*)\left(\frac{\xi_n^*}{d}\right)^{v_j} = f_j\left( (x_1^0+\xi_1^*)\left(\frac{\xi_n^*}{d}\right)^{v_1}, \ldots,\bar x_{n-1}^0+\xi_{n-1}^*;\mu \right), \]

\[ j=m+1,\ldots,n-1; \tag{6} \]

\[ \xi_n^* = f_n\left( (x_1^0+\xi_1^*)\left(\frac{\xi_n^*}{d}\right)^{v_1}, \ldots,\bar x_{n-1}^0+\xi_{n-1}^*;\mu \right). \]

The closeness of the point \(M^*\) to \(M^0\) and of \(TM^*\) to \(\overline M^0\) will hold if \(\xi_n^*>0\) and if \(\xi_1^*,\ldots,\xi_n^*\) are sufficiently small. Thus the problem of the birth of a periodic motion from the phase trajectory \(\Gamma_0\) reduces to finding conditions under which the mapping \(T\) found by us has a fixed point \(M^*\) tending to \(M^0\) as \(\mu\to0\).

1. Let us proceed to finding the fixed points of the transformation \(T\). In view of the smallness of the quantities \(\xi_1,\ldots,\xi_n\), the system (6) can be written in the form

\[ \sum_{p=1}^{m} \frac{\partial S(x_1^0,\ldots,x_m^0)}{\partial x_p}\,\xi_p^*+\cdots=0, \]

\[ \xi_i^* = A_i(\mu)+ \sum_{p=1}^{m} B_{ip}(x_p^0+\xi_p^*)\left(\frac{\xi_n^*}{d}\right)^{v_p} + \sum_{q=m+1}^{n-1}B_{iq}\xi_q^*+\cdots,\qquad i=2,\ldots,m; \]

\[ (\bar x_j^0+\xi_j^*)\left(\frac{\xi_n^*}{d}\right)^{v_j} = A_j(\mu)+ \sum_{p=1}^{m} B_{jp}(x_p^0+\xi_p^*)\left(\frac{\xi_n^*}{d}\right)^{v_p} + \]

\[ + \sum_{q=m+1}^{n-1}B_{jq}\xi_q^*+\cdots,\qquad j=m+1,\ n-1; \tag{7} \]

\[ \xi_n^* = A_n(\mu)+ \sum_{p=1}^{m} B_{np}(x_p^0+\xi_p^*)\left(\frac{\xi_n^*}{d}\right)^{v_p} + \sum_{q=m+1}^{n-1}B_{nq}\xi_q^*+\cdots, \]

where

\[ A_k(\mu)= \begin{cases} f_k(0,\ldots,0,\bar x_{m+1}^0,\ldots,\bar x_{n-1}^0;\mu)-x_k^0, & k=2,\ldots,m,\\ f_k(0,\ldots,0,\bar x_{m+1}^0,\ldots,\bar x_{n-1}^0;\mu), & k=m+1,\ldots,n; \end{cases} \]

\[ B_{ks}(\mu)= \frac{ \partial f_k(0,\ldots,0,\bar x_{m+1}^0,\ldots,\bar x_{n-1}^0;\mu) }{ \partial x_s }. \]

The Jacobian of the first \(n-1\) equations of this system at the point
\(\xi_1^*=\cdots=\xi_n^*=0\) is nonzero for all sufficiently small \(\mu\), if

\[ \Delta(\mu)= \left| \begin{array}{ccc} B_{m+1\,m+1} & \cdots & B_{m+1\,n-1}\\ \cdots & \cdots & \cdots\\ B_{n-1\,m+1} & \cdots & B_{n-1\,n-1} \end{array} \right| \tag{8} \]

for \(\mu=0\) is different from zero. Assuming that \(\Delta(0)\ne0\), from the first \(n-1\) equations (7) one can find \(\xi_1^*,\ldots,\xi_{n-1}^*\) in the form

\[ \xi_j^*=\alpha_j(\mu)+\sum \varphi_{js}\left(\frac{\xi_n}{d}\right)^{\nu_s(\mu)}+\cdots,\qquad j=1,\ldots,n-1 . \tag{9} \]

After substituting the found values \(\xi_1^*,\ldots,\xi_{n-1}^*\) into the last equation (7), we arrive at an equation for \(\xi_n^*\) of the form

\[ \xi_n=\Phi_n(\mu)+\sum_{s=1}^{m} C_s(\mu)\left(\frac{\xi_n}{d}\right)^{\nu_s(\mu)} +O\left[\left(\sum_{s=1}^{m}\left(\xi_n^{2\nu_s}+\xi_n^2\right)\right)^{1/2}\right], \tag{10} \]

where

\[ \Phi_n(\mu)=\Delta^{-1}(\mu) \left| \begin{array}{cc} \Delta(\mu) & A_{m+1}\\ & \cdots\\ & A_{n-1}\\ B_{n\,m+1}\ldots B_{n\,n-1} & A_n \end{array} \right|, \]

\[ C_s(\mu)=\frac{x_s^0}{\Delta(\mu)} \left| \begin{array}{cc} \Delta(\mu) & B_{m+1\,s}\\ & \cdots\\ & B_{n-1\,s}\\ B_{n\,m+1}\ldots B_{n\,n-1} & B_{n\,s} \end{array} \right|. \]

Let us note that in the case \(m=n-1\), \(\Phi_n(\mu)=A_n(\mu)\), where \(A_n(\mu)\) is the \(n\)-th coordinate of the point of intersection with the surface \(S\) of the phase curve issuing from the equilibrium state \(O\) in the region \(x_n>0\).

To each solution \(\xi_n^*(\mu)\) of equation (10) that tends to zero on the right as \(\mu\) tends monotonically to zero there corresponds the birth of a periodic motion from \(\Gamma_0\) under the reverse change of the parameter.

Case 1. Let \(\operatorname{Re}\nu_s(0)>1\) for \(s=1,\ldots,m\). In this case equation (10) has the unique solution \(\xi_n^*=\Phi_n(\mu)+\cdots\), tending to zero together with \(\mu\). This solution will correspond to a periodic motion if \(\Phi_n(\mu)>0\).

Case 2. Let \(\operatorname{Re}\nu_1(0)<1\). If \(\nu_1(0)\) is real and, for sufficiently small \(\mu>0\) (\(\mu<0\)), \(\Phi_n(\mu)C_1(0)<0\), then equation (10) has a unique solution tending to zero as \(\mu\to0\). On the contrary, when \(\nu_1(0)\) is a complex number, the number of solutions of equation (10) increases without bound as \(\mu\to0\). However, in this case there are no solutions tending to zero as \(\mu\to0\), and therefore the birth of a periodic motion from the closed phase trajectory \(\Gamma_0\) does not take place.

1. To investigate the stability of the fixed point \(M^*\) of the mapping \(T\), we study the behavior of the roots of the characteristic equation

\[ \chi(z)= \tag{11} \]

\[ \left| \begin{array}{ccc} \bar B_{ip}^*\left(\dfrac{\xi_n^*}{d}\right)^{\nu_1}-\delta_{ip}z & B_{iq}^* & \displaystyle \sum_{s=1}^{m} B_{is}^*(x_s^0+\xi_s^*)\left(\dfrac{\xi_n^*}{d}\right)^{\nu_s-1} \\[1.2em] \bar B_{jp}^*\left(\dfrac{\xi_n^*}{d}\right)^{\nu_1} & B_{iq}^*-\delta_{jq}\left(\dfrac{\xi_n^*}{d}\right)^{\nu_p}z & \displaystyle \sum_{s=1}^{m} B_{js}^*(x_s^0+\xi_s^*)\left(\dfrac{\xi_n^*}{d}\right)^{\nu_s-1} -(x_j^0+\xi_j^*)\left(\dfrac{\xi_n^*}{d}\right)^{\nu_j-1}z \\[1.2em] \bar B_{np}^*\left(\dfrac{\xi_n^*}{d}\right)^{\nu_1} & B_{nq}^* & \displaystyle \sum_{s=1}^{m} B_{ns}^*(x_s^0+\xi_s^*)\nu_s\left(\dfrac{\xi_n^*}{d}\right)^{\nu_s-1}-z \end{array} \right|=0, \]

\[ i=2,\ldots,m;\quad p=2,\ldots,m;\quad j=m+1,\ldots,n-1;\quad q=m+1,\ldots,n-1, \]

where
\[ B_{ks}^{*}=\left(\frac{\partial f}{\partial x_s}\right)_{T_0M^{*}},\qquad \bar B_{jp}^{*}=\left(\frac{\xi_n^{*}}{d}\right)^{\nu_p-\nu_1} B_{jp}^{*} -B_{j1}^{*}\left[\frac{\partial S}{\partial x_p}\right]_{M^{*}} \left[\frac{\partial S}{\partial x_1}\right]_{M^{*}}^{-1}. \]

For \(m=n-1\) and \(\operatorname{Re}\nu_s(0)>1\), all elements of the determinant \(\chi(0)\) vanish together with \(\mu\), while the coefficient of \(z^{n-1}\) is equal to \((-1)^{n-1}\). In view of this, all roots of equation (11) tend to zero as \(\mu\to0\), and therefore in the case under consideration the periodic motion that is born is stable. Conversely, if \(\nu_1(0)<1\) and \(C_1(0)\ne0\), then the coefficient of \(z^{n-2}\) tends to infinity as \(\mu\to0\). Consequently, instability occurs in this case.

When \(n-m-1>0\), because the coefficient of \(z^{n-1}\) tends to zero as \(\mu\to0\), and because the coefficient of \(z^m\) is representable in the form \(\Delta(0)+o(\mu)\) in the case when \(\operatorname{Re}\nu_i>1,\ i=1,\ldots,m\), and the coefficient of \(z^{m-1}\) is representable in the form
\[ [\nu_1 C_1+o(\mu)]\left(\frac{\xi_n^{*}}{d}\right)^{\nu_1-1}, \]
when \(\nu_1(0)<1\), it follows that equation (11) has a root \(z\) for which \(|z|\to\infty\) as \(\mu\to0\).

1. The results obtained can be formulated as the following theorems:

Theorem 1. If \(-\operatorname{Re}\lambda_i(0)>\lambda_n(0)\) \((i=1,2,\ldots,n-1)\), then for sufficiently small \(\mu>0\) \((\mu<0)\), for which
\[ f_n(0,\ldots,0,\bar x_{m+1}^{0},\ldots,\bar x_{n-1}^{0},\mu)>0, \]
there is born from \(\Gamma_0\) only one periodic motion, and it is stable.

Theorem 2. If \(-\operatorname{Re}\lambda_i(0)>\lambda_n(0)\) \((i=1,\ldots,m)\), \(m<n-1\), and \(\Delta(0)\ne0\), then for sufficiently small \(\mu>0\) \((\mu<0)\), for which \(\Phi_n(\mu)>0\), there is born from \(\Gamma_0\) a unique and unstable periodic motion.

Theorem 3. If \(-\lambda_1(0)<\lambda_n(0)\), \(\Delta(0)\ne0\), and \(C_1(0)\ne0\), then for sufficiently small \(\mu>0\) \((\mu<0)\), for which \(C_1(0)\Phi_n(\mu)<0\), there is born from \(\Gamma_0\) only one periodic motion, and it is unstable*.

Theorem 4. In the case when \(\lambda_1(0)\) is complex, \(\Gamma_0\) tends as \(t\to\infty\) into the saddle \(O\), touching the leading plane \(x_1x_2\), and \(C_1(0)\ne0\), no birth of a periodic motion from \(\Gamma_0\) takes place, despite the fact that in this case, for sufficiently small \(\mu\), there are periodic motions in a neighborhood of \(\Gamma_0\), and their number grows without bound as \(\mu\to0\).

Research Physico-Technical Institute
at Gorky State University
named after N. I. Lobachevsky

Received
18 VII 1964

CITED LITERATURE

1. A. A. Andronov, E. A. Leontovich, Uch. zap. Gorkovsk. gos. univ., vol. 6, 3 (1937).
2. A. A. Andronov, E. A. Leontovich, Matem. sborn., 48 (90), 3, 335 (1959).
3. L. P. Shilnikov, Matem. sborn., 61 (104), 4, 443 (1963).

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* Theorems 2 and 3 are also valid in the case when \(\Gamma_0\) enters the saddle as \(t\to-\infty\), touching not the leading coordinate axis corresponding to the simple root \(\lambda_j\) \((m+1\le j<n)\).