UDC 517.9

MATHEMATICS

L. P. SHILNIKOV

ON THE BIRTH OF PERIODIC MOTION FROM A TRAJECTORY ISSUING FROM AN EQUILIBRIUM STATE OF SADDLE–SADDLE TYPE AND RETURNING TO IT

(Presented by Academician A. Yu. Ishlinskii, November 22, 1965)

1. Consider a system of differential equations of order \((m+n+1)\)

\[ dx/dt=P(x,y,z,\mu),\qquad dy/dt=Q(x,y,z,\mu), \]
\[ dz/dt=R(x,y,z,\mu), \tag{1} \]

where \(P(P_1,P_2,\ldots,P_m)\), \(Q(Q_1,Q_2,\ldots,Q_n)\), and \(R\) are analytic or sufficiently smooth functions of the variables \(x(x_1,x_2,\ldots,x_m)\), \(y(y_1,y_2,\ldots,y_n)\), \(z\), and of the parameter \(\mu\) in some domain \(G\times(-\mu_0,\mu_0)\). Suppose that for \(\mu=0\) system (1) has an equilibrium state \(O\in G\), at which the characteristic equation has \(m\) roots with negative real parts, \(n\) roots with positive real parts, and one zero root.

By means of a linear nonsingular change of variables, system (1) can be brought to the form

\[ dx/dt=Ax+P_0(x,y,z,\mu),\qquad dy/dt=By+Q_0(x,y,z,\mu), \]
\[ dz/dt=R_0(x,y,z,\mu), \tag{2} \]

where \(P_0,Q_0,R_0\), for \(\mu=0\), vanish at \(O(0,0,0)\) together with their first derivatives; \(A\) is a matrix of order \(m\), whose characteristic polynomial has roots \(\lambda_1,\lambda_2,\ldots,\lambda_m\) with negative real parts, and \(B\) is a matrix of order \(n\), whose characteristic polynomial has roots \(\gamma_1,\gamma_2,\ldots,\gamma_n\) with positive real parts. Let

\[ x=u(z),\qquad y=v(z) \]

be a solution of the system

\[ Ax+P_0(x,y,z,0)=0,\qquad By+Q_0(x,y,z,0)=0 \]

for sufficiently small \(z\). Suppose that in the expansion

\[ R_0(u(z),v(z),z,0)=l_k z^k+l_{k+1}z^{k+1}+\cdots \tag{3} \]

the first nonzero quantity \(l_k\) has an even index and, for definiteness, is positive.

The behavior of phase trajectories in a neighborhood of such a complex equilibrium state was studied in the work of R. M. Mintz (¹); she called the equilibrium state itself an equilibrium state of type \(C\). This equilibrium state is formed by the coalescence of \(2k\) simple equilibrium states of saddle type; therefore, it seems to us more natural to call \(O\) a complex equilibrium state of saddle–saddle type.

Let us describe the behavior of integral curves in a neighborhood of \(O\). It is characterized first of all by the existence of an \((n+m)\)-dimensional separatrix

surface \(\pi\), passing through \(O\) and tangent to the \(x\)- and \(y\)-axes. This surface divides a neighborhood of \(O\) into two regions. In the region containing the half-axis \(z \leqslant 0\), there exists a unique \((m+1)\)-dimensional integral \(O^+\)-semisurface (a semisurface consisting of \(O^+\)-curves), which we shall denote by \(\pi^+\). In the region containing the half-axis \(z \geqslant 0\), there exists a unique \((n+1)\)-dimensional integral \(O^-\)-semisurface (a semisurface consisting of \(O^-\)-curves), which we shall denote by \(\pi^-\). The boundaries of the semisurfaces \(\pi^+\) and \(\pi^-\), which we shall denote by \(\Pi^+\) and \(\Pi^-\), are manifolds of dimensions \(m\) and \(n\), respectively. They lie in \(\pi\) and intersect at the point \(O\). All integral curves, except those that lie in \(\pi^+\) and \(\pi^-\), pass by \(O\).

The purpose of the present paper is to prove the following theorem.

Theorem. Suppose that for \(\mu = 0\) system (1) has the following properties: 1) there exists a trajectory \(\Gamma_0\) issuing from \(O\) and returning to it again as \(t \to +\infty\); 2) \(\Gamma_0\) does not lie in \(\Pi^+\) and \(\Pi^-\); 3) \(\pi^+\) and \(\pi^-\) intersect along \(\Gamma_0\), without being tangent.*

Then, for all sufficiently small \(\mu \ne 0\) for which the composite equilibrium state of saddle–saddle type disappears, a unique periodic motion of saddle type is born from \(\Gamma_0\).

The conditions for the disappearance of an equilibrium state of saddle–saddle type are completely analogous to the conditions for the disappearance of a composite equilibrium state of saddle–node type \((^4)\).

The proof of the theorem is based on the method of point mappings and consists in proving the existence of a fixed point of a mapping \(T\) of the plane \(z=-d\) (\(d>0\)) into itself.

We shall seek the mapping \(T\) in the form \(T=T_1T_0\), where \(T_0\) is the mapping of the plane \(z=-d\) into \(z=d\) in a neighborhood of the origin, and \(T_1\) is the mapping of \(z=+d\) into \(z=-d\) in a neighborhood of the global arc \(\Gamma_0\).

2. Construction of the mapping \(T_0\). Let us first consider system (2) for \(\mu=0\). As follows from the results of V. A. Pliss \((^2,^3)\), in a neighborhood of the origin \(O\) there exist two integral surfaces

\[ y=\psi(x,z), \qquad x=\varphi(y,z), \tag{4} \]

whose dimensions are \(m+1\) and \(n+1\), respectively. The functions \(\psi\) and \(\varphi\) are sufficiently smooth and vanish at the origin together with their first derivatives. On these surfaces \(O\) will be a composite equilibrium state of saddle–node type. From the uniqueness of the integral \(O^+\)-semisurface \(\pi^+\) and the \(O^-\)-semisurface \(\pi^-\) it follows that, in the region to the left of \(\pi\), the surface \(y=\psi(x,z)\) coincides with \(\pi^+\), while in the region to the left of \(\pi\) the surface \(x=\varphi(y,z)\) coincides with \(\pi^-\).

By means of a certain sufficiently smooth change of variables in a sufficiently small neighborhood of the origin, system (2) can be written in the form

\[ \begin{aligned} d\xi/dt &= A\xi + P'(\xi,\eta,z)\xi + \mu \alpha(z,\mu)R_0(z,\mu),\\ d\eta/dt &= B\eta + Q'(\xi,\eta,z)\eta + \mu \beta(z,\mu)R_0(z,\mu),\\ dz/dt &= R_0(z,\mu) + R_1'(\xi,\eta,z,\mu)\xi + R_2'(\xi,\eta,z,\mu)\eta, \end{aligned} \tag{5} \]

where \(P'\), \(Q'\), \(R_1'\), \(R_2'\) are small for sufficiently small \(\xi,\eta,z,\mu\), while \(R_0(z,0)=l_k z^k+\cdots\). In the new variables the equations of the surfaces (4) will be, respectively, \(\eta=0\) and \(\xi=0\), and the equations of \(\Gamma_0\) will be \(\xi=0,\ \eta=0\). It is obvious that the condition for the disappearance of the saddle–saddle will consist in the fulfillment of the inequality \(R_0(z,\mu)>0\). Suppose, for definiteness, that \((0,\bar\mu_0)\) is an interval of values of \(\mu\) for which this inequality is satisfied. A value of \(\mu\) from this interval will be denoted by \(\bar\mu\).

\[ \text{* From the works of V. A. Pliss }(^2,^3)\text{ it follows that } \pi^+ \text{ and } \pi^- \text{ are smooth surfaces.} \]

Introduce into consideration the closed surface \(S_{\bar\mu}\), consisting of the surfaces

\[ V \equiv \|\xi\|^{2}-R_{0}^{2}(z,\bar\mu)=0,\qquad W \equiv \|\eta\|^{2}-R_{0}^{2}(z,\bar\mu)=0, \]

\[ S_{0}: z=-d,\qquad S_{1}: z=d. \tag{6} \]

It is easy to see that inside \(S_{\bar\mu}\), for sufficiently small \(d\) and sufficiently small \(\mu\), the condition \(dz/dt>0\) is satisfied. After eliminating the time \(t\), system (5) inside \(S_{\bar\mu}\) can be written in the form

\[ \frac{d\xi}{dz}= \frac{A\xi+P_{1}(\xi,\eta,z,\bar\mu)\xi+\mu\alpha(z,\bar\mu)R_{0}(z,\bar\mu)} {R_{0}(z,\bar\mu)}, \]

\[ \frac{d\eta}{dz}= \frac{B\eta+Q_{1}(\xi,\eta,z,\bar\mu)\eta+\mu\beta(z,\bar\mu)R_{0}(z,\bar\mu)} {R_{0}(z,\bar\mu)}, \]

where \(P_{1}\) and \(Q_{1}\) are small inside \(S_{\bar\mu}\) for sufficiently small \(d\) and \(\bar\mu_{0}\).

Lemma 1. For every pair \(\xi_{0}\) and \(\eta_{1}\) satisfying the conditions
\[ \|\xi_{0}\|<a<R_{0}(-d,0),\qquad \|\eta_{1}\|<b<R_{0}(d,0), \]
there exists a solution of system (7)

\[ \xi(z)=\xi(z,\xi_{0},\eta_{1},\bar\mu),\qquad \eta(z)=\eta(z,\xi_{0},\eta_{1},\bar\mu), \tag{8} \]

lying inside \(S_{\bar\mu}\) and satisfying the conditions

\[ \xi(-d)=\xi_{0},\qquad \eta(d)=\eta_{1}, \]

and moreover

\[ \|\xi(z)\|<\|\xi_{0}\|\exp\left[\lambda\int_{-d}^{z}\frac{dz}{R_{0}(z,\bar\mu)}\right]+\bar K\mu, \]

\[ \|\eta(z)\|<\|\eta_{1}\|\exp\left[\gamma\int_{+d}^{z}\frac{dz}{R_{0}(z,\bar\mu)}\right]+\bar K\mu, \tag{9} \]

where
\[ \max_{1\le i\le m}\operatorname{Re}\lambda_i<\lambda<0,\qquad 0<\gamma<\min_{1\le i\le n}\operatorname{Re}\gamma_i, \]
and \(\bar K\) is some fixed constant.

Lemma 2. The estimates

\[ \left\|\frac{\partial\xi}{\partial\xi_{0}}\right\| + \left\|\frac{\partial\eta}{\partial\xi_{0}}\right\| < \exp\left[\lambda\int_{-d}^{z}\frac{dz}{R_{0}(z,\bar\mu)}\right], \]

\[ \left\|\frac{\partial\xi}{\partial\eta_{1}}\right\| + \left\|\frac{\partial\eta}{\partial\eta_{1}}\right\| < \exp\left[\gamma\int_{+d}^{z}\frac{dz}{R_{0}(z,\bar\mu)}\right] \]

hold for all \(z\in[-d,d]\) for sufficiently small \(d\) and \(\bar\mu_{0}\).

Consider the integral curve (8). Obviously, specifying \((\xi_{0},\eta_{1})\) makes it possible to determine uniquely
\[ \xi_{1}=\xi(d,\xi_{0},\eta_{1},\bar\mu) \quad\text{and}\quad \eta_{0}=\eta(-d,\xi_{0},\eta_{1},\bar\mu). \]
The mapping of the points \(M_{0}(\xi_{0},\eta_{0})\in S_{0}\) into the points
\(M_{1}(\xi_{1},\eta_{1})\in S_{1}\) obtained in this way will be denoted by \(T_{0}\).
At the same time, note that, since

\[ \lim_{\bar\mu\to 0}\int_{-d}^{d}\frac{dz}{R_{0}(z,\bar\mu)}=+\infty, \]

we have

\[ \|\xi_{1}\|+\|\eta_{0}\| +\|\partial\xi_{1}/\partial\xi_{0}\| +\|\partial\eta_{0}/\partial\xi_{0}\| +\|\partial\xi_{1}/\partial\eta_{1}\| +\|\partial\eta_{0}/\partial\eta_{1}\| \to 0 \tag{10} \]

as \(\bar\mu\to0\).

3. Construction of the mapping \(T_{1}\). In the variables \(\xi,\eta,z\), \(\Gamma_{0}\) intersects \(S_{0}\) and \(S_{1}\) at the points \(M_{0}^{0}(0,0)\) and \(M_{1}^{1}(0,0)\). As is known, for sufficiently small
\[ 0<a<\frac12 R_{0}(-d,0) \]
one can indicate such a
\[ 0<b<\frac12 R_{0}(d,0) \]
that, for all sufficiently small \(\mu\), the trajectories passing through the points \(M_{1}\in S_{1}\), lying in the \(b\)-neighborhood of \(M_{1}^{1}\), will intersect \(S_{0}\) at points
\(\bar M_{0}(\bar\xi_{0},\bar\eta_{0})\), lying in the \(a\)-neighborhood of \(M_{0}^{0}\). The resulting map-

We shall denote the mapping by \(T_1\). In the variables under consideration it will be written in the form

\[ \begin{gathered} \bar{\xi}_0=f_1(\xi_1,\eta_1,\mu)=A_1(\mu)+A_{11}(\mu)\xi_1+A_{12}(\mu)\eta_1+\ldots,\\ \bar{\eta}_0=g_1(\xi_1,\eta_1,\mu)=A_2(\mu)+A_{21}(\mu)\xi_1+A_{22}(\mu)\eta_1+\ldots \end{gathered} \tag{11} \]

Since, by assumption, \(\pi^+\) and \(\pi^-\) intersect along \(\Gamma_0\) without tangency, \(|A_{22}(0)|\ne0\).

1. The completion of the proof of the theorem is reduced to proving the existence for the mapping \(T=T_1T_0\) of a fixed point \(M^*(\xi_0^*,\eta_0^*)\), tending to \(M_0^0\) as \(\mu\to0\). Denote by \(\sigma(a,b)\) the image of the domain \(\|\xi_0\|<a,\ \|\eta_1\|<b\) under the mapping \((\xi_0,\eta_1)\to(\xi_0,\eta_0)\), where \(\eta_0=\eta(-d,\xi_0,\eta_1,\mu)\). It is easy to see, by virtue of the choice of \(a\) and \(b\), that the mapping \(T\) is defined on \(\sigma(a,b)\in S_0\) and maps \(\sigma(a,b)\) into \(S_0\).

Let us consider the equations for finding the coordinates of a fixed point. In the variables \(\xi_0^*,\eta_1^*\) they will be written in the form

\[ \begin{gathered} F(\xi_0^*,\eta_1^*,\bar{\mu})\equiv \xi_0^*-f_1(\xi(d,\xi_0^*,\eta_1^*,\bar{\mu}),\eta_1^*,\bar{\mu})=0,\\ G(\xi_0^*,\eta_1^*,\bar{\mu})= \eta(-d_0,\xi_0^*,\eta_1^*,\bar{\mu})- g_1(\xi(d,\xi_0^*,\eta_1^*,\bar{\mu}),\eta_1^*,\bar{\mu})=0. \end{gathered} \tag{12} \]

Since, using (10), \(F\) and \(G\) can be extended continuously to the boundary \(\bar{\mu}=0\), together with their derivatives, and since for \(\bar{\mu}=0\) the system under consideration has the unique solution \(\xi_0^*=0,\ \eta_1^*=0\), with Jacobian, at \(\xi_0^*=\eta_1^*=\bar{\mu}=0\), equal to \(-|A_{22}(0)|\), it follows that for sufficiently small \(\bar{\mu}>0\) we obtain that the system (12) has a unique solution \(\xi_0^*(\bar{\mu}),\eta_1^*(\bar{\mu})\), where \(\xi_0^*(\bar{\mu})\) and \(\eta_1^*(\bar{\mu})\) tend to zero together with \(\bar{\mu}\). Consequently, the mapping \(T\) has a fixed point \(M^*(\xi_0^*,\eta_0^*)\), tending to \(M_0^0\) as \(\bar{\mu}\to0\). We shall prove that \(M^*\) is a fixed point of saddle type. Obviously, the characteristic equation, using the estimates (10), can be written in the form

\[ \alpha_n(\bar{\mu})z^{m+n}+\ldots+\alpha_1(\bar{\mu})z^{m+1} +(|A_{22}(0)|+O(\bar{\mu}))z^m+\beta_m(\bar{\mu})z^{m-1}+\ldots+\beta_1(\bar{\mu})=0, \tag{13} \]

where \(\alpha_i(\bar{\mu})\) and \(\beta_i(\bar{\mu})\) tend to zero as \(\bar{\mu}\to0\). Since the coefficient of \(z^m\) is not equal to zero for sufficiently small \(\bar{\mu}\), we obtain that equation (13) has \(m\) roots with moduli less than unity and \(n\) roots with moduli greater than unity.

From the known relation between fixed points of a mapping and periodic solutions we obtain that from \(\Gamma_0\) there is born a unique periodic motion of saddle type.

Research Institute of Applied
Mathematics and Cybernetics
at Gorky State University
named after N. I. Lobachevsky

Received
21 XI 1965

CITED LITERATURE

1. R. M. Minz, DAN, 147, 31 (1962).
2. V. A. Pliss, Izv. AN SSSR, ser. matem., 28, No. 6, 1297 (1964).
3. V. A. Pliss, Differential Equations, 1, No. 1, 17 (1965).
4. L. P. Shilnikov, DAN, 143, 289 (1962); Matem. sborn., 61 (104), 443 (1963).