MATHEMATICS

L. P. SHILNIKOV

SOME CASES OF THE BIRTH OF PERIODIC MOTIONS IN \(n\)-DIMENSIONAL SPACE

(Presented by Academician L. S. Pontryagin on 2 XI 1961)

1. One of the basic problems of the qualitative theory of differential equations is the consideration of bifurcations of states of equilibrium and periodic motions \((^{1-6})\). In the present note, conditions are established for the birth of stable periodic motions from trajectories going from a state of equilibrium back into it. The results obtained are a generalization of the corresponding results of A. A. Andronov and E. A. Leontovich \((^{2,3})\) to the case of \(n\)-dimensional space.

2. Consider a system of \(n\) differential equations

\[ \frac{dx_i}{dt}=X_i(x_1,x_2,\ldots,x_n;\mu),\qquad i=1,\ldots,n, \tag{1} \]

depending on the parameter \(\mu\) \((|\mu|<\mu_0)\). With respect to the right-hand sides, assume that they are sufficiently smooth functions of the variables \(x_1,\ldots,x_n\) and \(\mu\). Suppose that for \(\mu=0\) system (1) has an equilibrium state \(O(0,\ldots,0)\), in which the roots \(\lambda_1,\ldots,\lambda_{n-1}\) of the characteristic equation

\[ \left|\left(\frac{\partial X_i}{\partial x_j}\right)_0-\delta_{ij}\lambda\right|=0 \tag{2} \]

have real parts of one sign, and \(\lambda_n=0\). By means of a nonsingular linear change, system (1) can be reduced to the form

\[ \begin{aligned} \frac{dx_i}{dt} &= \sum_{j=1}^{n-1} a_{ij}x_j+P_i(x_1,\ldots,x_n;\mu), \qquad i=1,\ldots,n-1;\\ \frac{dx_n}{dt} &= P_n(x_1,x_2,\ldots,x_n;\mu), \end{aligned} \tag{3} \]

where the functions \(P_i(x_1,\ldots,x_n;0)\), \(i=1,2,\ldots,n\), contain no linear terms with respect to \(x_1,\ldots,x_n\).

Consider the function

\[ \overline{P}_n(x_n;\mu) = P_n(U_1+\mu u_1,\ldots,U_{n-1}+\mu u_{n-1},x_n;\mu), \]

where

\[ x_i=U_i(x_n)+\mu u_i(x_n;\mu),\qquad i=1,\ldots,n-1, \]

satisfy the system of equations

\[ \sum a_{ij}x_j+P_i(x_1,\ldots,x_n;\mu)=0,\qquad i=1,\ldots,n-1. \]

Suppose that in the expansion \(\overline{P}_n(x_n;0)=l_m x_n^m+\ldots\) the first nonzero quantity \(l_m\) \((^{7})\) has an even index and is positive (if

\(l_m<0\), then, replacing \(x_n\) by \(-x_n\), we arrive at the case under consideration). In this case the equilibrium state \(O\) is called a saddle-node.

Let, for definiteness, \(\operatorname{Re}\lambda_i<0,\ i=1,2,\ldots,n-1\). The behavior of the integral curves of system (3) for \(\mu=0\) in a sufficiently small neighborhood of \(O\) is described by the following lemma, which is a generalization of a result of R. M. Mintz \((^8)\), established for \(n=3\).

Lemma. There exists an \((n-1)\)-dimensional separatrix surface \(I_{n-1}\), tangent to the plane \(x_n=0\), consisting of \(O^+\)-curves and dividing a neighborhood of \(O\) into nodal and saddle regions. In the nodal region every integral curve is an \(O^+\)-curve, while in the saddle region all integral curves, with the exception of one \(O^-\)-curve, pass at a finite distance from the saddle-node.

The condition that system (3) have no equilibrium states in a sufficiently small neighborhood of the origin is equivalent to the definiteness and positivity of the expression

\[ \bar P_n(x_n;\mu)=l_m x_n^m(1+q_1(x))+\mu q_2(x_n;\mu). \]

In particular, if \(q_2(0,0)>0\), then as the parameter \(\mu\) is varied the following bifurcation picture occurs: two simple equilibrium states—a saddle and a node, existing for \(\mu<0\), approach each other as \(\mu\) increases and then merge into one equilibrium state (a saddle-node), which disappears for \(\mu>0\). Studying the equation \(\bar P_n(x_n;\mu)=0\), one can show that sufficiently small values of the parameter \(\mu\) for which, in some fixed neighborhood of the origin, equilibrium states will be absent form intervals (one or two), one of whose endpoints will be the point with coordinate \(\mu=0\), corresponding to the existence of the saddle-node.* We denote the values of the parameter \(\mu\) from the indicated intervals by \(\bar\mu\).

According to Lemma 1, from the saddle-node there issues only one \(O^-\)-curve. Denote it by \(\Gamma_0\).

Theorem 1. If the trajectory \(\Gamma_0\) as \(t\to+\infty\) enters the saddle-node and does not lie on the separatrix surface, then, when the saddle-node disappears (as a result of a change of the parameter), from \(\Gamma_0\) there is born only one periodic motion, stable for \(\operatorname{Re}\lambda_i<0,\ i=1,\ldots,n-1\).

The proof of the theorem is based on the method of point transformations.

Denote by \(S_0\) and \(S_1\) the surfaces \(x_n=-d\) and \(x_n=d\), where \(d>0\). For sufficiently small \(d\), the trajectory \(\Gamma_0\) will intersect \(S_0\) and \(S_1\) at the points \(M_0^0\) and \(M_1^0\). One can show that for all sufficiently small \(\bar\mu\) (in absolute value) on the surface \(S_0\) there is a \(\sigma_0\)-neighborhood of the point \(M_0^0\) such that the trajectory passing at \(t=0\) through a point \(M^0\in\sigma_0\) intersects, for some \(t=t_1\), the surface \(S_1\) at a certain point \(M^1\). The point transformation, thus established, of the domain \(\sigma_0\) onto the surface \(S_1\) will be denoted by \(T_0\). It has also been proved that as \(\bar\mu\to0\) the domain \(T_0\sigma_0\) is uniformly contracted to the point \(M_1^0\), and the norm of the matrix of the linearized transformation \(T_0\) tends to zero. Since \(\Gamma_0\) intersects the surfaces \(S_1\) and \(S_0\) without tangency, it follows from the general theorems of the theory of point transformations \((^6)\) that for every point sufficiently close to the point \(M_1^0\) one can indicate a certain point on the surface \(S_0\). Moreover, one can indicate such a \(\sigma_1\)-neighborhood of the point \(M_1^0\) on \(S_1\) that its image under the indicated correspondence, which we denote by \(T_1\), will lie inside \(\sigma_0\). Consider the transformation \(T=T_1T_0\). The transformation \(T\) is contracting for all sufficiently small \(\bar\mu\). From the contractiveness of \(T\) and the known relation \((^9)\) between periodic motions of the dynamical—

* We note that the existence of two intervals \((-\mu^*,0)\) and \((0,\mu^*)\) is possible in the case when \(x_n=\mu=0\) is an isolated solution of the equation \(\bar P_n(x_n;\mu)=0\).

mical system and fixed points of the corresponding mapping, there follows the existence of a unique and stable periodic motion.

1. Let a system of the form (1), for \(\mu=0\), have a simple equilibrium state of saddle type, in which the roots \(\lambda_1,\ldots,\lambda_{n-1}\) of the characteristic equation are distinct and have negative parts, while \(\lambda_n>0\). With respect to the functions \(X_i\) we shall assume that they are twice continuously differentiable. By a change of variables, the system under consideration, for sufficiently small \(\mu\), can be brought to the form

\[ \begin{aligned} \frac{d x_i}{dt} &= \sum_{j=1}^{n-1} a_{ij}(\mu)x_j + P_i(x_1,\ldots,x_n;\mu), \qquad i=1,\ldots,n-1;\\ \frac{d x_n}{dt} &= \lambda_n(\mu)x_n + P_n(x_1,\ldots,x_n;\mu), \end{aligned} \tag{4} \]

where the functions \(P_i(x_1,\ldots,x_n;\mu)\), \(i=1,\ldots,n\), contain no linear terms in \(x_1,\ldots,x_n\) and, moreover,

\[ P_i(0,\ldots,0;\mu)\equiv 0. \]

As is known \((^{7,\,10-12})\), in this case there exist two integral manifolds \(I_1\) and \(I_{n-1}\), on which all the \(0^{-}\)-curves of the equilibrium state under consideration are situated.

The equations of the manifold \(I_1\), consisting of two \(0^{-}\)-curves, are written in the form

\[ x_i = F_i(x_n;\mu_1), \qquad i=1,\ldots,n-1, \]

where \(\partial F_i(0;\mu)/\partial x_n=0\), \(i=1,\ldots,n-1\), and the equation of the manifold \(I_{n-1}\), consisting of \(0^{+}\)-curves, is in the form

\[ x_n = F_n(x_1,\ldots,x_{n-1};\mu), \]

where \(\partial F_n(0,\ldots,0;\mu)/\partial x_i=0\), \(i=1,\ldots,n-1\). The manifold \(I_{n-1}\) divides a sufficiently small neighborhood of the origin into two regions, which we denote by \(D^{+}\) (\(D^{+}\) contains a segment of the positive semiaxis \(O x_n\)) and \(D^{-}\) (\(D^{-}\) contains a segment of the negative semiaxis \(O x_n\)). Denote by \(\Gamma_0\) the trajectory issuing from the saddle into the region \(D^{+}\).

Theorem 2. If: 1) for \(\mu=0\) and \(t\to+\infty\) the trajectory \(\Gamma_0\) enters the saddle \(O\), and for \(\mu>0\) (\(\mu<0\)) passes by the saddle, crossing \(D^{+}\); 2) \(-\operatorname{Re}\lambda_i>\lambda_n\), \(i=1,\ldots,n-1\), then, for sufficiently small \(\mu>0\) (\(\mu<0\)), from \(\Gamma_0\) there is born only one periodic motion, stable as \(t\to+\infty\).

One can specify a surface \(S_0\) whose equation is
\(S_0(x_1,\ldots,x_{n-1})=0\), where the function \(S_0\) is twice continuously differentiable, such that \(\Gamma_0\), for sufficiently small \(\mu\), will intersect \(S_0\). Denote the point of intersection of \(\Gamma_0\) with \(S_0\) by \(M_\mu^0\), and by \(\gamma\) the intersection of the surfaces \(S_0\) and \(I_{n-1}\). Obviously, the point \(M_0^0\) (the point \(M_\mu^0\) for \(\mu=0\)) lies on \(\gamma\). It can be proved that, for every sufficiently small and fixed \(\mu\), a trajectory of system (4) passing at \(t=0\) through a point \(M^0\), close to \(M_0^0\in S_0\) and lying in \(D^{+}\), for some \(t=t_1\) again intersects the surface \(S_0\). Denote the resulting mapping of the surface \(S_0\) onto itself by \(T\). If \(-\operatorname{Re}\lambda_i>\lambda_n\), \(i=1,\ldots,n-1\), then for sufficiently small \(\mu\), at points \(M^0\) close to \(\gamma\), the norm of the linearized mapping \(T\) will be less than the number \(q\), where \(q<1\).

We note that the mapping \(T\) can be completed by continuity on the curve \(\gamma\) itself and then continued, with preservation of smoothness, to that part of the surface \(S_0\) which lies in \(D^{-}\). This extended mapping, which we denote by \(\widetilde{T}\), for \(\mu=0\) will have a fixed point \(M_0^0\). By the implicit function theorem, since the Jacobian at \(M_0^0\) is less than uni-

boundary, it follows that for sufficiently small $\mu$ the mapping $\widetilde T$ will have a unique and stable fixed point $M_\mu^*$, tending to $M_0^0$ as $\mu \to 0$. It can be proved that the point $M_\mu^*$ will lie in $D^+$ and, consequently, is a fixed point of the mapping $T$, if $M_\mu^0$ also lies in $D^+$. Using the connection between periodic motions and fixed points, we consequently arrive at Theorem 2.

Research Institute of Physics and Technology
of the Gorky State University
named after N. I. Lobachevsky

Received
2 XI 1961

REFERENCES

¹ A. A. Andronov, L. S. Pontryagin, DAN, 14, No. 5, 247 (1937).
² A. A. Andronov, E. A. Leontovich, Scientific Notes of Gorky State University, issue 6, 3 (1937); A. A. Andronova, collection Proceedings of the Academy of Sciences of the USSR, 1956, p. 188.
³ A. A. Andronov, E. A. Leontovich, Mathematical Collection, 48 (90), 3, 335 (1959).
⁴ A. A. Andronov, A. A. Vitt, S. E. Khaikin, Theory of Oscillations, 1959.
⁵ Yu. I. Neimark, Proceedings of Higher Educational Institutions, Radiophysics, No. 1, 41 (1958); No. 2, 95 (1958); No. 5–6, 146 (1958).
⁶ Yu. I. Neimark, DAN, 129, 736 (1959).
⁷ A. A. Lyapunov, The General Problem of the Stability of Motion, 1935.
⁸ R. M. Minz, DAN, 111, No. 3, 539 (1956).
⁹ A. A. Andronov, Bulletin of the Academy of Sciences of the USSR, No. 6 (1944).
¹⁰ A. Poincaré, On Curves Defined by Differential Equations, Moscow–Leningrad, 1947.
¹¹ I. G. Petrovsky, Mathematical Collection, 41, 107 (1934).
¹² V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, 1949.
¹³ G. M. Fichtenholz, Course of Differential and Integral Calculus, Supplement, 1, Moscow–Leningrad, 1951.