MATHEMATICS

V. A. PLISS

ON INVARIANT SURFACES OF A SYSTEM OF TWO DIFFERENTIAL EQUATIONS

(Presented by Academician V. I. Smirnov, 18 XII 1959)

Consider a system of two differential equations

\[ \begin{aligned} \frac{dx}{dt} &= f_1(x,y,t)+\mu R_1(x,y,t,\mu),\\ \frac{dy}{dt} &= f_2(x,y,t)+\mu R_2(x,y,t,\mu), \end{aligned} \tag{1} \]

where the functions \(f_1, f_2, R_1\), and \(R_2\) are continuous, periodic in \(t\) with period \(\omega\), and uniformly analytic in \(x,y\) in a neighborhood of each point \(x,y\) for \(t \in [0,\omega]\) and sufficiently small \(|\mu|\).

We shall assume that for \(\mu=0\) system (1) has an invariant surface \(M_0\) homeomorphic to a torus, and we pose the question of when one can specify a \(\mu_0>0\) such that, for \(|\mu|<\mu_0\), system (1) has an invariant surface \(M_\mu\). An analogous question was solved by Levinson \((^1)\), Diliberto and Hufford \((^2)\), Marcus \((^3)\), and Coddington \((^4)\). All these authors assumed that either the rotation number on the toroidal surface \(M_0\) is irrational, or all solutions situated on this surface are periodic. We shall consider the case in which the rotation number on \(M_0\) is rational and not all solutions situated on \(M_0\) are periodic.

It is natural to consider system (1) in the toroidal space obtained by identifying the planes \(t=n\omega\) \((n=0,\pm1,\pm2,\ldots)\).

Let

\[ x=\varphi(x_0,y_0,t,\mu), \qquad y=\psi(x_0,y_0,t,\mu) \tag{2} \]

be the solution of system (1) with initial data \(x=x_0,\ y=y_0\) at \(t=0\). Suppose that the solution of system (1) with initial data \(x=x_0,\ y=y_0\) at \(t=0\) can be continued to the interval \(0\le t\le \omega\). Associate to the point \(x_0,y_0\) the point \(x=\varphi(x_0,y_0,\omega,\mu),\ y=\psi(x_0,y_0,\omega,\mu)\). The transformation obtained in this way will be denoted by \(I_\mu\).

In what follows we shall assume that there exists a closed Jordan curve \(\Gamma_0\), without self-intersections, invariant with respect to the transformation \(I_0\). Through the curve \(\Gamma_0\) draw all possible integral curves of system (1) for \(\mu=0\); then we obtain an invariant surface \(M_0\), homeomorphic to a torus. Suppose that on \(M_0\) there are closed integral curves (periodic solutions) of system (1) for \(\mu=0\). It is known \((^{5,6})\) that all these solutions have a common period \(k\omega\) (\(k\) natural). We shall assume that these periodic solutions have no characteristic exponents equal to zero. In addition, we shall suppose that the invariant surface \(M_0\) is asymptotically stable (for stability of invariant sets, see \((^7)\)).

1. Let us clarify somewhat more fully the structure of the surface \(M_0\). Note that on \(M_0\) there can be only a finite number of closed integral curves. From the stability of the set \(M_0\) it follows that one of the characteristic exponents of any periodic solution lying on \(M_0\) is negative.

To each periodic solution there correspond on \(\Gamma_0\) \(k\) points fixed with respect to \(I_0^k\). We shall say that a point \(p\), fixed with respect to \(I_0^k\), is unstable if, for every point \(q \in \Gamma_0\) lying sufficiently close to \(p\), the relation

\[ I_0^{-kn}(q) \longrightarrow p \quad n \to +\infty \tag{3} \]

holds.

In the opposite case we shall call the fixed point \(p\) stable. For a stable point the relation

\[ I_0^{kn}(q) \longrightarrow p, \quad n \to +\infty, \tag{4} \]

holds if \(q \in \Gamma_0\) and \(q\) is situated sufficiently close to \(p\). It is not difficult to see that on \(\Gamma_0\) the stable and unstable fixed points alternate.

Let \(p_j=(x_j,y_j)\) be a point of the curve \(\Gamma_0\) unstable and fixed with respect to \(I_0^k\). Then one of the characteristic exponents of the periodic solution \(x=\varphi(x_j,y_j,t,0)\), \(y=\psi(x_j,y_j,t,0)\) is positive. It can be shown that the roots of the characteristic equation corresponding to this periodic solution are positive. Then\({}^{(8)}\), in a neighborhood of the solution \(x=\varphi(x_j,y_j,t,0)\), \(y=\psi(x_j,y_j,t,0)\), system (1) can be represented in the form

\[ \frac{d\xi}{dt}=\lambda_1 \xi+F_1(\xi,\eta,t),\quad \frac{d\eta}{dt}=\lambda_2 \eta+F_2(\xi,\eta,t), \tag{5} \]

where \(\lambda_1>0\), \(\lambda_2<0\); \(F_1\) and \(F_2\) are series in powers of \(\xi\) and \(\eta\) with continuous and \(k\omega\)-periodic coefficients.

A. M. Lyapunov\({}^{(8)}\) proved that, in a neighborhood of the solution \(\xi=\eta=0\), system (5) has a one-parameter family of solutions

\[ \xi=ce^{\lambda_1 t}+\sum_{m=2}^{\infty} L_m(t)c^m e^{m\lambda_1 t},\quad \eta=\sum_{m=2}^{\infty} M_m(t)c^m e^{m\lambda_2 t}, \tag{6} \]

where the functions \(L_m(t)\) and \(M_m(t)\) are periodic, and the series on the right in (6) converge uniformly for sufficiently small \(|c|\) and all \(t \leq 0\).

Consider the curve \(\Lambda_j\) of the form

\[ \xi=c+\sum_{m=2}^{\infty} L_m(0)c^m,\quad \eta=\sum_{m=2}^{\infty} M_m(0)c^m \quad \text{for } |c|\leq \alpha, \tag{7} \]

where \(\alpha>0\) is sufficiently small. It is not difficult to prove that the curve \(\Lambda_j\) lies on \(\Gamma_0\). Introduce the set

\[ \mathcal{H}_j=\sum_{s=0}^{\infty} I_0^s(\Lambda_j). \tag{8} \]

This set consists of \(k\) open arcs lying on \(\Gamma_0\). By their ends these arcs adjoin stable points fixed with respect to \(I_0^k\). If, besides the solution passing through \(p_j\), system (1) for \(\mu=0\) has no periodic solutions on \(M_0\) with positive characteristic exponents, then the closure of the set \(\mathcal{H}_j\) coincides with \(\Gamma_0\). If, however, such solutions exist, then for each of them one should construct a set \(\mathcal{H}_j\), and the closure of the sum of all these sets will give \(\Gamma_0\).

1. Theorem 1. Suppose that:

1) for \(\mu=0\) system (1) has an invariant surface \(M_0\), homeomorphic to a torus;

2) the surface \(M_0\) is asymptotically stable;
3) on the surface \(M_0\) there are periodic solutions;
4) both characteristic exponents of each of the periodic solutions lying on \(M_0\) are nonzero.

Then there exists a \(\mu_0>0\) such that, for \(|\mu|\leqslant \mu_0\), system (1) has an invariant surface \(M_\mu\), homeomorphic to a torus.

The assumptions we have made make it possible to prove not only the existence of the surface \(M_\mu\), but also its stability.

Theorem 2. If the assumptions of Theorem 1 are satisfied, then there exists a \(\mu_0>0\) such that, for \(|\mu|\leqslant \mu_0\), the surface \(M_\mu\) is asymptotically stable.

1. Let us investigate the question of the structural stability of system (1) for \(\mu=0\). As follows from the results of E. A. Barbashin \((^7)\), under the assumptions made there exist two smooth surfaces \(S_1\) and \(S_2\), homeomorphic to a torus and such that: a) \(S_2\) lies inside the region bounded by \(S_1\); b) all solutions of system (1) for \(\mu=0\), as \(t\) increases, enter into the region \(G\) bounded by the surfaces \(S_1\) and \(S_2\); c) the surface \(M_0\) lies inside \(G\); d) any solution of system (1) with \(\mu=0\), beginning in \(G\), tends to \(M_0\) as \(t\to+\infty\).

Definition. We shall call system (1) with \(\mu=0\) structurally stable in \(G\) if, for every \(\varepsilon>0\), one can specify a \(\delta>0\) such that, for all \(|\mu_0|<\delta\), there exists a topological transformation \(T\) of the region \(G\) onto itself with the following properties: 1) the distance between the points \(p\in G\) and \(T(p)\) is less than \(\varepsilon\); 2) \(T\) transforms the integral curves of system (1) with \(\mu=0\) into the integral curves of system (1) with \(\mu=\mu_0\).

We note that this definition is a modification of the definition given by A. A. Andronov and L. S. Pontryagin \((^9)\).

Theorem 3. If the conditions of Theorem 1 are satisfied, then system (1) with \(\mu=0\) is structurally stable in \(\overline{G}\).

1. Under certain additional assumptions concerning the behavior of solutions of system (1) for \(\mu=0\), it is possible to prove the smoothness of the curve \(\Gamma_\mu\) and, consequently, of the surface \(M_\mu\).

Let \(q\in\Gamma_0\) be a stable fixed point with respect to the transformation \(I_0^k\). Through the point \(q\) passes the periodic solution \(x=\varphi(t)\), \(y=\psi(t)\). Suppose that the roots of the characteristic equation corresponding to this solution are positive and distinct. Then, in a neighborhood of the solution \(x=\varphi(t)\), \(y=\psi(t)\), system (1) can be represented in the form (5), where \(\lambda_2<\lambda_1<0\). We shall call the direction corresponding, in the plane \(x,y\) \((t=0)\), to the direction of the \(O\xi\) axis the principal direction.

Theorem 4. Suppose that the conditions of Theorem 1 are satisfied. Suppose, in addition, that:

1) the characteristic equation corresponding to each stable periodic solution lying on \(M_0\) has positive and distinct roots;
2) the curve \(\Gamma_0\) is smooth;
3) at each stable invariant point of the curve \(\Gamma_0\) with respect to \(I_0^k\), the curve \(\Gamma_0\) is tangent to the principal direction.

Then the surface \(M_\mu\) is smooth.

Leningrad State University
named after A. A. Zhdanov

Received
3 XII 1959

CITED LITERATURE

1. N. Levinson, Ann. of Math., 52, 727 (1950).
2. S. P. Diliberto, G. Hufford, Contributions to the Theory of Nonlinear Oscillations, 3, 1956.
3. M. Marcus, ibid.
4. P. Koosis, ibid.
5. A. Lyapunov, On Curves Defined by Differential Equations, Moscow–Leningrad, 1947.
6. V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Moscow–Leningrad, 1949.
7. E. A. Barbashin, Matem. sbornik, 29, no. 2 (1951).
8. A. M. Lyapunov, The General Problem of the Stability of Motion, Moscow–Leningrad, 1950.
9. A. A. Andronov, L. S. Pontryagin, DAN, 14, no. 5 (1937).