MATHEMATICS

L. N. BELYUSTINA

SMALL PERIODIC PERTURBATIONS OF A ROUGH AUTONOMOUS SYSTEM

(Presented by Academician L. S. Pontryagin on 7 VII 1962)

In the qualitative study of nonautonomous systems it is very important to single out in phase space special manifolds separating regions of solutions with different limiting behavior. In the present note special manifolds are singled out for nonautonomous periodic systems of the second order that are close to rough autonomous systems.

Let the autonomous system

\[ \frac{dx_1}{dt}=X_1(x_1,x_2), \qquad \frac{dx_2}{dt}=X_2(x_1,x_2) \tag{1} \]

be rough \(\left({}^{1,2}\right)\) in a region \(G\) of the plane \(x_1,x_2\).

Consider the nonautonomous system

\[ \frac{dx_1}{dt}=X_1(x_1,x_2)+\mu R_1(x_1,x_2,t); \qquad \frac{dx_2}{dt}=X_2(x_1,x_2)+\mu R_2(x_1,x_2,t), \tag{2} \]

defined in the region \(G_t\ \{(x_1,x_2)\subset G,\ -\infty<t<+\infty\}\) of the space \(x_1,x_2,t\). \(R_1\) and \(R_2\) are periodic functions of \(t\) of period \(\tau\). The functions \(X_1,X_2,R_1,R_2\) are assumed to be functions of class \(C^3\), and \(\mu\) is a parameter.

The solutions of system (2) generate a point transformation \(T^\mu_\tau\) of the plane \(t=t_0\) into the plane\(^*\) \(t=t_0+\tau\), coinciding for \(\mu=0\) with the transformation \(T^0_\tau\) generated by the solutions of system (1).

The special trajectories \(\left({}^{3}\right)\) of system (1) in the plane \(x_1x_2\), by virtue of its roughness, can only be rough equilibrium states, rough limit cycles, and separatrices that do not go from saddle to saddle.

Theorem 1. Let the rough system (1) have an equilibrium state \(A(x_1^0,x_2^0)\) of the type focus or node. Then, for sufficiently small \(\mu\ne0\), system (2) has a unique isolated periodic solution \(\Gamma_\mu\), \((x_1=\varphi_1(t,\mu), x_2=\psi_1(t,\mu))\) of period \(\tau\), which for \(\mu=0\) turns into the solution \(x_1=x_1^0,\ x_2=x_2^0\). The roots of the characteristic equations for the corresponding fixed points of the point transformations \(T^0_\tau,T^\mu_\tau\), generated by systems (1), (2), are pairwise close and are located simultaneously inside or outside the unit circle.

The uniqueness of the periodic solution of period \(\tau\) of system (2), generated by the solution \(x_1=x_1^0,\ x_2=x_2^0\) of system (1), follows from the fact that, for the generating solution, the roots \(|\lambda_i|\ne1\) \((i=1,2)\). The closeness of the roots \(\lambda_i^\mu\) and \(\lambda_i\) of the characteristic equations of the transformations \(T^\mu_\tau\) and \(T^0_\tau\) for the corresponding fixed points follows from the fact that, by continuity

\(^*\) By virtue of the periodicity in \(t\) of equations (2), the planes \(t=t_0+n\tau\) \((n=0,\pm1,\pm2,\ldots)\) are identified.

dependence on the parameter, for sufficiently small \(\mu\ne 0\) the point transformations \(T_\tau^\mu\) and \(T_\tau^0\) in neighborhoods of the corresponding fixed points are close \({}^{(5)}\).

Theorem 2. Suppose that the rough system (1) has an equilibrium state \(B(x_1^0,x_2^0)\) of saddle type. Then, for sufficiently small \(\mu\ne 0\), system (2) has a unique isolated periodic solution \(\Gamma_\mu\) of period \(\tau\), which for \(\mu=0\) turns into \(x_1=x_1^0,\ x_2=x_2^0\).

The roots of the characteristic equations of the point transformations \(T_\tau^0,\ T_\tau^\mu\), generated by systems (1), (2), for the corresponding fixed points are pairwise close and are located one inside and the other outside the unit circle.

Here, from the closeness of the corresponding point transformations it follows that if \(|\lambda_1|<1,\ |\lambda_2|>1\), then also \(|\lambda_1^\mu|<1,\ |\lambda_2^\mu|>1\).

A fixed point of the transformation \(T_\tau^\mu\) for which \(|\lambda_i|>1\) or \(|\lambda_i|<1,\ i=1,2\) \((|\lambda_1^\mu|<1,\ \text{and}\ |\lambda_2^\mu|>1)\) will henceforth be called a fixed point of focus or node type (a saddle fixed point).

Each cycle of the rough system (1) has one characteristic exponent different from zero.

Theorem 3. Suppose that the rough system (1) has, on the phase plane \(x_1,x_2\), a limit cycle \(l\). Then, for sufficiently small \(\mu\ne 0\), in the space \((x_1,x_2,t)\) there exists a unique toroidal integral surface \(L^\mu\), filled with solutions of system (2), homeomorphic to a torus. The surface \(L^\mu\) has, in the section \(t=t_0\), a closed curve \(l^\mu\), which is an invariant curve of the point transformation \(T_\tau^\mu\). As \(\mu\to 0\), the curve \(l^\mu\) contracts in the plane \(x_1,x_2\) to the limit cycle \(l\).

This theorem follows as a special case for \(n=2\) from the results established by N. Levinson in \({}^{(6)}\) (see also \({}^{(7-10)}\)).

Periodic solutions of system (2) on the surface \(L^\mu\) (if they exist) have \({}^{(11,12)}\) a common period \(k\omega\) (\(k\) natural). In this connection, depending on the number of fixed points of the transformation \((T_\tau^\mu)^k\) (\(k\) natural), the closed curve \(l^\mu\) on the surface \(L^\mu\) may be filled with solutions that are periodic or almost periodic, or with a finite number of periodic solutions and nonperiodic ones tending to them.

Suppose that system (1) has an equilibrium state \(B(x_1^0,x_2^0)\) of saddle type. The separatrix arcs of the saddle \(B\), located in some neighborhood \(\sigma\) of it, will be denoted by \(s_i\) \((i=1,2,3,4)\). The fixed point \(B\) of the transformation \(T_\tau^0\) generates (by Theorem 2) a saddle fixed point \(B^\mu\) of the transformation \(T_\tau^\mu\).

Theorem 4. For sufficiently small \(\mu\ne 0\), in some neighborhood \(\sigma_1\subset\sigma\) of the saddle fixed point \(B^\mu\) of the transformation \(T_\tau^\mu\), there exist smooth open arcs \(s_i^\mu\) \((i=1,\ldots,4)\) adjoining the point \(B^\mu\), which are invariant arcs of the transformation \(T_\tau^\mu\) and contract, as \(\mu\to 0\), to the parts of the separatrix arcs \(s_i\) \((i=1,2,3,4)\) lying in this neighborhood.

Indeed, according to Hadamard’s theorem \({}^{(12)}\), in some neighborhood \(\sigma_1\) of the saddle fixed point \(B^\mu\) there exist two and only two invariant curves \(C_1^\mu\) and \(C_2^\mu\) passing through the fixed point \(B^\mu\), having a continuous \({}^{(13,14)}\) tangent. The fixed point \(B^\mu\) divides the curves \(C_1^\mu\) and \(C_2^\mu\) into arcs \(s_i^\mu\) \((i=1,2,3,4)\). It is established that, as \(\mu\to 0\), the arcs \(s_i^\mu\) contract in the plane \(x_1,x_2\) to the arcs \(s_i\).

The separatrices of the saddles of system (1), by virtue of its roughness, cannot go from saddle to saddle. Let \(S\) denote one of the separatrices of the saddle \(B\), and \(s\) the part of it belonging to the neighborhood \(\sigma\) of the point \(B\).

Theorem 5. Suppose that the rough system (1) on the phase plane \(x_1,x_2\) has an equilibrium state \(B\) of saddle type and a separatrix \(S\) of this saddle,

tending as \(t\to+\infty\) \((t\to-\infty)\) to the equilibrium state \(A\) of node or focus type. Then, for sufficiently small \(\mu\ne0\), in the space \((x_1,x_2,t)\) there exists a unique integral surface \(\Omega^\mu\), filled with nonperiodic solutions of system (2), tending as \(t\to-\infty\) \((t\to+\infty)\) to the isolated periodic solution generated by the equilibrium state \(B\) of saddle type and, as \(t\to+\infty\) \((t\to-\infty)\), to the isolated periodic solution generated by the equilibrium state \(A\) of focus or node type. The curve \(S^\mu\) of the section \(t=t_0\) of the surface \(\Omega^\mu\) is an invariant curve of the transformation \(T_\tau^\mu\), going from the saddle fixed point \(B^\mu\) to the fixed point \(A^\mu\) of focus or node type. As \(\mu\to0\), the curve \(S^\mu\) contracts in the plane \(x_1,x_2\) to the separatrix \(S\).

For the proof of the theorem, on the invariant separatrix arc \(S^\mu\) of the saddle point \(B^\mu\) (see Theorem 4) certain points are distinguished: \(P_0\); \(P_1=T_\tau^\mu P_0\) and \(P_2=T_\tau^\mu P_1\). The totality of the preceding \((P_{-n}P_{-n+1}=T^{-n}P_0P_1)\) and succeeding \((P_nP_{n+1}=T^nP_0P_1)\) arcs for the arc \(P_0P_1\), obtained as a result of the transformations \(T_\tau^\mu\) and \(T_{-\tau}^\mu\) of this arc, form a continuous curve \(S^\mu\) with a continuous tangent.

By virtue of the roughness of system (1) and the continuous dependence of the solutions of system (2) on the parameter and on the initial conditions, the sequences of subsequent points for the points of the curve \(S^\mu\) tend to the fixed point \(A^\mu\), generated by the equilibrium state \(A\) of focus or node type. It is established that, as \(\mu\to0\), the curve \(S^\mu\) contracts in the plane \(x_1,x_2\) to the separatrix \(S\).

Theorem 6. Let the rough system (1) in the phase plane \(x_1,x_2\) have an equilibrium state \(B\) of saddle type and a separatrix \(S\) of this saddle, tending as \(t\to+\infty\) \((t\to-\infty)\) to the limit cycle \(l\). Then, for sufficiently small \(\mu\ne0\), in the space \(x_1,x_2,t\) there exists a unique integral surface \(\Omega^\mu\), filled with nonperiodic solutions of system (2), tending as \(t\to-\infty\) \((t\to+\infty)\) to the periodic solution generated by the equilibrium state \(B\) of saddle type and, as \(t\to+\infty\) \((t\to-\infty)\), to an integral surface homeomorphic to a torus, whose section \(t=t_0\) \((l^\mu)\), as \(\mu\to0\), contracts in the plane \(x_1,x_2\) to the cycle \(l\).

The curve obtained in the section \(t=t_0\) of the surface \(\Omega^\mu\) is an invariant curve of the point transformation \(T_\tau^\mu\), contracting as \(\mu\to0\) in the plane \(x_1,x_2\) to the separatrix \(S\).

The proof of Theorem 5 is analogous to the proof of Theorem 4.

Definition 1. The invariant curves \(S^\mu\) defined by Theorems 4 and 5 will be called invariant separatrix curves, and the surfaces \(\Omega^\mu\)—integral separatrix surfaces.

Definition 2. The fixed points, invariant closed curves, and invariant separatrix curves of the point transformation \(T_\tau^\mu\) will be called special invariant curves of the transformation \(T_\tau^\mu\).

Definition 3. The integral manifolds formed in the space \(x_1,x_2,t\) by the solutions of system (2) whose initial values belong to the special invariant curves (isolated periodic solutions, toroidal and separatrix integral surfaces) will be called special integral manifolds of system (2).

A consequence of the theorems given above is the following

Theorem 7. Let the special trajectories of the rough system (1) partition the domain \(G\) of the plane \(x_1,x_2\) into cells filled with nonspecial trajectories. Then, for sufficiently small \(\mu\ne0\), the special integral manifolds of sys-

systems (2) divide the domain \(G_t\) of the space \(\dot{x}_1, x_2, t\) into cells filled with solutions of system (2) having the same asymptotic behavior. The cells that arise are close to the cells of system (1) interpreted in the space \(x_1, x_2, t\), and coincide with them when \(\mu = 0\).

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

Received
30 VI 1962

CITED LITERATURE

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