UDC 517.925

MATHEMATICS

G. Ya. SHAFRANOV

THE BIRTH OF PERIODIC MOTIONS FROM A STATE OF EQUILIBRIUM

(Presented by Academician L. S. Pontryagin, 24 XII 1969)

In the present note we consider conditions for the appearance of limit cycles from the system

[
\dot{x}=X(x),
\tag{A}
]

which has an equilibrium state with two purely imaginary roots (the remaining ones having nonzero real parts) and a Lyapunov quantity (g_{2m+1}\ne 0), in passing to a (\delta)-close, up to rank (2m+4) (see ((^1))), system

[
\dot{x}=X(x)=P(x)=\widetilde{X}(x).
\tag{(\widetilde{A})}
]

A relation is established between the number of generated limit cycles and the order of the Lyapunov quantity (g_{2m+1}); a complete classification of the trajectories of system ((\widetilde{A})) is given. Here (x) is a vector; (X(x), P(x)) are vector-functions belonging to the class (C^N) ((N \ge 2m+4)) (to the analytic class).

In ((^1)) conditions for the appearance of limit cycles from an equilibrium state of a two-dimensional system were studied; in ((^2,{}^3)), conditions for the birth of one periodic motion from a system having two purely imaginary roots and (n) negative roots, (g_3\ne 0). The general case has not hitherto been considered by anyone. The study is carried out by the method of point mappings; for its construction the system is reduced to a special form, and the neighborhood is split into a noncritical region, in which there are no closed trajectories, and a critical one.

§ 1. Lemma 1. The system ((\widetilde{A})), by a nondegenerate change of variables and time, can be reduced to the form

[
\begin{aligned}
\dot{x}&=-y+g_{2m+1}x(x^2+y^2)^m+X(x,y,z,w)+P(x,y,z,w),\
\dot{y}&=x+g_{2m+1}y(x^2+y^2)^m+Y(x,y,z,w)+Q(x,y,z,w),\
\dot{z}&=A^-z+Z(x,y,z,w)+L(x,y,z,w),\
\dot{w}&=A^+w+W(x,y,z,w)+M(x,y,z,w),
\end{aligned}
\tag{1}
]

where (x,y) are scalars; (z,w) are vectors; (A^-), (A^+) are matrices whose characteristic roots have negative and, respectively, positive real part. (X,Y,Z,W,P,Q,L,M) belong to the class (C^{N-1}) (to the analytic class); (X,Y,W) vanish for (x=0,\ y=0,\ w=0); (X(x,y,0,0)), (Y(x,y,0,0)), (Z(x,y,0,0)), (W(x,y,0,0)) vanish for (x=y=0) together with their derivatives up to order (2m+1); (X,Y,Z,W) are obtained from the right-hand sides of system ((A)), while (P,Q,L,M) are obtained from the additions.

For the proof, the ideas of the monographs ((^4,{}^5)) are used.

§ 2. Lemma 2 (on an invariant surface). Suppose that in some neighborhood (G_x\times G_y) of the fixed point (x=0,\ y=0), where (x,y) are vectors, a point mapping is given

[
\bar{x}=Ax+X(x,y), \qquad \bar{Y}=By+Y(x,y),
\tag{2}
]

where the spectral radius of the constant matrix (B), (|B|_{\mathrm{sp}}=r_2), matrices

[
A^{-1}|A^{-1}|_{\mathrm{sp}}^{-1}=r_1,\quad r_1>r_2,\quad r_1>1;\quad X(x,y),\ Y(x,y)
]
are vector functions that vanish, together with their first-order derivatives, at (x=0), (y=0), and belong to the class (C^k), (k\ge 1).

Then, in some neighborhood of the origin, there exists an invariant surface (y=\varphi(x)) belonging to the class (C^{k-1}), (\varphi(0)=0) ((\varphi_x'(0)=0) for (k>1)), satisfying the Poincaré functional equation
[
\varphi[Ax+X(x,\varphi(x))]=B\varphi(x)+Y(x,\varphi(x)).
\tag{3}
]

Lemma 3. Suppose that (\det A\ne 0), (|A|{\mathrm{sp}}=r_1), (|B^{-1}|=r_2), (r_1<r_2), (r_1<1).}}^{-1

Then the mapping (2) has an invariant surface (x=\psi(y)) (uniqueness may fail), (\psi(0)=0), (\psi(y)\in C^{N_0-1}), where (1\le N_0\le k).

Lemmas 2 and 3 are proved by constructing a metric space of surfaces and then applying the contraction mapping principle.* On the basis of Lemma 3, one establishes the existence of smooth invariant surfaces (E_1: w=\psi(r,z)), (\psi(0,0)=0), and (E_2: z=\chi(r,w)), (\chi(0,0)=0), of dimensions (p+1) and (q+1), respectively, defined in some neighborhood of the fixed point of the point mapping
[
\begin{aligned}
\bar r&=r_0+f_1(r_0,z_0,w_0),\
\bar z&=s^-z_0+f_2(r_0,z_0,w_0),\
\bar w&=s^+w_0+f_3(r_0,z_0,w_0),
\end{aligned}
\tag{T}
]
where (r_0,\bar r) are scalars; (z_0,\bar z) are (p)-dimensional vectors; (w_0,\bar w) are (q)-dimensional vectors; (s^-) is a constant ((p\times p)) matrix with spectrum inside the unit circle, (\det s^-\ne 0); (s^+) is a constant ((q\times q)) matrix with spectrum outside the unit circle; (f_1,f_2,f_3) are nonlinearities belonging to the smoothness class (C^{N-1}). Considering the mapping (T) on one of the surfaces of the form (w=\psi(r,z)) and then selecting the invariant surface (z=\psi_1(r)), (\psi_1(0)=0), we obtain
[
\bar r=r_0+\bigl(G_k+f_4(r_0)\bigr)r_0^k,\quad k=2,\ldots,N-1,
\tag{4}
]
where (f_4(0)=0), and suppose (G_k\ne 0) is a quantity which we shall call Lyapunov. An invariant manifold containing the fixed point (O) will be called incoming (outgoing) if (T^nM) tends asymptotically to (O) as (n\to\infty) ((n\to-\infty)). Denote by (C^1_{i,j}), (i+j=p+q+1), a saddle fixed point with an (i)-dimensional incoming and a (j)-dimensional outgoing manifold. Then the following holds.

Lemma 4. 1) Suppose (G_k>0), (k) odd; then we have a saddle (C^1_{p,q+1}); 2) (G_k<0), (k) odd; we have a saddle (C^1_{p+1,q}); 3) (k) even; then the fixed point has a ((p+1))-dimensional incoming manifold with boundary and a ((q+1))-dimensional outgoing manifold with boundary.

We shall denote such a saddle point by (C^1_{p+1/2,q+1/2}), and in what follows we shall regard it as the merging of the saddles (C^1_{p+1,q}) and (C^1_{p,q+1}). Trajectories ((T^nM) for increasing and decreasing (n)) that do not belong to the incoming and outgoing invariant manifolds will be saddle trajectories (cf. ((^{11}))).

To construct the point mapping in the critical region
[
|w|^2\le \sigma r^2,\quad \text{where } r^2=x^2+y^2,\quad \sigma>0,\quad |w|^2=w_1^2+\cdots+w_q^2,
\tag{5}
]
where (w_i) are the components of the vector (w), in equations (1) we make the substitution
[
x=r\cos\varphi,\quad y=r\sin\varphi,\quad z=z,\quad w=r\bar w
\tag{6}
]
in the region (5), (|w|^2\le \sigma r^2), which leads to regular expressions.

* Invariant surfaces of point mappings were studied in ((^{6-8})). On invariant surfaces of differential equations, see ((^{10})).

The mapping of the surface (\varphi=0) onto the surface (\varphi=2\pi) will take the form

[
\begin{aligned}
\bar r&=sr_0+f_1(r_0,z_0,w_0),\
\bar z&=s^{-}z_0+f_2(r_0,z_0,w_0),\
\bar w&=s^{+}w_0+f_3(r_0,z_0,w_0).
\end{aligned}
\tag{7}
]

Here, when the additions are absent, (s=1); (s^{-}=\exp(2\pi A^{-})) is a ((p\times p))-matrix, (s^{+}=\exp(2\pi A^{+})) is a (q)-dimensional matrix whose characteristic roots for (s^{-}) lie inside, and for (s^{+}) outside, the unit circle; (\det s^{-}\ne0). The fixed points of the mapping (7) (with the exception of the trivial one (r_0=0,\ z_0=0,\ w_0=0), corresponding to the equilibrium state) correspond to limit cycles of equation (1). Reducing the problem on fixed points of the mapping (7) to the problem of determining the roots (r_0^*) of a defining scalar equation, we see that there will be no more than (m) positive roots if (g_{2m+1}\ne0) is a Lyapunov quantity.

Lemma 5. In some neighborhood, independent of the particular choice of a system ((\tilde A)) that is (\delta)-close up to order (2m+4) to the system ((A)), and of the point mapping ((T)), there exists an invariant surface of the form (w=\psi(r,z)), passing through all fixed points of the mapping.

Denote by (C_{i,j}) a cycle in the ((p+q+2))-dimensional space, (p+q+2=i+j-1), which is the intersection of the (i)-dimensional incoming and the (j)-dimensional outgoing manifolds. Under the point mapping, the cycle (C_{ij}) corresponds to a saddle fixed point (C_{i-1,j-1}^{1}). Restricting ourselves to cycles of integer multiplicity, we obtain the main result.

Theorem 1. Let a system ((A)) of class (C^N) (of analytic class), with two purely imaginary roots, (p) roots with negative real part, and (q) roots with positive real part, have the Lyapunov quantity (g_{2m+1}\ne0) ((2m+4\le N)).

Then for a system ((\tilde A)) of class (C^N) (of analytic class) the following holds: a) there exist (\varepsilon_0>0,\ \delta_0>0) such that, whatever system ((\tilde A)), (\delta)-close up to order (2m+4), we take, in the (\varepsilon_0)-neighborhood of (O) (the equilibrium state) there will be no more than (m) closed trajectories; b) for any (\varepsilon<\varepsilon_0) and (\delta<\delta_0) one can specify a system ((\tilde A)) for which, in the (\varepsilon)-neighborhood of the point (O), there will lie a prescribed number (k) ((0\le k\le m)) of limit cycles; c) the cycles (C_{p+2,q+1}) and (C_{p+1,q+2}) will alternate (will be adjacent in (r_0)); adjacent to the equilibrium state (O_{p+2,q}) will be the cycle (C_{p+1,q+2}), and adjacent to (O_{p,q+2}) will be the cycle (C_{p+2,q+1}); d) a two-dimensional manifold whose trajectories, as (t\to\infty), tend to one limit cycle (C_{p+2,q+1}(O_{p+2,q})), and as (t\to-\infty) tend to the cycle (C_{p+1,q+2}(O_{p,q+2})), is the intersection of the incoming ((p+2))-dimensional manifold of one cycle and the ((q+2))-dimensional manifold of the other cycle (or equilibrium state); e) all other trajectories not belonging to invariant incoming and outgoing manifolds pass at a finite distance from the generated cycles (and the equilibrium state).

The author thanks E. A. Leontovich-Andronova for posing the problem.

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

Received
22 XII 1969

CITED LITERATURE

1. A. A. Andronov, E. A. Leontovich, Matem. sborn., 40, (82), 179 (1956).
2. Yu. I. Neimark, Radiofizika, 1, No. 1, 41 (1958); 1, No. 2, 95 (1958); 1, No. 5–6, 146 (1958); DAN, 129, No. 4, 736 (1959).
3. N. N. Brushlinskaya, DAN, 139, No. 1, 9 (1961).
4. A. M. Lyapunov, The General Problem of the Stability of Motion, Collected Works, Vol. 2, Acad. Sci. USSR Press, 1956.
5. I. G. Malkin, Theory of Stability of Motion, Moscow, 1966.
6. J. Hadamard, Bull. Soc. Math. France, 29, 224 (1901).
7. S. Lattes, Ann. mat., 13, ser. 3, 1 (1907).
8. D. V. Anosov, Scientific Reports of Higher School, Physical-Mathematical Sciences, 3, No. 1, 3 (1959).
9. Yu. I. Neimark, Radiofizika, 10, No. 3, 311 (1967).
10. A. Kelley, J. Diff. Equat., 3, No. 4, 546 (1967).
11. R. M. Mintz, DAN, 147, No. 1, 31 (1962).