UDC 517.92

MATHEMATICS

K. ESHMURADOV

CONDITIONS FOR STRUCTURAL STABILITY AND CLASSIFICATION OF CELLS OF A THREE-DIMENSIONAL DYNAMICAL SYSTEM WITHOUT CLOSED TRAJECTORIES

(Presented by Academician A. Yu. Ishlinskii, 1 III 1968)

In the works of A. A. Andronov, L. S. Pontryagin, E. A. Leontovich, and A. G. Maier (¹), conditions for structural stability were established and a classification was given of the cells of a two-dimensional dynamical system. In the present paper we formulate the results of solving these same questions for a dynamical system of third order, described in a certain domain \(G\) (bounded by a certain smooth contact-free surface \(S\)) by the equations

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

under the essential restriction that closed trajectories are absent.

Necessary and sufficient conditions for structural stability with respect to small perturbations, together with the first derivatives, of the right-hand sides of the dynamical system (1) can be formulated in the form of the following theorem:

Theorem 1. The dynamical system (1) without closed trajectories is structurally stable in the domain \(G\) if and only if in this domain it:

1) has only such equilibrium states for which the corresponding characteristic equations have no roots with zero real part, and the number of such equilibrium states is finite;

2) has no semitrajectories stable in the sense of Poisson other than equilibrium states;

3) has only a finite number of trajectories going from a saddle or saddle-focus into another saddle or saddle-focus, along which the separatrix surfaces of the corresponding saddles or saddle-foci intersect without tangency;

4) has no trajectories going from a saddle to a saddle, except for the trajectories listed in the preceding item;

5) has no limiting set of semitrajectories other than an equilibrium state.*

From the structural-stability conditions 1–5 it follows that a structurally stable dynamical system of third order without closed trajectories can have phase trajectories only of the following types**:

I. Equilibrium state:

1) a stable or unstable node (focus);

2) a stable or unstable saddle (saddle-focus)***.

II. From an unstable (stable) saddle or saddle-focus:

3) tending as \(t\to-\infty\) (\(t\to+\infty\)) to an unstable (stable) node or focus;

4) leaving the domain \(G\) as \(t\to-\infty\).

* Conditions 1–5 agree with Smale’s well-known hypothesis (²). The proofs of items 1 and 2 are based on the results of (³) and on the lemma on closures of (⁴). In the proof of items 3 and 4, considerations stated in (⁵) are used.

** Without loss of generality it is assumed that trajectories intersecting the contact-free boundary surface \(S\), with increasing time \(t\), enter the interior of the domain \(G\).

*** A saddle or saddle-focus will be called stable (unstable) if the corresponding characteristic equation has two roots with negative (positive) real parts.

III. The separatrix of a stable (unstable) saddle or saddle-focus:

5) tending as \(t\to-\infty\) (\(t\to+\infty\)) to an unstable (stable) node or focus;

6) leaving the domain \(G\) as \(t\to-\infty\);

7) tending as \(t\to\infty\) (\(t\to+\infty\)) to an unstable (stable) saddle or saddle-focus whose separatrix surfaces intersect without tangency.

IV. A trajectory tending as \(t\to+\infty\) to a stable node or focus;

8) tending as \(t\to-\infty\) to an unstable node or focus;

9) leaving the domain \(G\) as \(t\to-\infty\).

With respect to the singular and ordinary trajectories defined in accordance with paper \((^1)\), the following theorem holds.

Theorem 2. Trajectories of types I–III are singular, and trajectories of type IV are ordinary.

The set of points belonging to ordinary trajectories is open and may decompose into a finite or infinite number of connected domains, which are called cells \((^1)\). The number and the basic characteristics of the cells are determined by the following theorem.

Theorem 3. If a dynamical system without closed trajectories (1) satisfies in the domain \(G\) conditions 1–5 of Theorem 1, then:

1) the set of singular trajectories divides the domain \(G\) into a finite number of cells;

2) each interior cell consists of entire ordinary trajectories and has in its boundary exactly one sink and one source;

3) each cell adjacent to the surface \(S\) consists of \(\omega\)-orbitally stable positive semitrajectories and has in its boundary exactly one sink and a connected domain of the surface \(S\).

The behavior of singular trajectories on the boundary of a cell is described by

Theorem 4. Every nonsingular separatrix of an unstable (stable) saddle that is part of the boundary of an interior cell, as \(t\to+\infty\) (\(t\to-\infty\)) tends to the sink (source) that is part of the boundary of the given cell. Every whisker of a stable (unstable) saddle that is part of the boundary of an interior cell, as \(t\to+\infty\) (\(t\to-\infty\)) tends to the sink (source) that is part of the boundary of the given cell. Every nonsingular separatrix of an unstable (stable) saddle that is part of the boundary of a cell adjacent to the surface \(S\), as \(t\to+\infty\) (\(t\to-\infty\)) tends to the sink that is part of the boundary of the given cell (leaves the domain \(G\)). Every whisker of a stable (unstable) saddle that is part of the boundary of a cell adjacent to the surface \(S\), as \(t\to+\infty\) (\(t\to-\infty\)) tends to the sink that is part of the boundary of the given cell (leaves the domain \(G\)).

Table 1

| | \multicolumn{2}{c}{\(k=0\)} | \multicolumn{2}{c}{\(k=1\)} | \multicolumn{4}{c}{\(k=2\)} |
|---|---|---|---|---|---|---|---|---|
| | \(n=0\) | \(n\ge 1\) | \(n=0;\ 1\) | \(n\ge 2\) | \(n=0;\ 1\) | \(n=2\) | \(n=3;\ 4\) | \(n\ge 5\) |
| \(\alpha\) | 0 | \(n\) | 0 | \(n-2\) | 0 | 1 | \(2n-4\) | \(3n-8\) |
| \(\beta\) | 1 | \(n+1\) | 0 | \(n-1\) | 0 | 1 | \(4n-8\) | \(4n-8\) |

Special separatrices are separatrices going from a saddle to a saddle; all the remaining separatrices are called nonsingular.

The type of a cell is characterized to a considerable extent by the numbers \(n\) and \(k\) of saddles and special separatrices that are part of the boundary of the cell. In Table 1, for a number of the first values of \(k\) and arbitrary \(n\), the numbers \(\alpha\) and \(\beta\) of topological—

* In the presence, in a third-order dynamical system, of closed trajectories, as shown by the example constructed in paper \((^6)\), the number of cells can be infinite.

sketches of different types of internal cells and cells adjacent to the surface \(S\).

Below is a clarification of the composition and character of the cell boundaries.

For any \(n \geq 1\) and \(k=0\), the boundary of an internal cell includes \(n\) nodes and all separatrices of each saddle that is part of the boundary of the given cell; and the boundary of a cell adjacent to the surface \(S\), in addition, includes an \(n\)-connected region of the surface \(S\). For \(n=k=0\), the boundary of a cell adjacent to the surface \(S\) consists of a sink and the surface \(S\).

Fig. 1

Fig. 2

For any \(n \geq 3\) (for cells adjacent to the surface \(S\), also for \(n=2\)) and \(k=1\), the boundary of an internal cell includes one special separatrix, \((n+2)\) nodes, and all separatrices of each saddle that is part of the boundary of the given cell; and the boundary of a cell adjacent to the surface \(S\), in addition, includes an \((n-1)\)-connected region of the surface \(S\).

For \(k=2\), the boundary of a cell includes two special separatrices \(\gamma_1\) and \(\gamma_2\). If \(\gamma_1\) and \(\gamma_2\) go from one and the same saddle to one and the same saddle, then we shall call the given cell a cell of the first kind; if from one and the same saddle to different saddles—a cell of the second kind; and, finally, if from different saddles to different saddles—a cell of the third kind.

For any \(n \geq 2\) and \(k=2\), the boundary of an interior cell of the first kind includes \(n\) whiskers, all separatrices of \((n-2)\) saddles, and part of the nonspecial separatrices of two saddles that are contained in the boundary of the given cell; and the boundary adjoining the surface \(S\) of a cell of the first kind, in addition, includes an \((n-1)\)-connected region of the surface \(S\).

For any \(n \geq 3\) and \(k=2\), the boundary of an interior cell of the second kind includes \((n+1)\) whiskers, all separatrices of \((n-1)\) saddles, and part of the nonspecial separatrices of one saddle; and the boundary adjoining the surface \(S\) of a cell of the second kind, in addition, includes an \((n-2)\)-connected region of the surface \(S\). For \(n=k=2\) there are no cells of the second kind.

For \(n \geq 5\) (for cells adjoining the surface \(S\), also for \(n=4\)) and \(k=2\), the boundary of an interior cell of the third kind includes \((n+4)\) whiskers and all separatrices of each saddle contained in the boundary of the given cell; and the boundary adjoining the surface \(S\) of a cell of the third kind, in addition, includes an \((n-2)\)-connected region of the surface \(S\). For \(n=2,3\) and \(k=2\) there are no cells of the third kind, and for \(n=4\) and \(k=2\) there are no interior cells of the third kind.

The cells described above are rough.

In Figs. 1 and 2b there are schematically shown an interior cell with \(k=0\), a cell with \(k=0\) adjoining the surface \(S\), and an interior cell with \(k=1\). In Fig. 2a there is shown an element whose insertion into the interior of a cell with \(k=0\) gives a cell with \(k=1\). In these figures the following notation is introduced: \(O^+\)—sink; \(O^-\)—source; \(\Pi^0\)—region of the surface \(S\) contained in the boundary of the cell; \(C_i^-\)—unstable saddle; \(C_i^+\)—stable saddle; \(m\)—the number of unstable saddles and \((n-m)\)—the number of stable saddles contained in the boundary of the given cell; \(\gamma\)—a trajectory going from saddle to saddle (a special separatrix).

I express my sincere gratitude to Prof. Yu. I. Neimark for posing the problem and for valuable guidance.

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

Received
28 II 1968

CITED LITERATURE

1. A. A. Andronov, L. S. Pontryagin, DAN, 14, No. 5 (1937); A. A. Andronov, A. A. Vitt, S. E. Khaikin, Theory of Oscillations, Moscow, 1959; A. A. Andronov, E. A. Leontovich et al., Qualitative Theory of Dynamical Systems of the Second Order, Moscow, 1966.
2. S. Smale, Proc. Sympos. Ordin. Diff. Equat., Mexico City, 1961, p. 195.
3. R. M. Minsh, DAN, 124, No. 6 (1959).
4. C. C. Pugh, Bull. Am. Math. Soc., 70, No. 4 (1964).
5. Yu. N. Neimark, Izv. vyssh. uchebn. zaved., Radiofizika, 1, Nos. 1, 2, 5–6 (1958).
6. L. E. Reizin, Izv. AN LatvSSR, ser. phys.-techn. sciences, No. 4 (1964).