Remarks on the Realization of Integral Curves of Equations of the Form \(\dot x=f(t,x,x(t-\tau))\) on Certain Two-Dimensional Manifolds

V. R. Petukhov

Let us consider the question of sufficient conditions for the realizability of integral curves of the equation

\[ \dot x(t)=f(t,x(t),x(t-\tau(t,x(t)))),\quad \tau(t,x)\geqslant 0 \tag{1} \]

on the sphere, cylinder, and torus. Here \(f(t,x,y)\) is a function continuous in the aggregate of its arguments for \(0\leqslant t<+\infty\), \(-\infty<x,y<+\infty\), and continuously differentiable with respect to \(x\) and \(y\) in the indicated domain; \(\tau(t,x)\geqslant 0\) is a function continuous in the aggregate of its arguments for \(0\leqslant t<+\infty\), \(-\infty<x<+\infty\), and continuously differentiable with respect to \(x\) in the indicated domain. The situation is analogous also in the case where equation (1) contains \(n\) delays \(\tau_1,\ldots,\tau_n\) of a similar form.

1. Let \(\tau(0,x)\equiv 0\), \(\tau(t,0)\equiv \tau(t,T)\equiv 0\); \(f(t+T,x,y)\equiv f(t,x,y)\); \(\tau(t+T,x)\equiv \tau(t,x)\); \(\tau(t,x+T)\equiv \tau(t,x)\); \(t-\tau(t,x)\geqslant 0\), \(t\geqslant 0\); \(f(t,x+T,y)=f(t,x,y+T)=f(t,x,y)\); \(f(t,0,0)=0\); \(T>0\) is some number. Under these conditions the integral curves of equation (1) can be realized on the sphere by taking, for example, the line \(t=0\) \((0<x<T)\) as a certain zero meridian and posing on it the initial problem \(x(0)=x_0\), \(0<x_0<T\pmod T\). The direction field on this meridian, as well as at the poles \(x=0\) and \(x=T\), is determined uniquely by the right-hand side. Owing to the assumptions made on the functions \(f\) and \(\tau\), every integral curve will be uniquely determined by a point on the zero meridian (see [1] and [2]). It should also be noted that, under our assumptions on the functions \(f\) and \(\tau\), the two poles of the sphere (\(x=0\) and \(x=T\)) may be identified and regarded as one pole.

2. Let the functions \(f\) and \(\tau\) be \(T\)-periodic in \(t\), and let \(\tau(0,x)\equiv 0\), \(t-\tau(t,x)\geqslant 0\) for \(t\geqslant 0\), \(-\infty<x<+\infty\). Under these conditions the integral curves of equation (1) can be realized on the cylinder, and \(\tau\) may even be a functional of \(x\).

3. Let \(f(t,x,y)\) and \(\tau(t,x)\) be \(T\)-periodic functions in each variable separately, and consequently also jointly, while the remaining requirements concerning continuity in \(t,x,y\) and continuous differentiability in \(x\) and \(y\) are assumed to be fulfilled. Further, let \(\tau(0,x)=\tau(t,0)=0\); \(t-\tau(t,x)\geqslant 0\), \(t\geqslant 0\). Under these conditions the integral curves of equation (1) can be realized on the torus by taking, for example, the line \(t=0\) as a certain circle with a plane perpendicular to the axis of the torus, and identifying on it the lines \(x=0\) and \(x=T\). However, the meridians and parallels on the torus may be interchanged without any change in the conditions formulated above.

Let us study in more detail the behavior of trajectories on the torus under a monotone increase of \(t\) (only such a change of \(t\) can be considered). For the initial data \(x(0,x_0)=x_0\), consider the function \(\psi(x_0)=x(T,x_0)\). Here we assume that all solutions of equation (1) beginning at \(t=0\) exist on the interval \([0,T]\); moreover, each such solution is unique and the function \(\psi(x_0)\) depends continuously on \(x_0\). All these assumptions are fulfilled under the conditions formulated earlier (see [1]). It is also obvious that \(\psi(x_0+T)=\psi(x_0)+T\).

By the continuous dependence on the initial data, any bounded connected set on the line \(t=0\) is mapped by the function \(\psi(x_0)\) into a bounded and connected set on the line \(t=T\). Hence, in particular, it follows that the circle \(t=0\) on the torus is mapped onto the circle \(t=T\), i.e., onto itself, though possibly not one-to-one. Form the functions \(\psi_2(x_0)=\psi(\psi(x_0)),\ldots,\psi_n(x_0)=\psi(\psi_{n-1}(x_0))\), \(n>1\). Obviously, \(\psi_n(x_0+T)=\psi_n(x_0)+T\). Consider

\[ \mu=\lim_{n\to+\infty}\frac{\psi_n(x_0)}{Tn}, \tag{2} \]

if this limit exists, and call \(\mu\) the rotation number corresponding to the given \(x_0\). We establish some properties of the number \(\mu\).

1) If for some \(x_0\) and some natural number \(m\), \(\psi_m(x_0)=x_0\pmod T\), i.e., the curve closes on the torus after \(m\) revolutions, then the number \(\mu\) exists and is rational.

2) If \(\mu\) is one and the same for all \(x_0\) and is rational, then there exists at least one trajectory on the torus that closes after a finite number of revolutions.

The proofs of these assertions are analogous to their proofs for ordinary differential equations (see [2], [3]).

We note that if, on the half-plane \((t\geqslant 0,x)\), the deviation from one another (in the sense of Hausdorff [4]) of curves corresponding to different initial data (one may take, for example, \(0<x_0<T\)) is bounded for \(t\geqslant 0\), then the rotation numbers of all curves on the torus, if they exist, are identical.

On the two-dimensional torus one can also realize equations of neutral type:

\[ \dot{x}(t)=f(t,x(t),x(t-\vartheta_1(t,x)),\dot{x}(t-\vartheta_2(t,x))), \]

where \(f\) is continuously differentiable here with respect to the last three arguments, continuous in the aggregate of its arguments, and \(T\)-periodic in each of the first three arguments; the equation \(z=f(0,x_0,x_0,z)\) must be uniquely solvable for any \(x_0\in[0,T]\); \(f(t,0,0,z)\equiv0\), \(t\geqslant0\), \(-\infty<z<+\infty\); \(t-\vartheta_1\geqslant0\), \(t-\vartheta_2\geqslant0\) for \(t\geqslant0\), and \(\vartheta_1\) and \(\vartheta_2\) are continuous functions in the aggregate of their arguments, \(T\)-periodic in each argument, and continuously differentiable with respect to \(x\),

\[ \vartheta_i(0,x)=\vartheta_i(t,0)=0\quad (i=1,2). \]

The existence, uniqueness, and also continuous dependence of solutions of this equation on the initial data can be proved under certain conditions on the function \(f\), by reducing this equation to an equation not containing derivatives of the unknown function, applying Euler’s polygonal method and the method of differential and functional inequalities, similarly to how this was done in [5].

On a torus of a larger number of dimensions, under suitable conditions, systems of equations of the form (1) can also be realized.

As an example, consider the equation

\[ \dot{x}(t)=\sin x\bigl(t-\delta|\sin t\sin x|\bigr)+\varepsilon\sin t,\qquad 0<\delta<1,\quad 0<\varepsilon<1. \]

This equation has at least one \(2\pi\)-periodic solution, since for \(x=\dfrac{\pi}{2}\), \(\dot{x}>1-\varepsilon\), while for \(x=\dfrac{3\pi}{2}\), \(\dot{x}<-1+\varepsilon\). (For \(\varepsilon=0\) we have the obvious solutions \(x=k\pi\).) Moreover, \(t-\delta|\sin t\sin x|\ge 0\) for \(t\ge 0\), and the integral curves of this equation are realizable on the torus \((T=2\pi)\). The rotation number is the same for all integral curves and is equal to 1, since the deviations of the integral curves from one another are bounded.

Remark 1. In this paper, the inequality \(t-\tau(t,x)\ge 0\) for \(t\ge 0\) could be assumed to hold only on the integral curves of the equations under study.

Remark 2. If there exists an ordinary differential equation with a right-hand side smooth in \(x\) and continuous in \((t,x)\), for which the entire set of integral curves coincides with the entire set of integral curves of equation (1) under the condition that \(t-\tau(t,x)\ge 0\), \(t\ge 0\), \(x(0)=c\), then, for example, the behavior of trajectories on the torus can be studied by means of this ordinary differential equation.

References

1. Driver R. D. Contributions to Differential Equations 1, 1963, 317—336.
2. Bliss V. A. Nonlocal Problems in the Theory of Oscillations. Publishing House “Nauka,” Moscow—Leningrad, 1964.
3. Poincaré A. On Curves Defined by Differential Equations. GITTL, 1947.
4. Barbashin E. A. DAN SSSR, 111, No. 1, 1956.
5. Driver R. D. International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics. Academic Press. New York, 1963, pp. 474—484.

Received by the editors
February 16, 1965.