Full Text
Preamble
DIFFERENTIAL EQUATIONS
APRIL 1967, VOLUME III, No. 4
DETERMINATION OF CAPTURE REGIONS FOR SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS
Introduction
Consider a system of equations:
$$\frac{dx_i}{dt} = \sum_{j=1}^{n} \phi_{ij}(t + \tau) x_j, \quad i = 1, 2, \dots, n \tag{1}$$
where $\phi_{ij}$ are real continuous periodic functions with period $\omega$, and $\tau$ is a parameter. The solution to this system can be written in the form:
$$x_i(t, \tau, x_{1,0}, x_{2,0}, \dots, x_{n,0}) = \sum_{j=1}^{n} u_{ij}(t, \tau) x_{j,0} \tag{2}$$
where the quantities $u_{ij}$ are defined as:
$$u_{ij}(t, \tau) = x_i(t, \tau, 0, \dots, 1, \dots, 0)$$
We define the capture region with respect to the variables $x_1, x_2, \dots, x_n$ as the set of all points in $n$-dimensional space such that, for any $t > 0$, the formulas (2) yield a point in $(p+1)$-dimensional space lying within the $(p+1)$-dimensional parallelepiped:
$$|x_j| \le a_j, \quad j = i, i+1, \dots, i+p$$
where $a_j$ are given positive constants.
In the present work, we calculate quantities that generally provide a lower bound for the capture regions for the cases $n=2$ and $n=3$. Furthermore, a non-trivial example is presented where the calculated value, under certain known conditions, yields the exact magnitude of the capture region.
§ 1. INTEGRALS OF MOTION OF A SYSTEM OF TWO EQUATIONS
Consider a system of equations given by:
$$\frac{dx_i}{dt} = \sum_{j=1}^{2} \phi_{ij}(t + \tau) x_j, \quad i = 1, 2 \tag{1.1}$$
where $\phi_{ij}$ are continuous, periodic real-valued functions with a period of $\omega$, and $\tau$ is a parameter. It is well known \cite{1} that in cases where the characteristic roots of (1.1) are distinct, the matrix of linearly independent solutions for this system can be expressed as follows:
$$W(t, \tau) = \exp \left[ \int_{0}^{t} \Phi(\xi + \tau) \, d\xi \right] \tag{1.2}$$
The solution can be represented using the following transformation:
$$\Psi(z) = [f(z)] \Phi(z) + [g(z)] \Phi^*(z) \tag{1.3}$$
where the square brackets denote a diagonal matrix; $f, g$ are complex-valued functions, and the asterisk denotes complex conjugation. Here, $\Psi(z)$ represents a periodic solution to the nonlinear equation:
$$\mathcal{L} \Psi = 0 \tag{1.5}$$
We proceed by separating the equation into its real and imaginary parts:
$$w_{21} = u_{21} + i v_{21} \tag{1.6}$$
and we shall assume that $k > 0$. In place of the function $u$, we introduce another function $v$ according to the relation:
$$v = \frac{u}{\phi_{11} - \phi_{22}} \tag{1.7}$$
By substituting (1.6) into (1.5) and subsequently separating the real and imaginary parts, we find that the function $v$ satisfies the equation:
$$v'' - \frac{\phi'_{12}}{\phi_{12}} v' + (\phi_{11} - \phi_{22}) v = 0 \tag{1.8}$$
and the function $v$ is expressed in terms of $u$ as follows: $u = v(\phi_{11} - \phi_{22})$ (1.9). Equation (1.8) generally possesses complex periodic solutions. However, we shall assume that the coefficients $\phi_{ij}$ are such that the function is real-valued. Let us introduce the notation:
$$\eta = \phi_{11} + \phi_{22}$$
The solution to (1.1) is written using (1.2) as follows:
$$x(t) = W^{-1}(z + t) W(z) x_0 \tag{1.12}$$
By applying (1.3) and the subsequent formulas, we find that (1.12) is equivalent to the equation:
$$s_{1} + i s_{2} = S_{1}(x, t) + i S_{2}(x, t) \tag{1.14}$$
By equating the moduli and phases of the complex quantities on the right and left sides of (1.14), we obtain:
$$s_{1}^2(x, z, t) + s_{2}^2(x, z, t) = \exp \left( \int_{0}^{t} (\phi_{11} + \phi_{22}) \, d\xi \right) \tag{1.15}$$
$$\partial_t \sigma(X, Z, t) = \sigma(X, z) \tag{1.16}$$
where $\sigma(X, z) = \sigma(X, z, 0)$ (1.18). In the following, we shall refer to (1.15) as the first integral and to (1.16) as the second integral of the system (1.1).
§ 2. STUDY OF THE FIRST INTEGRAL OF THE SYSTEM
We solve equations (1.13) with respect to $t$, and let $u_{\pm}(x, z, t) = v(z \pm t) \pm s_2(x, z, t)$:
$$u(x, z, t) = \Phi u'(z + t) + \Phi u'(x, z, t) \tag{2.1}$$
Let us assign specific values to $z$ and $t$, and choose the components of the vector $x$ such that the expression $s_1^2(x, z) + s_2^2(x, z)$ remains constant. Then the values of $(x, z, t)$ will correspond to the coordinates of the points on the circle (1.15) in the plane. To find the maximum absolute values of these functions on this circle, it is necessary to solve the relative extremum problem (see \cite{2}). As a result, for the maximum of $|u|$, we obtain:
$$|u(x, z, t)| = \exp \left( - \int_{0}^{t} (\phi_{11} + \phi_{22}) \, d\zeta \right) \sqrt{v^2(z + t) + s^2(x, z)} \tag{2.2}$$
We shall now determine two regions of values for the quantity $Q = \sqrt{S_1^2(x, z) + S_2^2(x, z)}$ (2.4) such that the values from the first region, when substituted into (2.2), and those from the second region, when substituted into (2.3), yield $|x_i| < \epsilon$ ($i = 1, 2$) (2.5) for all $t$ in the interval $0 < t < \infty$ (2.6).
Let us denote by $M_1(z)$ the minimum of the function $v(z + t) \exp \{ -\int_0^t \phi_1(\tau) d\tau \}$ and by $M_2(z)$ the minimum of the function related to the second component. The required intervals are then given by ($i = 1, 2$) (2.10). The formula for the area of the region in the $(x, z, t)$ plane is:
$$A_i(z, t) = \pi M_i^2(z) \exp \left\{ \int_0^t \phi_i(\tau) d\tau \right\} \tag{2.11}$$
By setting $t = 0$ in (2.11), we obtain a value that provides a lower bound for the area of the capture region:
$$J_i(z, 0) = \pi M_i^2(z) \tag{2.12}$$
The final capture region area is limited by:
$$K(z) = \min [ V_1(z, 0), V_2(z, 0) ] \tag{2.13}$$
Consider the special case \cite{3}: $\phi_{11}(z) = \phi_{22}(z) = 0, \phi_{12}(z) = 1, \phi_{21}(z) = -g(z)$. The regions in the plane will be the interiors of the ellipses:
$$\frac{x_1^2}{v^2(z)} + \frac{(v(z)x_2 - v'(z)x_1)^2}{1} \le \epsilon^2 \tag{2.16}$$
§ 3. STUDY OF THE SECOND INTEGRAL OF SYSTEM (1.1)
We write the function $w(x, z, t)$ for the case (2.14):
$$w(x, z, t) = P_i \cos \theta(x_1, x_2, z) \tag{3.1}$$
where
$$\tan \theta(x_1, x_2, z) = -\frac{1}{v(z)} \frac{x_2 v(z) - x_1 v'(z)}{x_1} \tag{3.3}$$
Consider the case where $v^*(z) = c_0 (1 - 2b \cos 2\pi z)$ (3.4). For simplicity, let $z = 0$, $b = 1/2$, and $c_0 = -1/2$. We obtain:
$$u(x, 0, t) = p(x_1, x_2, 0) \sqrt{2 - \cos(2\pi t)} \sin(2\pi t + \sigma) \tag{3.11}$$
The function (3.11) is periodic. It can be deduced that the maximum of $|\chi(x, 0, t)| < 1$ over a single period. Consequently, there exist points on the ellipse (3.9) that satisfy the condition $|u(x, 0, t)| < a$ for all $t$. If the function (3.11) were not periodic—which occurs when $\omega/\pi$ is irrational—the capture region would coincide exactly with the region bounded by the ellipse (3.9).
§ 4. INTEGRALS OF MOTION FOR THE SYSTEM OF FOUR EQUATIONS
Let there be a system of equations given by:
$$\phi_k = \phi_k(z) \quad (k = 1, 2, 3, 4) \tag{4.1}$$
where $\phi_k(z)$ are periodic real-valued functions with period 1. The matrix of linearly independent solutions is:
$$W(z) = \exp \left[ \int \Phi(z, \xi) \, d\xi \right]; \quad \Phi = (\phi_{ik}) \tag{4.2}$$
The elements of the matrix are periodic solutions of the nonlinear system:
$$\dot{\Phi}_{ik} = \Phi_{ik} \tag{4.4}$$
We separate the real and imaginary parts: $\Phi_{ik} = u_{ik} + i v_{ik}$ (4.8). We assume the determinant $\Delta \neq 0$ (4.9). We transform to the variables $s_1, s_2, s_3, s_4$ (4.10). This leads to the first integrals:
$$|s|^2 + |\sigma|^2 = Q^2 \exp\left(2 \int \gamma \, d\xi\right) \tag{4.22}$$
$$s f^* + s^* f = Q \exp\left(2 \int \gamma \, d\xi\right) \tag{4.23}$$
and the second integrals:
$$\phi(x, z, t) = \int \omega \, d\xi + \Phi(x, z, 0) \tag{4.24}$$
$$\phi(x, z, t) = \Phi(x, z, 0) \tag{4.25}$$
§ 5. INVESTIGATION OF THE FIRST INTEGRALS OF SYSTEM (4.1)
We solve equations (4.10) for $s_\ell + a_\ell$:
$$s_\ell + a_\ell = \frac{j_\ell + a_\ell}{(z + t)s_\ell} \tag{5.1}$$
We seek the maximum value $m_i(x, z)$ subject to constraints (4.22) and (4.23):
$$m_i(x, z) = Q_1 \sqrt{a_1^2(z + t) + a_1^2} + Q_2 \sqrt{a_2^2(z + t) + a_2^2} \tag{5.2}$$
[FIGURE:1]
The region where every point satisfies inequality (2.5) for all $t$ is bounded by the coordinate axes and a curve [FIGURE:2]. Let $M(x, z)$ denote the minimum of $Q$ within the interval (2.7). The parametric equation of the boundary curve is:
$$Q_1 = Q_1(x, z), \quad Q_2 = Q_2(x, z) \tag{5.7}$$
[FIGURE:2]
The four-dimensional volume corresponding to the values of $Q$ and $\bar{Q}$ is:
$$V = \frac{Q_s(z)}{\Phi} \int \Phi(Q, r) Q \, dQ \tag{5.16}$$
By setting $\xi = 0$ in (5.16), we obtain the volume:
$$Q_i(z) = \int_{0}^{\infty} \dots \int_{0}^{\infty} Q \, dQ \tag{5.17}$$
which serves as a lower bound for the measure of the capture region. In conclusion, I express my gratitude to A. D. Myshkis and Yu. S. Bogdanov for valuable discussions.
References
- Sharshanov, A. A. Mathematical Physics. Naukova Dumka, Kiev.
- Fikhtengol'ts, G. M. Course of Differential and Integral Calculus. Moscow-Leningrad, Gostekhizdat, 1947, pp. 536–544.
- Courant, E., Snyder, H. Annals of Physics, 3, 1–48, 1958.
Received by the editorial office
Physico-Technical Institute of the Academy of Sciences
March 10