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Sufficient Conditions for Isochronism of Canonical Systems of Two Differential Equations
A. P. Vorob’ev
Consider the canonical system
\[ \frac{dx}{dt}=-\frac{1}{2}\frac{\partial}{\partial y}(u^2+v^2),\qquad \frac{dy}{dt}=\frac{1}{2}\frac{\partial}{\partial x}(u^2+v^2), \tag{1} \]
where \(u\) and \(v\) are functions of the variables \(x\) and \(y\). In what follows we shall assume that the functions \(u(x,y)\) and \(v(x,y)\) are defined and continuous, together with their first-order partial derivatives, in some domain \(D\) containing the origin; that they ensure uniqueness of solutions of system (1) at every point of this domain; and that at the point \(x=0,\ y=0\) they have the following properties:
\[ u(0,0)=v(0,0)=u_y(0,0)=v_x(0,0)=0, \]
\[ u_x(0,0)=v_y(0,0)=1. \tag{2} \]
We shall require of the functions \(u(x,y)\) and \(v(x,y)\) that generate system (1) that they carry closed trajectories of this system enclosing the point \(x=0,\ y=0\) into circles, i.e., that the transformation
\[ u=u(x,y),\qquad v=v(x,y), \]
reduce system (1), in some neighborhood of the origin, to the form
\[ \frac{du}{dt}=-v,\qquad \frac{dv}{dt}=u. \]
For this, as in the case of holomorphic \(u\) and \(v\) [1], it is necessary and sufficient that the functions \(u(x,y)\) and \(v(x,y)\) satisfy the equation
\[ u_xv_y-u_yv_x=1. \tag{3} \]
It follows from what has been said that finding sufficient conditions for isochronism of the canonical system (1)—conditions under which the periods of the closed trajectories around the origin do not depend on the initial data—reduces to finding solutions of equation (3).
It is easy to verify directly that the following functions \(u(x,y)\) and \(v(x,y)\) possess properties (2) and satisfy equation (3):
\[ \text{a) }\quad u=z[P(z)+R(x)]-\int_0^z P(z)\,dz+a,\qquad v=P(z)+R(x), \tag{4} \]
where
\[ z=yF(x)+\Phi(x),\quad R(x)=\pm\left(\beta^2-2\int_0^x \frac{dx}{F(x)}\right)^{\frac12}, \]
where \(F(x)\) and \(\Phi(x)\) are arbitrary functions, continuous together with their first derivatives in some neighborhood of the point \(x=0\), \(\alpha\) and \(\beta\) are arbitrary constants, and \(P(z)\) is an arbitrary function, continuous together with its first derivative in some neighborhood of the point \(z=\Phi(0)\), except that these functions and constants satisfy the following relations:
\[ P[\Phi(0)]+R(0)=\Phi(0)+F(0)R(0)=\int_0^{\Phi(0)} P(z)\,dz-\alpha=0, \]
\[ P'[\Phi(0)]F(0)=\Phi(0)R'(0)=\Phi'(0)R(0)=1; \]
\[ \text{b) } \quad u=P(z)+R(x),\quad v=z, \tag{5} \]
where
\[ z=yF(x)+\Phi(x),\quad R(x)=\int_0^x \frac{dx}{F(x)}, \]
and \(F(x)\), \(\Phi(x)\), \(P(x)\) are arbitrary functions, continuous together with their first derivatives in some neighborhood of the point \(x=0\), except that they satisfy the following conditions:
\[ F(0)=1,\quad \Phi(0)=\Phi'(0)=P(0)=P'(0)=0; \]
\[ \text{c) } \quad u=xP(z),\quad v=y/P(z), \tag{6} \]
where \(z=xy\), and \(P(z)\) is an arbitrary function, continuous together with its first derivative in some neighborhood of the point \(z=0\), with \(P(0)=1\).
Suppose that the functions \(u(x,y)\) and \(v(x,y)\), defined by formulas (4), (5), and (6), ensure the uniqueness of solutions of system (1) at every point of some domain \(D\) containing the origin.
We now formulate sufficient conditions for the isochronism of system (1).
Theorem. For the isochronism of the canonical system (1), generated by the functions \(u(x,y)\) and \(v(x,y)\), it is sufficient that these functions have the form a), b), or c).
Remark 1. From the functions \(u(x,y)\) and \(v(x,y)\) defined by formulas (4), (5), and (6), without first subjecting them to conditions (2), one can obtain other functions ensuring the isochronism of system (1), using the following: if there are functions \(u_0(x,y)\) and \(v_0(x,y)\) satisfying equation (3), then the inverse functions to them and the functions \(u(x,y)=v_0(y,x)\) and \(v(x,y)=u_0(y,x)\), as well as the functions \(u(x,y)=au_0(x,y)+bv_0(x,y)\) and \(v(x,y)=cu_0(x,y)+dv_0(x,y)\) \((a,b,c,d\) are constants, \(ad-bc=1)\), also satisfy equation (3).
Remark 2. Taking in formulas (4), (5), and (6) the functions \(F(x)\), \(\Phi(x)\), and \(P(z)\) to be holomorphic, we obtain sufficient conditions for the isochronism of system (1) with holomorphic right-hand sides; moreover, if in formula (5) we set \(F(x)\equiv 1\), and take polynomials as \(\Phi(x)\) and \(P(z)\), then we obtain sufficient conditions for the isochronism of system (1) with polynomial right-hand sides.
It is easy to show that the canonical system (1), generated by functions \(u\) and \(v\) of type a), may have singular points in the domain \(D\) distinct from the origin, whereas the canonical system (1), generated by functions \(u\) and \(v\) of type b) or c), has the unique singular point \(x=0,\ y=0\).
We note that the qualitative picture of the behavior of the integral curves of the canonical system (1) having an isochronous center in the domain \(D\) is determined completely in this domain on the basis of [2].
References
- Vorob’ev A. P. DAN BSSR, 7, No. 8, 1963.
- Vorob’ev A. P. Differential Equations, 1, No. 4, 1965.
Received by the editors
January 21, 1965.
Institute of Mathematics
Academy of Sciences of the BSSR