Piecewise linear differential systems without equilibria produce limit cycles?

Abstract

In this article, we study the planar piecewise differential systems formed by two linear differential systems separated by a straight line, such that both linear differential have no equilibria, neither real nor virtual. When the piecewise differential system is continuous, we show that the system has no limit cycles. But when the piecewise differential system is discontinuous, we show that it can have at most one limit cycle.

Appendix extended complete Chebyshev system

The functions $(f_0, \ldots , f_n)$ defined on an interval I form an extended Chebyshev system if and only if any nonzero linear combination of these functions has at most n zeros in I taking into account their multiplicities and this number is reached.

The functions $(f_0, \ldots , f_n)$ form an extended complete Chebyshev system if and only if for any $k\in \{0, 1, \ldots , n\}$, $(f_0, \ldots , f_k)$ form an extended Chebyshev system.

Theorem 5

Let $f_0, \ldots , f_n$ be analytic functions defined on an open interval $I\subset {\mathbb {R}}$. Then $(f_0, \ldots , f_n)$ is an extended complete Chebyshev system on I if and only if for each $k\in \{0, 1, \ldots , n\}$ and all $y\in I$ the Wronskian

$$\begin{aligned} W(f_0, \ldots , f_k)(y)= \left| \begin{array}{cccc} f_0(y) &{} f_1(y) &{} \cdots &{} f_k(y)\\ f_0'(y) &{} f_1'(y) &{} \cdots &{} f_k'(y)\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ f_0^{(k)}(y) &{} f_1^{(k)}(y) &{} \cdots &{} f_k^{(k)}(y) \end{array} \right| \end{aligned}$$

is different from zero.

For a proof of Theorem 5, see [16].