4.4 Lienard systems

Although the Poincaré-Bendixson theorem can be used to infer the existence of at least one periodic solution, it provides no information on the stability property of the periodic solution. We consider, in this section, some physical systems and establish some stronger results.

The following Van der Pol equation

$$x'' + \mu (x^{2}-1) x' + x = 0 \, ,\quad \mu > 0 \, .$$ (4.14)

was studied by a Dutch scientist Balthasar Van der Pol in 1922 in a paper on radio circuits. (4.14) can be rewritten as

$$\left[ x' +\mu \left( \frac{x^{3}}{3} - x \right) \right]' + x = 0 \, .$$

Thus with x₁ = x, $x_{2} = x' + \mu \left( \frac{x^{3}}{3} - x \right)$, the Van der Pol equation reduces to

$$\left\{ \begin{array}{l} x_{1}' = x_{2} - \mu \left( \frac{x_... ...\right) \, , \\ \\ x_{2}' = - x_{1} \, . \end{array} \right.$$ (4.15)

The origin is the only equilibrium point of (4.15) and thus by Theorem 4.3.2, any periodic orbit must enclose the origin. Further, ${\dss\frac{\partial f_{1}}{\partial x_{1}} + \frac{\partial f_{2}}{\partial x_{2}}} = \mu ( 1 - x_{1}^{2})$ and hence by Theorem 4.2.1, any periodic orbit must include a portion outside the strip x₁² = 1, as well as a portion inside this strip.

Lienard Systems Simulation

Theorem 4.4.1:     There exists a unique isolated periodic orbit for (4.15) which is asymptotically stable.

Proof: (omitted)

In 1928, a French scientist, Alfred Liénard, was also interested in self-sustained oscillations and considered a more general equation

 

x'' + f(x) x' + g(x) = 0 (4.25)

which is now known as Liénard's equation.

Let $F(x) = {\dss\int_{0}^{x}} f(s)ds$ and $G(x) = {\dss\int_{0}^{x}} g(s)ds$. Then (4.25) is equivalent to

$$\left\{ \begin{array}{l} x_{1}' = x_{2} - F(x_{1}), \\ x_{2}' = - g(x_{1}) . \end{array} \right.$$ (4.26)

Theorem 4.4.2: (Liénard's theorem)      Assume that

(i)

$f, g \in C^{1} (R)$, F and g are odd functions of x, x g(x) > 0 for $x \neq 0$;

(ii)

F(0) = 0, F' (0) < 0 and F has a single positive zero at x₁ = a;

(iii)

F increases monotonically to infinity for $x \geq a$ as $x_{1} \raro \infty$.

Then system (4.26) has exactly one isolated periodic orbit which is asymptotically stable.

Proof: (omitted)

The Liénard's equation has been studied by many mathematicians for the past sixty years and still remains an object of research.

In 1923, Dulac showed that a planar system with f(x) analytic in $\Omega \subset R^{2}$ cannot have an infinite number of limit cycles in $\Omega$ provided $\Omega$ is bounded, and if f(x) is a polynomial system, then (4.14) has at most a finite number of limit cycles in R².

Theorem 4.4.3: (Dulac)     Let f(x) be analytic on $\Omega$. Then system (4.14) has at most a finite number of isolated periodic orbits in $\Omega$ if $\Omega$ is bounded. If f(x) is a polynomial, then (4.14) has at most a finite number of isolated periodic orbits in R².

Errors were recently found in Dulac's original proof and are corrected in 1988, independently, by a group of French mathematicians and the Russian mathematician Y. Ilyashenko.

In 1900, David Hilbert presented a list of 23 mathematical problems to the 2$^{\rm nd}$ International Congress of Mathematicians. The 16$^{\rm th}$ is the following question: What is the maximum number of isolated periodic orbits of an autonomous ODE in R² if the vector field f(x) is a polynomial function? This problem is still unsolved even for the case of quadratic polynomials. In 1962, the Russian mathematician N.V. Bautin proved that any quadratic system has at most three local limit cycles. And it had been believed that the upper bound was three until in 1979, the Chinese mathematicians S.L. Shi, L.S. Chen and M.S. Wang produced examples of quadratic systems with four limit cycles. For a detailed discussion of this topic, see the monograph ``Theory of limit cycles'' by Yanqian Ye published in 1986.