4.2 Dulac's criteria

In this section, we consider the 2-dimensional system

 

x' = f(x) (4.7)

and establish conditions under which (4.7) has no periodic solutions.

Theorem 4.2.1: (Bendixson's Criterion)    Let $\Omega \subset R^{2}$ be a simply connected open set and $f \in C^{1} (\Omega )$. If the divergence of the vector field ${\rm div} f = {\dss\frac{\partial f_{1}}{\partial x_{1}} + \frac{\partial f_{2}}{\partial x_{2}}}$ is nonzero and does not change sign on $\Omega$, then the system (4.7) has no periodic orbit lying entirely in $\Omega$.

Proof: (omitted)

Example 4.2.1:     Consider the damped Hamiltonian system in R²

$$\left\{ \begin{array}{l} x_{1}' = x_{2}, \\ x_{2}' = - \alpha x_{2} - v' (x_{1}), \quad \alpha > 0 \, . \end{array} \right.$$ (4.8)

$${\rm div} f = \frac{\partial}{\partial x_{1}} (x_{2}) + \frac... ... v' (x_{1})) = - \alpha < 0, \quad \forall \; x \in R^{2} \, .$$

Thus by Theorem 4.2.1, the system (4.8) admits no periodic orbits, irrespective of the form of the potential function v(x₁).

The following result due to Dulac is a generalization of Theorem 4.2.1.

Theorem 4.2.2: (Dulac's criterion)     Let $\Omega \subset R^{2}$ be a simply connected open set and $f \in C^{1} (\Omega )$. If there exists a function $B \in C^{1} (\Omega )$ such that ${\rm div} (Bf)$ is nonzero and does not change sign in $\Omega$ then (4.7) has no periodic orbits lying entirely in $\Omega$.

Example 4.2.2:     Consider the population growth model

$$\left\{ \begin{array}{l} x_{1}' = x_{1} - x_{1}^{2} - x_{1} x... ...} x_{2}^{2} - \frac{3}{4} x_{1} x_{2} \, . \end{array} \right.$$ (4.9)

$$\begin{eqnarray*}{\rm div} f & = & \frac{\partial}{\partial x_{1}} (x_{1} - x_{1... ... \\ & = & \frac{3}{2} - \frac{11}{4} x_{1} - \frac{3}{2} x_{2} \end{eqnarray*}$$

which does not have constant sign.

Let B(x₁, x₂) = x₁^-1 x₂^-2. Then

$$\begin{eqnarray*}{\rm div} (Bf) & = & \frac{\partial}{\partial x_{1}} (x_{2}^{-2... ...x{for all} \quad x_{1} > 0 \quad \mbox{and} \quad x_{2} > 0 \, . \end{eqnarray*}$$

Thus by Theorem 4.2.2, system (4.9) has no periodic solutions in the first quadrant.

The next criterion for excluding periodic orbits, which is valid in Rⁿ, $n \geq 2$, is the Lyapunov function approach.

Example 4.2.2 Simulation

Theorem 4.2.3:     Let $V \in C^{1} (\Omega )$, where $\Omega \subset R^{n}$ is an open set. If
$\dot{V} (x) = \nabla V (x) \cdot f (x) \leq 0$, V(x) is bounded from below on $\Omega$, and the set $E = \{ x \in \ol{\Omega} ; \; \dot{V} (x) = 0 \}$ contains no whole orbits except possibly equilibrium points of (4.7), then the system (4.7) has no periodic solutions lying entirely in $\Omega$.

Proof:     It follows from Theorem 3.7.2.

Example 4.2.3:     Consider the nonlinear system

$$\left\{ \begin{array}{l} x_{1}' = x_{2}, \\ x_{2}' = - \alph... ...1}^{2} - 2x_{1} \, , \quad \alpha > 0 \, . \end{array} \right.$$ (4.10)

Let V(x₁, x₂) = 2x₁² + x₂² + 5. $\dot{V} (x_{1}, x_{2}) = - 2 \alpha x_{1}^{2} x_{2}^{2}$. $E = \{ (x_{1}, x_{2}) \in R^{2}; \, x_{1} =0$ or $x_{2} = 0 \}$ contains (0,0) as the only whole orbit. Thus by Theorem 4.2.3, the system (4.10) has no periodic solution in R².

Example 4.2.3 Simulation