📂Dynamics

The Van der Pol Oscillator

Duffing Equation¹

$$ \ddot{x} + \delta \dot{x} + \alpha x + \beta x^{3} = \gamma \cos \left( \omega t \right) $$

Variables

• $t$: Represents time.
• $x$: Represents position in (for example, a particle’s) $1$-dimensional space.
• $\dot{x}$: Represents (a particle’s) velocity.
• $\ddot{x}$: Represents (a particle’s) acceleration.

Parameters

• $\delta$: Controls damping, similar to friction.
• $\alpha$: Controls stiffness, similar to a spring constant, affecting potential energy.
• $\beta$: Controls nonlinear restoring force; when $\beta = 0$, it describes a forced damped harmonic oscillator.
• $\gamma$: Controls the magnitude of the driving force; when $\gamma = 0$, it becomes a self-sustaining system and is called Unforced Duffing Oscillator.
• $\omega$: Controls angular frequency.

Classical Mechanics Significance

The Duffing oscillator is a non-autonomous system named after Georg Duffing, illustrating the oscillations of an object with both kinetic and potential energy. $x$ could represent, for example, the length of a suspended spring, and given the various parameters, it can describe more complex scenarios than that.

The physical interpretation of the Duffing equation comes into play when there is a classical mechanics interest. In dynamics, where there is an interest in mathematical properties, the system is preferably represented by the following set of coupled ordinary differential equations.

Duffing Oscillator System²

$$ \begin{align*} \dot{x} =& y \\ \dot{y} =& \gamma \cos ( \omega t ) - \alpha x - \beta x^{3} - \delta y \end{align*} $$

Variables

• $x(t)$: Represents position in (for example, a particle’s) $1$-dimensional space.
• $y(t)$: Represents (a particle’s) velocity.

Parameters

• $\delta$: Controls damping, similar to friction.
• $\alpha$: Controls stiffness, similar to a spring constant, affecting potential energy.

$\beta$: Controls nonlinear restoring force; when $\beta = 0$, it describes a forced damped harmonic oscillator.

• $\gamma$: Controls the magnitude of the driving force; when $\gamma = 0$, it becomes a self-sustaining system and is called Unforced Duffing Oscillator.
• $\omega$: Controls angular frequency.

Dynamical Analysis

As mentioned before, when $\gamma = 0$, the Duffing oscillator becomes a self-sustaining system. Textbook examples simplify it to the form of Unforced Damped Duffing Oscillator, as seen in equations $\alpha = -1$ and $\beta = 1$. $$ \begin{align*} \dot{x} =& y \\ \dot{y} =& x - x^{3} - \delta y \end{align*} $$ In dynamics, the interest is not so much in the solution of these differential equations themselves. Rather, if they can be solved easily, it is precisely that reason which might make it unworthy of analysis using the tools of dynamics. The interest lies in how the system changes with variations in parameter $\delta$. Mechanically, for the system to be meaningful, the damping represented by $\delta$ cannot be negative, hence it must be $$ \begin{align*} \dot{x} =& y \\ \dot{y} =& x - x^{3} - \delta y \qquad , \delta \ge 0 \end{align*} $$ Of course, the system’s equilibrium points $(\dot{x} , \dot{y}) = (0,0)$ are given as follows, which is obvious. $$ (x,y) = (0,0) , (\pm 1 , 0) $$ Thus, for dynamic analysis of this system, it is necessary to represent the equation as a nonlinear function $f,g$. $$ \begin{align*} \dot{x} =& f(x,y) \\ \dot{y} =& g(x,y ; \delta) \qquad , \delta \ge 0 \end{align*} $$ This can be expressed in vector form as $\mathbb{x} ' = \mathbb{f} (\mathbb{x})$, and if there is no confusion with the notations, even simply as $\dot{x} = f(x)$. Of course, in such a case, $x,y$ appears and can be very confusing, so it is necessary to clearly state that some nonlinear function $\mathbb{f} : \mathbb{R}^{2} \to \mathbb{R}^{2}$ is represented as $\mathbb{x} ' = \mathbb{f} (\mathbb{x})$ in the form of $\mathbb{x} = (x,y)$. When using this expression, the system’s Jacobian is obtained as follows. $$ D \mathbb{f} = \begin{bmatrix} 0 & 1 \\ 1 - 3 x^{2} & - \delta \end{bmatrix} $$ Now, if we want to know about the Lyapunov stability for the three equilibrium points $(0,0) , (\pm 1 , 0)$, we can simply substitute them one by one. $$ D \mathbb{f} (0,0) = \begin{bmatrix} 0 & 1 \\ 1 & - \delta \end{bmatrix} \\ D \mathbb{f} (\pm1,0) = \begin{bmatrix} 0 & 1 \\ -2 & - \delta \end{bmatrix} $$ By finding the eigenvalues of these matrices, looking at the real parts varying with $\delta$ gives us the stability of the fixed points. Although dynamics covers a vast range of systems, its analytical studies are mostly confined to this traditional method.