#### Case 1: Assume that the resistor is voltage-controlled ($i = g(v)$)

The state equations can then be written as

\begin{equation} \frac{dq}{dt} = -g(v) = -g(q/C) \end{equation}

The trajectories can be visualized on the dynamic route, where we plot $\frac{dq}{dt}$ against $q$ and the arrows indicate how the trajectory will evolve:

#### Case 2: assume that the resistor is current controlled ($v = f(i)$).

We have $v = f(i) = f(-dq/dt)$. Since $v = q/C$ we get $q = Cf(-dq/dt)$. Using the dynamic route interpretation and depending on the shape of $f$, it is possible to find*impasse*points, where the trajectory does not know what to do.

How do we resolve such nonphysical situations? By adding parasitic elements. Suppose we add a parasitic inductor

*L*in series to the nonlinear resistor.

The resulting state equations will be given by: \[ \begin{array}{lcl}\frac{dq}{dt} & = & i_L = \frac{\phi}{L} \\ \frac{d\phi}{dt} & = & v_L = v_R - v_C = f(i_R) - q/c = f(\phi/L) - q/c \end{array} \]

By introducing the variable $i_C = -i_L = -\phi/L$ we get the state equations: \[ \begin{array}{lcl}\frac{dq}{dt} & = & i_C \\ \frac{di_C}{dt} & = & -\frac{1}{L}(q/C - f(-i_C)) \end{array} \] By drawing the phase portrait, we see that at the curve $q = Cf(-i_C)$, $di_C/dt = 0$. In fact, we have an oscillator.

For the nonlinear resistor with a constituency relation as shown above, the existence of the limit cycle can be proved more rigorously by using the Poincaré -Bendixson theorem and the fact that the system has a bounded trapping region and a single equilibrium point that is unstable.

For $f$ an odd function, a second order system in this form is a Liénard equation [5] and the Levinson-Smith theorem [6] allows us to prove the existence of a stable limit cycle when the nonlinear resistor is of a certain form:

**Theorem [Levinson-Smith 1942]:**Consider the second order equations

\[ \begin{array}{lcl} dx/dt & = & y - F(x) \\ dy/dt & = & -g(x) \end{array} \] Suppose that

- $F$ is an odd $C^1$ function (i.e. $F(-x) = -F(x)$),
- $g$ is an odd $C^0$ function with $xg(x)>0$ for all $x\neq 0$,
- There is a constant $a>0$ such that $F(x) < 0$ on $0 < x < a$, $F(x) > 0$ on $x > a$, and $F'(x) > 0$ on $x>a$,
- $\int_0^x g(s)ds \rightarrow\infty$ as $|x|\rightarrow \infty$ and $F(x) \rightarrow\infty$ as $x\rightarrow \infty$.

Note that our circuit above satisfies the hypothesis of the Levinson-Smith theorem if our nonlinear resistor has the form $v = f(i)$ where $f$ is a cubic polynomial of the form $x^3-ax$ for $a>0$. In particular, we get exactly the Van der Pol equation $x'' + \alpha (x^2-1)x' + x = 0$ [8], an equation used to model oscillations occurring in a circuit involving vacuum tubes.

Another condition for guaranteeing the existence of a unique periodic solution is the following result [7]:

**Theorem:**Consider the second order equations

\[ \begin{array}{lcl} dx/dt & = & y - F(x) \\ dy/dt & = & -g(x) \end{array} \] Suppose that

- $F$ is an odd $C^1$ function and is zero only at $x=0$,$x=a$ and $x=-a$ for some $a>0$,
- $F(x) \rightarrow \infty$ monotonically as $x\rightarrow \infty$ for $x >a$.
- $g$ is an odd $C^0$ function with $xg(x)>0$ for all $x\neq 0$,

**References:**

[1] L. O. Chua, Introduction to nonlinear network theory, McGraw-Hill, 1969.

[2] L. O. Chua, C. A. Desoer, E. S. Kuh, Linear and nonlinear circuits, McGraw-Hill, 1987.

[3] L. O. Chua, Memristor-The missing circuit element, IEEE Transactions on Circuit Theory, Sept. 1971, vol. 18, no. 5, pp. 507-519.

[4] D. B. Strukov, G. S. Snider, D. R. Stewart1 & R. S. Williams, The missing memristor found, Nature 453, 1 May 2008, pp. 80-83.

[5] A. Liénard, Étude des oscillations entretenues, Revue génér. de l'électr. 23, (1928), 901 - 902; 906 - 954.

[6] N. Levinson, O. Smith, A general equation for relaxation oscillations Duke Math. J. 9, (1942), 382 - 403.

[7] D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press, 1977.

[8] Van der Pol, B. and Van der Mark, J., Frequency demultiplication, Nature, vol. 120, pp. 363-364, 1927.