Josephson junctions are a combination of two superconductors weakly coupled by a conductive material. The Josephson Junction is modeled by the system of a nonuniform oscillator.
Solving 4.6.5 d)
After defining the equation
$$to describe the circuit-current relations in a Josephson Junction containing circuit where $\phi_{k}$ is the phase difference of the JJ. Finding an explicit equation for $\dot{\phi}_{k}$ the entire equation was summed for $K=1,2,\dots,N$.\sum_{i=1}^{N}b = I_{c}\sum_{i=1}^{N}\sin(\phi_{i})+\frac{h}{2er}\sum_{i=1}^{N} \dot{\phi_{i}} + \sum_{i=1}^{N}\underbrace{\left( \frac{h}{2eR}\sum_{k=1}^{N} \dot{\phi}{k} \right)}{\rho}
because $\sum\_{i=1}^{N}b = Nb$ and the last term $\rho$ is not dependent on k, we write $N\rho$. Through further substitution, non-dimensionalization with $\tau, \Omega, a$ we retrieve\frac{d\phi_{k}}{d\tau} = \Omega +a\sin \phi_{k} + \frac{1}{N}\sum_{j=1}^{N} \sin \phi_{j}
with $\Omega = \frac{RI\_{b}}{NrI\_{c}}, a= -\frac{R+Nr}{Nr}, \tau = \frac{2Ner^2I\_{c}}{h(R+Nr)}$