We theoretically study how the dynamics of the resistive state in narrow superconducting channels shunted by an external resistor depends on channels length $L$, the applied current $j$, and parameter $u$ characterizing the penetration depth of the e
lectric field in the nonequilibrium superconductors. We show that changing $u$ dramatically affects both the behaviour of the current-voltage characteristics of the superconducting channels and the dynamics of their order parameter. Previously, it was demonstrated that when $u$ is less than the critical value $u_{c1}$, which does not depend on $L$, the phase slip centers appear simultaneously at different spots of the channel. Herewith, for $u>u_{c1}$ these centres arise consecutively at the same place. In our work we demonstrate that there is another critical value for $u$. Actually, if $u$ does not exceed a certain value $u_{c2}$, which depends on $L$, the current-voltage characteristic exhibits the step-like behaviour. However, for $u>u_{c2}$ it becomes hysteretic. In this case, with increase of $j$ the steady state, which corresponds to the time independent distribution of the order parameter along the channel, losses its stability at switching current value $j_{sw}$, and time periodic oscillations of both the order parameter and electric filed occur in the channel. As $j$ sweeps down, the periodic dynamics ceases at certain retrapping current value $j_r<j_{sw}$. Shunting the channel by a resistor increases the value of $j_r$, while $j_{sw}$ remains unchanged. Thus, for some high enough conductivity of the shunt $j_r$ and $j_{sw}$ eventually coincide, and the hysteretic loop disappears. We reveal dynamical regimes involved in the hysteresis, and discuss the bifurcation transitions between them.
We have investigated the properties of the resistive state of the narrow superconducting channel of the length L/xi=10.88 on the basis of the time-dependent Ginzburg-Landau model. We have demonstrated that the bifurcation points of the time-dependent
Ginzburg-Landau equations cause a number of singularities of the current-voltage characteristic of the channel. We have analytically estimated the averaged voltage and the period of the oscillating solution for the relatively small currents. We have also found the range of currents where the system possesses the chaotic behavior.
The current-voltage characteristics of long and narrow superconducting channels are investigated using the time-dependent Ginzburg-Landau equations for complex order parameter. We found out that the steps in the current voltage characteristic can be
associated with bifurcations of either steady or oscillatory solution. We revealed typical instabilities which induced the singularities in current-voltage characteristics, and analytically estimated period of oscillations and average voltage in the vicinity of the critical currents. Our results show that these bifurcations can substantially complicate dynamics of the order parameter and eventually lead to appearance of such phenomena as multistability and chaos. The discussed bifurcation phenomena sheds a light on some recent experimental findings.
