Chaos in the  \(\varphi _0\) SFS Josephson Junction

The occurrence of chaos is studied in the superconductor-ferromagnet-superconductor Josephson junction, known as the \(\varphi _0\) junction. We use the Landau-Lifshitz-Gilbert equation to model the net magnetic moment of the ferromagnet and the resistively-capacitively-shunted-junction model for the superconducting junction. The system dynamics is examined near the ferromagnetic resonance region, where the Josephson frequency is reasonably close to the precession frequency of the magnetic moment about the effective field. In this system, we find that the presence of the ferromagnetic layer effectively adds two extra degrees of freedom, compared to the ordinary Josephson junction, and this can lead to chaos, even without any external time-dependent excitation. Detailed analyses of the full spectrum of Lyapunov exponents, as well as the current-voltage characteristics and the maximum value of the magnetization components, show the presence of chaos with two zero Lyapunov exponents. However, the system is not capable of hyperchaos (two positive Lyapunov exponents) for any combination of parameters, despite it being four-dimensional and dissipative. These particular properties are studied by plotting two-dimensional projections of typical chaotic trajectories in the system and through careful examination of the basins of attraction on the unit sphere of the magnetic moment. High-resolution analyses of two-dimensional bifurcation diagrams reveal a very complex landscape of regular, quasi-periodic, and chaotic, regions. Within the chaotic regions, we observe many islands of periodicity, known as shrimp structures.