Bifurcation cascade, self-similarity, and duality in the three-rotor problem

Area of Science:

• Classical and quantum dynamics
• Nonlinear systems and chaos theory
• Mathematical physics

Background:

• The three-rotor system, modeling coupled Josephson junctions, exhibits complex order-chaos-order dynamics.
• It features stable periodic orbits, including pendula and breathers, with known stability transitions.
• A specific energy band (5.33g≲E≲5.6g) is characterized by seemingly global chaos.

Purpose of the Study:

• To investigate the nature of stability transitions in the three-rotor system's periodic orbits.
• To identify and characterize new families of periodic orbits emerging from these transitions.
• To explore self-similarity and scaling laws associated with bifurcations.

Main Methods:

• Analysis of stability transitions using isochronous and period-doubling bifurcations.
• Employing an efficient search algorithm initiated by transverse perturbations to discover new orbit families.
• Numerical validation of scaling constants and asymptotic behaviors.

Main Results:

• Identified fork-like bifurcations driving the creation of new periodic orbit families.
• Demonstrated an asymptotic duality between bifurcation energies and orbit shapes, described by Lamé equations.
• Found and validated scaling constants for self-similarity in orbit stability and shapes near E=4g.

Conclusions:

• The study elucidates the mechanisms behind stability transitions and the birth of new periodic orbits.
• New orbit families exhibit self-similar properties, particularly as energy approaches E=4g.
• The findings reinforce the presence of global chaos in the 5.33g≲E≲5.6g energy range.