Hamilton’s equations of motion described the evolution of a physical system that conserves energy. Hamiltonian systems are used to describe molecular dynamics, plasma physics, stellar astronomy phenomena and more. Until only the last decade, it was thought that coupling two or more Hamiltonian systems would not lead to synchronization of each separate system’s oscillations.

Recent studies have revealed weak coupling in various linearly coupled systems, but evidence presented in Chaos: An Interdisciplinary Journal of Nonlinear Science reveals synchronization in nonlinearly coupled Hamiltonian systems.

The report’s authors found that two nonlinearly coupled one-dimensional oscillators exhibit a phenomenon known as measure synchronization (MS). Measure refers to the volume in n-dimensional phase space covered by the oscillator’s orbit. For Hamiltonian systems, the measure is invariant. MS occurs when the individual phase space domains exhibited by the two separate oscillators merge into a single domain as coupling strength varies.

The specific system studied here used the Pullen-Edmonds potential, which has a bound potential well with quadratic terms and a quartic coupling term. Understanding measure synchronization in this type of system could lead to methods for controlling the behavior of plasma related phenomena like the motion of charged particles in chaotic magnetic fields.

The research reported here makes a distinction between MS and the type of synchronization observed in dissipative systems. MS is not characterized by the merging of trajectories, but rather, takes place when the trajectories occupy the same phase space domains and, thus, come arbitrarily close to one another.

The authors found an order parameter, M, that is used to identify both the transition to measure synchronization and the extent to which system behavior can be controlled. They also showed that measure synchronization is more stable against variation in coupling strength in chaotic regions compared to quasiperiodic regions.

Source: “Exploring the route to measure synchronization in non-linearly coupled Hamiltonian systems,” by Shraddha Gupta, Sadhitro De, M. S. Janaki, and A. N. Sekar Iyengar, Chaos (2017). The article can be accessed at https://doi.org/10.1063/1.4996814.