vT(T) over bar-deformed oscillator inspired by ModMax

Inspired by a recently proposed Duality and Conformal invariant modification of Maxwell theory (ModMax), we construct a one-parameter family of two-dimensional dynamical systems in classical mechanics that share many features with the ModMax ? theory. It consists of a couple of root T(T) over bar-deformed oscillators that nevertheless preserves duality (q -> p, p -> -q) and depends on a continuous parameter gamma , as in the ModMax case. Despite its nonlinear features, the system is integrable. Remarkably, it can be interpreted as a pair of two coupled oscillators whose frequencies depend on some basic invariants that correspond to the duality symmetry and rotational symmetry. Based on the properties of the model, we can construct a nonlinear map dependent on gamma that maps the oscillator in 2D to the nonlinear one, but with parameter 2 gamma . The reason behind the existence of such map can be revealed through a construction of two Lax pairs associated with the system. The dynamics also shows the phenomenon of energy transfer and we calculate a Hannay angle associated to geometric phases and holonomies.

Automated summary

Inspired by ModMax theory, we construct a one-parameter family of 2D dynamical systems consisting of root T(T) over bar-deformed oscillators that preserve duality and depend on a continuous parameter gamma. Despite its nonlinearity, the system is integrable and can be interpreted as two coupled oscillators. We also construct a nonlinear map that transforms the 2D oscillator to the nonlinear one, with a parameter 2 gamma, and reveal the reason behind its existence through the construction of two associated Lax pairs.