On regular and chaotic dynamics of a non- PT -symmetric Hamiltonian system of a coupled Duffing oscillator with balanced loss and gain

Abstract

A non- PT -symmetric Hamiltonian system of a Duffing oscillator coupled to an anti-damped oscillator with a variable angular frequency is shown to admit periodic solutions. The result implies that PT -symmetry of a Hamiltonian system with balanced loss and gain is not necessary in order to admit periodic solutions. The Hamiltonian describes a multistable dynamical system—three out of five equilibrium points are stable. The dynamics of the model is investigated in detail by using perturbative as well as numerical methods and shown to admit periodic solutions in some regions in the space of parameters. The phase transition from periodic to unbounded solution is to be understood without any reference to PT -symmetry. The numerical analysis reveals chaotic behaviour in the system beyond a critical value of the parameter that couples the Duffing oscillator to the anti-damped harmonic oscillator, thereby providing the first example of Hamiltonian chaos in a system with balanced loss and gain. The method of multiple time-scales is used for investigating the system perturbatively. The dynamics of the amplitude in the leading order of the perturbation is governed by an effective dimer model with balanced loss and gain that is non- PT -symmetric Hamiltonian system. The dimer model is solved exactly by using the Stokes variables and shown to admit periodic solutions in some regions of the parameter space.

Classical Hamiltonian Systems with Balanced loss and gain

“…The points P ± 2 are not stable anywhere in the parameter-space. These results are confirmed by perturbative and numerical analysis [27] and the system admits periodic solutions around the equilibrium points. The bifurcation diagram is plotted in Fig.…”

“…• The quantum problem is not well-defined on the real line and a well-defined ground state does not exist. However, if the quantum Hamiltonian is defined in appropriate Stokes wedges, the system may admit well defined bound states [14,19,23,24,25,27]. This is consistent from the viewpoint of axiomatic foundations of quantum mechanics.…”

“…[14] and examples considered in Refs. [19,23,24,25,26,27] are bounded from below/above within some regions in the parameter-space. These examples admit stable periodic solutions at the classical level and quantum bound states.…”

“…Similarly, a two particle Calogero-type model with BLG admitting periodic solutions may be reproduced for [19]. There are many examples of systems with BLG admitting periodic solutions in certain region of the parameters space [20,21,22,23,24,25,26,27].…”

“…It has been shown recently that a coupled Duffing-oscillator system with BLG [27] admits chaotic solutions. The equations of motion of the system are,…”