On the Lagrangians and potentials of a two coupled damped Duffing oscillators system and their application on three-node motif networks

Abstract

<jats:p>In this work, we investigate the Lagrangians and potentials for two coupled damped Duffing oscillators both directionally and bi-directionally. We show that, although it is not always possible to define a potential in dissipative systems, the potential of our model can be defined if the damping coefficient has a logarithmic derivative form. It is possible to apply these results to the analysis of the dynamics of complex networks based on three-node motif configurations. As an example, we study numerically the dynamics for one of the thirteen different possible configurations. Here, the phenomenon of synchronization is observed in terms of the coupling parameter.</jats:p>

Dynamics of a ring of three unidirectionally coupled Duffing oscillators with time-dependent damping

“…Although dissipative systems cannot be described by a proper potential [24], in some cases the potential can still be found. For example, in the case of time-dependent damping, the system can be viewed as an undamped oscillator but with a variable mass and therefore the corresponding potential can be found [25,26]. In addition, analytical studies of transitions which occur between three possible dynamical states (cluster synchronization, complete synchronization, and instability) were performed in a ring of N diffusely coupled Duffing oscillators [27,28].…”

“…It is well known that, in the general case, damped systems do not always have a defined potential [24]. However, the potential for a dissipative system can be found if the damping coefficient is written as a logarithmic derivative of some function (see, e.g., [25,26]). More precisely, a damped harmonic oscillator in the form…”

Dynamics of a ring of three unidirectionally coupled Duffing oscillators with time-dependent damping

“…Although dissipative systems cannot be described by a proper potential [25], however, in some cases the potential can still be found. For example, in the case of a linear time-dependent damping term, the system can be viewed as an undamped oscillator but with a variable mass and therefore the corresponding potential can be obtained [26,27]. In addition, analytical studies of transitions which occur between three possible dynamical states (cluster synchronization, complete synchronization, and instability) were performed in a ring of N diffusely coupled Duffing oscillator [28,29].…”

“…It is well known that in general the damped systems do not always have a defined potential [25]. However, it is possible to find potentials for dissipative systems if the damping coefficient can be written as logarithmic derivative of certain function (see, e.g., [26,27]). More explicitly, if we have a motion equation of the type…”

“…However, we will show in the next sections that different types of time dependence of the damping coefficient can change the dynamics of the ring-coupled Duffing oscillatory system. The ring of three unidirectionally coupled Duffing oscillators is described as follows [22,27]…”