Dynamics of the chain of forced oscillators with long-range interaction: From synchronization to chaos

Abstract

We consider a chain of nonlinear oscillators with long-range interaction of the type 1/l 1+α , where l is a distance between oscillators and 0 < α < 2. In the continues limit the system's dynamics is described by the Ginzburg-Landau equation with complex coefficients. Such a system has a new parameter α that is responsible for the complexity of the medium and that strongly influences possible regimes of the dynamics. We study different spatial-temporal patterns of the dynamics depending on α and show transitions from synchronization of the motion to broadspectrum oscillations and to chaos. 1A chain of nonlinear interacting oscillators is a model of the wide-spread investigation of different physical phenomena such as synchronized behavior of the system, bifurcations to different regimes, spatial-temporal turbulent or chaotic dynamics, different instabilities, appearance of defects, and many others. The long range interaction between oscillators leads to a new qualitative dynamics and thermodynamics. Our consideration is related to the power-law long-range interaction that makes it possible to find new regimes of the system and to establish a link of the corresponding equations of motion to the equations with fractional derivatives.

Review of Some Promising Fractional Physical Models

“…Similar features were observed in the lattice models with power-like long-range interactions [45,46,47,48,49]. As it was shown [50,51,52,53,54,55,56,57,58], the equations with fractional derivatives can be directly connected to chain and lattice models with long-range interactions.…”

“…The effects of synchronization, breather-type and solution-type solutions for the systems with nonlocal interaction of power-law type 0 < β < 2 (β = 1) were investigated [53,54,55,56]. Nonequilibrium phase transitions in the thermodynamic limit for long-range systems are considered in [57].…”

Lattice model of fractional gradient and integral elasticity: Long-range interaction of Grünwald–Letnikov–Riesz type

“…x is the Riesz fractional derivative of order α − 2, which is equivalent by (11) to the Grünwald-Letnikov-Riesz fractional derivative GLR D α−2…”

“…Discrete system of long-range interacting particles serve as a model for numerous applications in mechanics and physics [1,2,3,4,5,6,7,8,9,10,11]. Long-range interactions are important type of interactions for complex media with non-local properties (see references in [12]).…”

On universality in fractional dynamics

“…Fractional differential equations are used in applications to describe systems with long-range interactions [1], [2], [3] or systems with power-law memory. Systems with memory include: Hamiltonian systems where memory is the result of stickiness of trajectories in time to the islands of regular motion [4], [5]; dielectric materials where electromagnetic fields are described by equations with time fractional derivatives due to the universal response -the power-law frequency dependence of the dielectric susceptibility in a wide range of frequencies [6], [7], [8], [9]; viscoelastic materials and materials with rheological properties where non-integer order differential stress-strain relations give a minimal parameter set concise description of polymers and other viscoelastic materials with non-Debye relaxation and memory of strain history [10], [11], [12], [13], [14].…”