Coupled pendula chains under parametricPT-symmetric driving force
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We consider a chain of coupled pendula pairs, where each pendulum is
connected to the nearest neighbors in the longitudinal and transverse
directions. The common strings in each pair are modulated periodically by an
external force. In the limit of small coupling and near the 1:2 parametric
resonance, we derive a novel system of coupled PT-symmetric discrete nonlinear
Schrodinger equation, which has Hamiltonian symmetry but has no gauge symmetry.
By using the conserved energy, we find the parameter range for the linear and
nonlinear stability of the zero equilibrium. Numerical experiments illustrate
how destabilization of the zero equilibrium takes place when the stability
constraints are not satisfied. The central pendulum excites nearest pendula and
this process continues until a dynamical equilibrium is reached where each
pendulum in the chain oscillates at a finite amplitude.
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