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E. Destyl, S.P. Nuiro, D.E. Pelinovsky, and P. Poullet

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Stabilization of the coupled pendula chain under parametric PT -symmetric driving force

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**Abstract:**

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.

**Keywords**:

PT-symmetry, discrete nonlinear Schrodinger equation, Hamiltonian structure,
global existence, apriori energy bounds.