GendelmanProf. Oleg V. Gendelman
Faculty of Mechanical Engineering - Technion-Isreal Institute of Technology University
Exact solutions for Hamiltonian and forced/damped discrete breathers in vibro-impact chain
Tuesday, Feb 25, 2014, 12:00pm to 1:00pm | Room 5-314

Discrete breathers (DBs), or intrinsic localized modes (ILMs) are well-known in many mechanical and physical systems, including chains of mechanical oscillators, superconducting Josephson junctions, nonlinear magnetic metamaterials, electrical lattices, michromechanical cantilever arrays, antiferromagnets and  Bose – Einstein condensates. Generically, these response regimes appear due to interplay between discreetness of the system and its nonlinearity; therefore, analytic description of this sort of phenomena poses essential challenge.

Current work considers the DBs in vibro-impact (VI) lattice models and is based on important peculiarity of these systems – they exhibit extreme (actually, the strongest possible) nonlinearity, but it reveals itself only at the moments of impacts; between the impacts the system obeys linear equations of motion, if other sources of nonlinearity are absent. Consequently, the VI models can offer relatively simple description of complicated nonlinear phenomena.

Exact solutions for symmetric discrete breathers (DBs) are obtained in Hamiltonian and forced – damped linear chain with on-site vibro-impact constraints. The damping in the system is caused by inelastic impacts; the forcing functions should satisfy conditions of periodicity and antisymmetry. The family of vibro-impact models allows explicit computation of a monodromy matrix and makes it possible to study the stability globally. For instance, the forced/damped DB can lose its stability either by pitchfork, or through Neimark – Sacker bifurcations. The pitchfork bifurcation is related to internal dynamics of each individual oscillator. It is revealed that the coupling can suppress this type of instability. To the contrary, the Neimark – Sacker bifurcation occurs for relatively large values of the coupling, presumably due to closeness of the excitation frequency to a boundary of propagation zone of the chain. Both bifurcation mechanisms seem to be generic for considered type of forced – damped lattices. Some unusual phenomena, like non-monotonous dependence of the stability boundary on the forcing amplitude, are revealed analytically for the initial system and illustrated numerically for small periodic lattices.

Short Bio

Born in 1969 in Kharkov, Ukraine. Obtained MSc from Moscow Institute of Physics and Technology, PhD (1995) and habilitation (2000) in Institute of Chemical Physics in Moscow. Since 2003 - with Faculty of Mechanical Engineering, Technion, Israel, now - Professor.  Main scientific interests: nonlinear dynamics, vibration absorption and mitigation, heat transfer at nanoscale, plasticity in amorphous media. Authored about 120 papers and 2 monographs.