# Centre for Nonlinear Dynamics Zagreb

## Extended Dynamical Systems

Focus of research:

Extended dynamical systems are equations on unbounded domains. For example, in the discrete space / continous time setting, they are systems of infinitely many equations - infinite lattices in 1d or higher dimensions. In the continous space/time setting, we consider partial differential equations on unbounded domains.

Such systems are important, as they model a number of physical phenomena for which the size of the domain is very large as compared to the local dynamics.

We developed a number of tools to deal with such systems. For example, for "extended gradient systems", i.e. systems which resemble gradient systems (more precisely, whose restriction to any finite domain has a Lyapunov function), we (jointly with Th. Gallay) described up to a large extend possible dynamics.

Recent results include application of these ideas to the Navier-Stokes equation, as well as to driven Frenkel-Kontorova model.

Key publications:

1. Th. Gallay, S. Slijepčević, Energy flow in formally gradient partial differential equations on unbounded domains, J. Dynam. Differential Equations 13 (2001), 757-789.

2. S. Slijepčević, Monotone gradient dynamics and Mather's shadowing, Nonlinearity 12 (1999), 969-986.

3. S. Slijepčević, Aubry-Mather theorem for generalized driven elastic chain, to appear in Discrete Cont. Dynam. Syst.; arXiv:1305.1109.

4. Th. Gallay, S. Slijepčević, Distribution of energy and convergence to equilibria in extended dissipative systems, to appear in J. Dyn. Diff. Equations, arXiv:1212.1573.

5. S. Slijepčević, The energy flow of discrete entended gradient systems, Nonlinearity 26 (2013), 2051-2079

6. J. P. Milišić, B. Duering, D. Matthes, A gradient flow scheme for nonlinear fourth order equations", Discrete Contin. Dyn. Syst. Ser. B 14(3) (2010), 935-959.