Kyoto University, Kyoto 606-8502, Japan

The dynamics of large assemblies or extended fields of coupled nonlinear elements depends crucially on the interaction range involved. Most foregoing studies have been confined to the limiting cases of either local coupling or global coupling, while systematic studies of more general nonlocally coupled systems remain few. In this series of lectures, I will discuss some remarkable features in the dynamics which can arise peculiarly to nonlocally coupled systems, working mainly with oscillatory dynamics. The first lecture will be devoted to general discussions on what are essentially new with nonlocally coupled systems as contrasted with locally coupled systems. Some points to be discussed include the following:

- 1.
- Effective nonlocality in coupling arising from locally coupled systems (typically reaction-diffusion systems) as a result of elimination of some variables.
- 2.
- Proposal of a 3-component reaction-diffusion model as a canonical model covering the local, global and nonlocal regimes of the effective coupling, thus providing an ideal model for the study of the effects of nonlocality on the dynamics.
- 3.
- Pointing out the fact that nonlocal coupling has its own asymptotic regeme, not being merely something intermediate between the local and global.
- 4.
- Applicability of center-manifold reduction and phase reduction leading to some new forms of universal equations.
- 5.
- General conditions under which the effects of nonlocality become overt.
- 6.
- Mean-field picture applicable to nonlocally coupled systems.
- 7.
- Onset of spatial discontinuity of patterns as a general feature of nonlocally coupled systems.

Kyoto University, Kyoto 606-8502, Japan

In this second lecture, the impact of the nonlocality in coupling will be demonstrated by showing a variety of curious dynamics exhibited by the canonical model introduced in the first lecture or by its reduced forms. Our emphasis will be placed upon how the types of behavior observed are peculiar to the coupling nonlocality and how they can naturally be understood in terms of the general notions developed in the first lecture. In particular, the mean-field picture valid for nonlocally coupled elements turns out quite useful for interpreting certain features of behavior. Some problems involving stochasticity are also discussed for which a theory can be formulated by virtue of the same mean-field picture. The types of behavior discussed are:

- A.
- Self-sustained pacemakers in monostable excitable media whose origin is completely different from those proposed in the past.
- B.
- Two-dimensional spiral waves with a strongly turbulent core which is initiated by a breakdonw of synchronization of a small group of central oscillators to the periodic internal forcing.
- C.
- Coexistence of coherent and incoherent domains. This is a generalization of case B.
- D.
- Spatio-temporal chaos with multi-scaling properties for which the pattern is fractalized with its fractal dimension changing continuously with the system parameter.
- E.
*Soft-mode turbulence*which occurs right at the Turing instability in the presence of a Goldstone mode.- F.
*Sawtooth turbulence*characterized by persistent creations and annihilations of wave sources and sinks.- G.
- Transmission of waves in random media for which a statistical theory can be formulated in terms of a Boltzmann type kinetic equation.