Single parameter dissipativity and attractors in discrete time asynchronous systems * P. E. Kloeden FB Mathematik, Johann Wolfgang Goethe Universit¨ at, D-60054 Frankfurt am Main, Germany E-mail: kloeden@math.uni-frankfurt.de V. S. Kozyakin † Institute for Information Transmission Problems Russian Academy of Sciences Bolshoj Karetny lane, 19, 101447 Moscow, Russia E-mail: kozyakin@iitp.ru Abstract Discrete time nonautonomous dynamical systems generated by nonautonomous difference equations are formulated as discrete time skew-product systems consisting of co- cycle state mappings that are driven by discrete time autonomous dynamical systems. For- wards and pullback attractors are two possible generalizations of autonomous attractors to such systems. Their existence follows from appropriate forwards or pullback dissipativity conditions. For discrete time nonautonomous dynamical systems generated by asynchronous systems with frequency updating components such a dissipativity condition is usually known for a single starting parameter value of the driving system. Additional conditions that then ensure the existence of a forwards or pullback attractor for such an asynchronous system are investigated here. AMS Subject Classification: 34C35, 93D09 Key words: cocycle mapping, forwards and pullback attractors, forwards and pullback dissipativity conditions, asynchronous systems 1 Introduction A discrete time nonautonomous dynamical system on a state space R d can be formulated in terms of a parameterized state space system that is driven by an autonomous dynamical * This work was supported by the DFG Forschungsschwerpunkt “Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme”. † The author was partially supported by grants No.00-15-96116 and 0001-00571 of the Russian Foundation for Basic Research. 1