Complexity in Physics and Chemistry

A heteroclinic network is a sequence of trajectories connecting saddles in a topological network. Although this particular nonlinear dynamics looks exceptional, it is frequently found in ordinary differential equations under certain constraints like symmetries or delay. Accordingly it is predicted in models of coupled phase oscillators, pulse-coupled oscillators and models of winnerless competition. The applications range from social systems to ecological systems, fluid mechanics, chemostats, computation to neuronal networks.

In this talk we will present theoretical and experimental findings on pattern formation in bistable chemical networks. Effects of feedbacks on self-organization phenomena will also be discussed. For regular trees, an approximate analytical theory for localized stationary patterns under application of global feedbacks is constructed. Using it, properties of such patterns in different parts of the parameter space are discussed. We will also show that localized oscillatory patterns can be formed in these networks if the global feedback is applied with certain time delay.

As early as in 1983 a coexistence of an ordered and disordered distribution of charged
colloidal dispersions [1,2], as well as the formation of void structures [3,4] were reported.
These phenomena imply that there are long-ranged repulsive and attractive potentials, of
the order of thousands or even tens of thounsands of angstroms. However, molecular
interaction potentials are relativelly short ranged [5]. Even for charged fluids, at low
colloidal concentration, the particles’ correlation is at most a few hundreds of angstroms.

We use molecular dynamics simulations to investigate the displacement of a periodically folding molecular motor in a viscous environment. We observe two different time regimes. For slow foldings (regime I) the
diffusion evolves very slowly with τ, while for rapid foldings (regime II) the diffusion increases strongly with
τ, suggesting two different physical mechanisms.

The coupling between the nonlinear kinetics and the transport phenomena leads to a complex non-trivial evolution in a closed unstirred Ferroin-catalyzed Belousov Zhabotinsky reaction. An initially periodic phase evolves into a chaotic one following a Ruelle-Takens-Newhouse scenario, which itself evolves back into another periodic phase which lasts until the reaction reaches thermodynamic equilibrium following an inverse Ruelle-Takens-Newhouse scenario [1].