## Regularities in Firm Dynamics: The Basic of the Complex Economy

Firm Dynamics is a field defined in mathematical economics and econophysics. The research question of my work is rather simple: what are the regularities of company performance? How a company evolves over time and what is the relevance to the macroeconomy. In this study, I use empirical evidence rather than hypothesis to draw a big picture of company activities in the economy. With data from Amadeus, Compustat and Company House, covering 20M companies in EU and US, the conclusion is drawn from empirical data in an unprecedented level of resolution. To analyse the detail, I further developed a high-resolution probability estimator which enables me to recover the true empirical distribution at the best...

## Self-Organised Criticality

Self-Organised Criticality describes the tendency of some non-equilibrium systems to display scale-invariant, intermittent behaviour. As the name suggests, these systems seem to organise themselves to a classical critical point, where they experience all features of a second order phase transition. The key characteristics of these systems are: slow drive (separation of time scales), strong interaction, non-linear relaxation and thresholds, displaying bursts of activities (avalanches) whose distribution is self-similar (power laws) and only cut off by the system size. All scaling in such systems is finite size (and finite time) scaling. Our research into self-organised criticality (SOC) is three pronged: The basic principles of SOC (field theory, scaling, mapping to established critical phenomena). Numerical investigation of SOC using large scale computer simulations. Applying the concepts of SOC to understand natural...

## Synchronisation by time delay

Synchronisation is a very widespread phenomenon observed in flashing fireflies, applauding audiences and the neuronal network of the brain. Hitherto, one major branch of research has focussed on the exchange of instantaneous, sudden pulses which are exchanged when an oscillator reaches a threshold, triggering sudden, discontinuous relaxations. A second branch focussed on smooth interaction that vanishes in the synchronised state, best known as the Kuramoto Model. We changed this setup, studying smooth, continuous interaction that never disappears. At first, very basic considerations suggest that such a system cannot synchronise. Numerics, however, seems to suggest otherwise. It turns out that this clash is caused by an effective time delay built into the numerics: Time delay causes synchronisation on a time scale that is inversely proportional to the time lag. On a more technical level, if the Kuramoto Model is $latex \dot{\theta}_{1,2}=\omega + J \sin(\theta_2-\theta_1)$ then we study, for example, \$latex \dot{\theta}_{1,2}=\omega + J (\sin(\theta_{2,1}(t-\delta t)-\sin(\theta_{1,2}(t-\delta...