Destructive role of competition and noise for control of microeconomical chaos

The problem of control of chaos in a microeconomical model describing two competing firms with asymmetrical investment strategies is studied. Cases when both firms try to perform the control simultaneously or when noise is present are considered. For the first case the resulting control efficiency depends on the system parameters and on the maximal values of perturbations of investment parameters for each firm. Analytic calculations and numerical simulations show that competition in the control leads to ‘parasitic’ oscillations around the periodic orbit that can destroy the expected stabilization effect. The form of these oscillations is dependent on non-linear terms describing the motion around periodic orbits. An analytic condition for stable behaviour of the oscillation (i.e. the condition for control stability) is found. The values of the mean period of these oscillations is a decreasing function of the amplitude of investment perturbation of the less effective firm. On the other hand, amplitudes of market oscillations are increasing functions of this parameter. In the presence of noise the control can be also successful provided the amplitude of allowed investment changes is larger than some critical threshold which is proportional to the maximal possible noise value. In the case of an unbounded noise, the time of laminar epochs is always finite but their mean length increases with the amplitude of investment changes. Computer simulations are in very good agreement with analytical results obtained for this model.