A heterogeneous Bertrand duopoly game with bounded rational and adaptive players manufacturing differentiated products is subject of investigation. The main goal is to demonstrate that participation of one bounded rational player in the game suffices to destabilize the duopoly. The game is modelled with a system of two difference equations. Evolution of prices over time is obtained by iteration of a two dimensional nonlinear map. Equilibria are found and local stability properties thereof are analyzed. Complex behavior of the system is examined by means of numerical simulations. Region of stability of the Nash equilibrium is demonstrated in the plane of the speeds of adjustment. Period doubling route to chaos is presented on the bifurcation diagrams and on the largest Lyapunov characteristic exponent graph. Lyapunov time is calculated. Chaotic attractors are depicted and their fractal dimensions are computed. Sensitive dependence on initial conditions is evidenced.
The chaotic phenomena of coronary artery systems are hazardous to health and may induce illness development. From the perspective of engineering, the potential harm can be eliminated by synchronizing chaotic coronary artery systems with a normal one. This paper investigates the chaos synchronization problem in light of the methodology of sliding mode control (SMC). Firstly, the nonlinear dynamics of coronary artery systems are presented. Since the coronary artery systems suffer from uncertainties, the technique of derivative-integral terminal SMC is employed to achieve the chaos synchronization task. The stability of such a control system is proven in the sense of Lyapunov. To verify the feasibility and effectiveness of the proposed method, some simulation results are illustrated in comparison with a benchmark.
Many nonlinear dynamical systems can present a challenge for the stability analysis in particular the estimation of the region of attraction of an equilibrium point. The usual method is based on Lyapunov techniques. For the validity of the analysis it should be supposed that the initial conditions lie in the domain of attraction. In this paper, we investigate such problem for a class of dynamical systems where the origin is not necessarily an equilibrium point. In this case, a small compact neighborhood of the origin can be estimated as an attractor for the system. We give a method to estimate the basin of attraction based on the construction of a suitable Lyapunov function. Furthermore, an application to Lorenz system is given to verify the effectiveness of the proposed method.
Electromagnetic mill installation for dry grinding represents a complex dynamical system that requires specially designed control system. The paper presents model-based predictive control which locates closed loop poles in arbitrary places. The controller performs as gain scheduling prototype where nonlinear model – artificial recurrent neural network, is parameterized with additional measurements and serves as a basis for local linear approximation. Application of such a concept to control electromagnetic mill load allows for stable performance of the installation and assures fulfilment of the product quality as well as the optimization of the energy consumption.