Using Fisher information approach in nonlinear dynamical systems

The aim of this thesis is to investigate nonlinear dynamical systems that exist in various fields such as engineering and science. Nonlinear dynamical systems permit the understanding and development of models of simple and complex phenomena. Specifically, this thesis includes an investigation of the following systems; the logistic model, the Gompertz model, predator-prey model, and three species model. In addition, we perform a comparison between the two most popular growth models; logistic and Gompertz models from the viewpoint of variability. The main focus is on the use of Fisher information as a measure of variability/sustainability which depends on the gradient of Probability Density Function (PDF).
In this work, we present two case studies for each dynamical system. The first case study describes the analysis of these systems in their deterministic conditions whereas the second one presents the investigation of these systems in their indeterministic conditions (perturbed conditions), where the model parameters involve perturbations, elucidating the effects of these perturbations on the behaviour of the system. The variation in the model parameter values is considered in order to observe the behaviour of the different dynamical systems and detect dynamical changes in the behaviour of each species.
Since Fisher information is considered as a measure of an intrinsic accuracy of the dynamical systems, therefore, we obtain Fisher information for different values of parameters in order to select the optimal parameter value where a peak of Fisher information is observed, which indicates to less variability in the behaviour of the system. Thus, the existence of Fisher information peak which linked to the narrowest PDF is investigated in the frame of time trace analysis.
In summary, the main contribution of this work to the field is to assess the significance of Fisher information index for nonlinear dynamical systems including perturbations in the model parameters. Applying this measure to more complicated systems and comparing the results to other widely used measures would be of interest for future work.