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Unfoldings and Bifurcations of Polynomial Differential Systems

Periodic Reporting for period 1 - UBPDS (Unfoldings and Bifurcations of Polynomial Differential Systems)

Reporting period: 2016-06-30 to 2017-06-29

The main objective of this project is to obtain the properties of invariant sets in some dynamical models, to investigate the influence of small perturbations for the systems, and to analyze dynamics of real world models in biology and chemistry. For these purposes, we investigated the problems of isochronicity, linearizability and critical period bifurcation of planar differential systems, limit cycles of discontinuous piecewise perturbations of a linear center, global dynamics and unfoldings of planar smooth and piecewise smooth quasi-homogeneous differential systems, unfoldings of odd Lienard systems, and global dynamics of epidemic models and autocatalator models, respectively.

In this project, we not only solve some classical and difficult mathematical problems in the field of dynamical systems, but also our research on general differential systems can be applied in some mathematical models of real word phenomena. For example, the theoretical results in our investigation of practical models provide an intuitive basis for understanding the evolution of the epidemic and the nonlinear chemical reaction steps of the prototype autocatalator, which allow us to predict the outcomes of control strategies during the course of the epidemic and the autocatalator.
The work objectives and research problems were not performed independently. Actually, one research usually includes several topics.

For the problems of isochronicity, linearizability and critical period bifurcation of polynomial differential systems (PDSs), we investigated a family of cubic complex planar systems and a class of cubic generalized Riccati systems. We gave a classification of linearizable /isochronous systems obtaining conditions for linearizability/ isochronicity in terms of parameters. The coexistence of isochronous centers in the systems with degenerate infinity and the global structure of the generalized Riccati systems with an isochronous center at the origin were studied. Furthermore, we determined the order of weak center and proved that the maximum number of critical periods is reachable. For the investigation we have used the approach based on the modular calculations of the set of solutions of polynomial systems, which can be efficiently applied to study various mathematical models for the problem of solving polynomial equations.

For the problems of unfoldings and classification, we discussed the versal unfolding and the classification of a nilpotent Lienard equilibrium within the odd Lienard family. We proved that the nilpotent Lienard equilibrium is of degeneracy of codimension 2 and then we can use two parameters to display all possible bifurcations, such as pitchfork bifurcation, saddle-center bifurcation and homoclinic (heteroclinic) loop bifurcation. Although the technique of versal unfolding is developed and applied effectively to nilpotent equilibria, there are still great difficulties in studying the cases of higher codimension, referred to degenerate Bogdanov-Takens bifurcations, because those involved terms of higher degree produce more equilibria and heteroclinic (homoclinic) loops. On the other hand, we studied global dynamics and unfoldings of planar piecewise smooth quadratic quasi-homogeneous systems and smooth quasi-homogeneous systems with given degrees. We presented sufficient and necessary conditions for the existence of a center in piecewise quadratic quasi-homogeneous systems, and proved that the center is global and non-isochronous if it exists, which cannot appear in smooth quadratic quasi-homogeneous systems. The classification of global structures and unfoldings of both piecewise smooth and smooth quasi-homogeneous were studied. We investigated limit cycle bifurcation of the piecewise smooth quadratic quasi-homogeneous center and gave the maximal number of limit cycles bifurcating from the periodic orbits of the center.

For the dynamics of practical models, we considered compartmental SIRS epidemic models with asymptomatic infection and seasonal succession and autocatalator models. We defined and evaluated the basic reproduction number, and obtained uniform persistence of the disease and threshold dynamics for epidemic models. Besides, we judged the number of all possible equilibria and analyzed all local properties of equilibria for autocatalator models, even though the coordinates of equilibria are difficult to be calculated.

For discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center, we analyzed the maximum number of limit cycles of these discontinuous piecewise polynomial perturbations of the linear center, which can be obtained by using the averaging theory. The results show that the discontinuous quadratic and cubic polynomial perturbations of the linear center have more limit cycles than the ones found for continuous and discontinuous linear perturbations.

The results of the study have been published or submitted in leading peer-review journals and presented on important international conferences and workshops. The obtained knowledge will help to understand better the dynamics of biological and chemical processes and this can be helpful for research at the development departments of some companies dealing with such studies. The developed tools and ideas are useful for the planned project in the field of informatics and biochemistry aimed on creation a symbolic calculation platform for automating of analysis of biochemical models described by dynamical systems.
The project contributes to establish European excellence and competitiveness through developing new knowledge in the field of dynamical systems and through establishing of a long-time collaboration, since the theory of dynamical systems is one of the most important tools for analysis of numerous models of technical and applied sciences.

We have established contacts with non-academic sphere to explain importance and potential applications of our study for developing of modern technology and improving living environment. We plan to write a number of articles for Slovenian and Chinese daily newspapers or popular magazines explaining and promoting our research for general public.