Research On Dynamics Of A 4D Hyperchaotic Segmented Disc Dynamo And Integrability Of The Pehlivan System

As a complex motion phenomenon in nonlinear system,the study of chaos has attracted the great interest of scholars from various fields and areas,especially in the fields of neural network,astronomical satellite,nonlinear circuit and secure communication.In the past half century,hyperchaos which is based on chaos theory has been more in-depth studied.Compared to the chaotic system,hyperchaotic system has at least two positive Laypunov exponents,the system's randomness and uncertainty has enhanced a lot.So the dynamic behavior of hyperchaotic system is much more complex.In recent years,due to the practical prospect of hyperchaotic systems in the field of science and engineering applications,hyperchaos has become one of the frontiers and hotspots of nonlinear science research.In this paper,a new four-dimensional hyper-chaotic segmented dynamo was proposed.The new system only had one trigonometric function term,but it showed very complex dynamic behavior.The system has not only hidden attractors,but also has the phenomena of multistabilty and megastability.The hyperchaotic attractors and various attractors are studied by phase,Laypunov exponents,bifurcation and Poincare mapping.The conditions for the existence of fork bifurcations,Hopf bifurcations and Zero-Zero-Hopf bifurcations are given.In addition,the existence of chaos in the Pehlivan system is tested by Monte Carlo null hypothesis test,which can be regarded as a supplementary means to the traditional method of studying the existence of chaos.The paper also gives all the Darboux polynomials of the Pehlivan system,and proves that there is no exponential factor.Finally,it is proved that Pehlivan system is neither polynomial nor Darboux integrable.In chapter 1,the research background and the significance of this paper are presented,introduces the basic concepts of chaos theory and the related research methods of analyzing chaotic systems.The significance of studying the integrability of systems and Darboux integrability theory are introduced.In chapter 2,a new 4D hyperchaotic system which is generated by feedback controller based on segmented disc dynamo.The equilibrium stability of the system is studied and the Laypunov exponents and hyperchaos phenomena are numerically analyzed.It is found that the new system has complex dynamics behaviors such as multistability and megastability when appropriate parameters are taken.In chapter 3,fork bifurcations,Hopf bifurcations and Zero-Zero-Hopf bifurcations are studied,and the conditions for the existence of various bifurcations are given,include numerical simulation is given.For Hopf bifurcation and Zero-Zero-Hopf bifurcation,the conditions for judging the stability of periodic solutions are further given.In chapter 4,the existence of chaos is tested by hypothesis test,and then the integrability of Pehlivan system is discussed.In this paper,all irreducible Darboux polynomials of the system are completely classified,and it is proved that there is no exponential factors,and the system is neither Darboux nor polynomial integrable.