| As a new interdisciplinary subject,research and development of nonlinear science make people's understanding of nature from the linear field to the nonlinear field.Bifurcation and chaos are the important contents of nonlinear science research.Studies of bifurcation are the phenomenon of some topological changes,while homoclinic orbits and heteroclinic orbits play a key role in these phenomena.Chaos generally exists in nature,and its complex dynamics has become one of the frontier topics.Compared with chaotic attractor orbits,hyperchaos will separate in more directions,so its dynamic behavior is more complex.The unpredictability and randomness are also stronger,which make hyperchaos have a very broad research prospect.The Stretch-Twist-Fold(STF)flow is a three-dimensional Stokes flow for the study of electromagnetic fields.Because magnetic fields are widely present within and between astral bodies,studying them in astrophysics has important implications.The STF flow describes the mechanism by which the magnetic field flow is stretched,twisted,and folded to generate magnetic energy in a magnetic field.In this paper,the three-dimensional STFLike flow system is first studied.Hypothesis testing is used to test whether the system is chaotic.Then the system's heteroclinic bifurcation,Hopf bifurcation,pitchfork bifurcation and circuit simulation are studied.Then five-dimensional STF flows with cubic terms are proposed.The system not only has three positive Lyapunov exponents,but also has three types of coexisting hidden attractors.At the same time,the Hopf bifurcation,pitchfork bifurcation and circuit simulation of the system are studied.The main content of this article is as follows:The first chapter is the introduction.It mainly introduces the research background and development status.It briefly introduces bifurcation theory,chaos theory and hidden attractors.Finally,the main research contents of this paper are introduced.In the second chapter,based on the STF stream,the STF-like system is proposed.Chaos is studied by Monte Carlo null hypothesis test,also known as surrogate data method.This method uses hypothesis test to check the existence of chaos in the system.It can be used as a supplement to the traditional test methods such as Lyapunov exponent Poincarèsectional graph and branch graph.Two precise heteroclinic orbits of the system are obtained.Their heteroclinical orbital bifurcations are studied separately,and the existing conditions and existing regions are given.The Hopf bifurcation of the system is studied,the stability of the bifurcated periodic solution is obtained,the pitchfork bifurcation is studied,and the conditions for the existence of bifurcation and the direction of bifurcation are obtained.Numerical simulation confirmed the theoretical results.Finally,a digital analog circuit was designed to verify the feasibility of the system.In chapter 3,a five dimensional hyperchaotic STF flow system is proposed.The system not only contains cubic terms,but also has three positive Lyapunov exponents.under di?erent parameter conditions,the system has three types of hidden attraction factors coexisting:(1)Coexistence of periodic attractors and hidden hyperchaotic attractors at linear singularities.(2)Coexistence of hidden chaotic and hyperchaotic attractors under stable singularities.(3)The quasi-periodic attractor coexists with hidden hyperchaotic attractor and the hidden hyperchaotic attractor coexists with chaotic attractor without singularities.The Hopf bifurcation of the system is also studied.Not only the condition and direction of the bifurcation are obtained,but also the stability of the periodic solution of the bifurcation is obtained.For the pitchfork bifurcation of the system,the condition and direction of bifurcation are also given.The numerical simulation verifies the theoretical results.Finally,a digital analog circuit is designed to verify the feasibility of the system. |
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