Research On Novel No-equilibrium Chaotic Systems And Their Coexisting Hidden Attractors

The no-equilibrium chaotic systems can not only yield hidden attractors,but also can generate coexisting hidden attractors and even coexisting an infinite number of hidden attractors.Hidden attractors and coexisting hidden attractors are currently a new research hotspot.The hidden attractor is a newly defined type of attractor.It is not excited by unstable equilibrium points.Its attracting basin is generally small,so it has good hiding characteristics,which makes it huge in the field of chaos.Coexisting hidden attractors,meaning multiple hidden attractors coexist,are named hidden multistability,which can make the system more flexible,thereby it is particularly significant to study and reveal the intrinsic characteristics and evolutionary mechanisms of coexisting hidden attractors in the system.The main work of this dissertation focuses on constructing a series of new pointless chaotic systems.And the complex dynamics of hidden attractors and coexisting hidden attractors are analyzed.The specific research contents and results are as follows:1)Designing a new type of three-dimensional no-equilibrium Jerk-like chaotic system with coexisting asymmetric hidden attractorsBy using strict mathematical calculation and system search method,a novel no-equilibrium Jerk-like chaotic system is constructed and explored.Particularly,owing to the absence of the equilibria,such a new system can be categorized as a system with hidden attractors.More interestingly,this system holds three conspicuous characteristics.The first one is that various asymmetric coexisting hidden attractors and complicated transient chaos behaviors are obtained.The second one is the new finding of the periodic bursting oscillation and unusual phenomenon of transient periodic bursting oscillation in the system.The third one is the observation of the amazing and rare phenomenon of one to two full Feigenbaum remerging trees,namely,antimonotonicity.With the help of phase portraits,time series,bifurcation diagram,Lyapunov exponents,chaotic dynamical diagram,basin of attraction and so forth,the rich hidden dynamical properties of this system are systematically analyzed and investigated.Additionally,a hardware electronic circuit on a breadboard is carried out.The experimental results are in good agreement with the Matlab numerical simulation results.2)Constructing a new four-dimensional no-equilibrium augmented Lü system with coexistence of infinite hidden attractorsBy using a simple state feedback controller in a three-dimensional improved Lü system,a novel 4D chaotic system is derived.And depending on the different values of the constant term,this system has a line of equilibrium points or no equilibrium points,indicating that this is a hidden attraction subsystem and covers two types of hidden attractors.By choosing different parameters and initial conditions,the system can not only display one to four wings hidden attractors,various coexisting hidden attractors,complex transient transition behaviors but also can produce an infinite number of interesting phenomena of coexisting hidden attractors.Finally,the new system is implemented by an electronic circuit.And a very good agreement is observed between the experimental results and the numerical simulations of the same system on the Matlab platform,which verifies the rationality of the system construction and the correctness of the circuit design.3)Creating a new type of no-equilibrium charge-controlled memristive chaotic system with coexistence of infinitely many hidden attractorsBy introducing a tiny perturbation into a memristive chaotic oscillator,a new memristive chaotic system without equilibrium is proposed.Therefore,the resulting attractors are all hidden.Particularly,as the initial conditions of the memristor change,an interesting phenomenon of coexisting an infinite number of hidden attractors can be observed.The complex dynamical behaviors are numerically analyzed.Furthermore,a hardware experiment is carried out.A good similarity between numerical simulations and experimental results is obtained,which verifies that the system can produce rich hidden attractors and further demonstrates the effectiveness of this small perturbation method.4)Designing a new type of four-dimensional no-equilibrium fractional hyperchaotic system with coexistence of infinitely many hidden attractorsBy utilizing simple state feedback control technique,a novel 4D fractional-order hyperchaotic system is introduced.Compared with other fractional-order hyperchaotic systems,this system possesses three unique features:(i)The amazing and interesting phenomenon of the coexistence of infinitely many hidden attractors with respect to same system parameters and different initial conditions are observed,meaning hidden extreme multistability arises.(ii)By varying the initial conditions and selecting appropriate system parameters,the striking phenomenon of antimonotonicity is first discovered,especially in such no-equilibrium fractional-order hyperchaotic system.(iii)An attractive and special feature of the convenience of offset boosting control of the system is also revealed.The complex hidden extreme multistability dynamical behaviors of this system are investigated by using phase portraits,bifurcation diagram and Lyapunov exponents.The spectral entropy complexity algorithm is used to calculate the complexity of the system.And it is found that the complexity of the system increases with the decrease of the fractional order.The hardware electronic circuit is designed to further confirm the hidden hyperchaotic characteristics of the system and the special behavior of coexisting an infinite number of hidden attractors.