Generation of a family of fractional order hyper-chaotic multi-scroll attractors

Abstract An unified method to yield a family of fractional-order (FO) hyper-chaotic multi-scroll (HCMS) systems in Rn is proposed. Firstly, a new simple 3-dimensional (3-D) FO unstable linear system is introduced. Afterwards, additional variables are added and one nonlinear controller with adjustable parameters is included to generate HCMS attractors. A guideline to construct HCMS systems of any dimension is presented, that is verified along within the dynamics of three examples, namely 4-D, 5-D and 10-D FO HCMS systems. Phase portraits, Poincare maps and two positive Lyapunov exponents are calculated. Moreover, a circuit of 0.96-order is also designed to realize one 4-D FO HCMS system. Numerical simulations and circuit simulation results show the feasibility of the novel approach.

[1]  Elena Grigorenko,et al.  Chaotic dynamics of the fractional Lorenz system. , 2003, Physical review letters.

[2]  I. Podlubny.  Fractional differential equations , 1998 .

[3]  Ahmed Sadek Hegazi,et al.  Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system , 2011, Appl. Math. Lett..

[4]  C. Lamarque,et al.  Dynamics Investigation of Three Coupled Rods with a Horizontal Barrier , 2003 .

[5]  Marius-F. Danca,et al.  Hidden chaotic attractors in fractional-order systems , 2018, 1804.10769.

[6]  Shuyi Shao,et al.  Circuit Implementations, Bifurcations and Chaos of a Novel Fractional-Order Dynamical System* , 2015 .

[7]  G. Leonov,et al.  Hidden attractors in dynamical systems , 2016 .

[8]  Yigang He,et al.  Generation and circuit implementation of fractional-order multi-scroll attractors , 2016 .

[9]  Simin Yu,et al.  Generating hyperchaotic Lü attractor via state feedback control , 2006 .

[10]  Chunguang Li,et al.  Chaos and hyperchaos in the fractional-order Rössler equations , 2004 .

[11]  Henry Leung,et al.  Experimental verification of multidirectional multiscroll chaotic attractors , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[12]  Guanrong Chen,et al.  Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications , 2006 .

[13]  Wajdi M. Ahmad.  Generation and control of multi-scroll chaotic attractors in fractional order systems , 2005 .

[14]  Guanrong Chen,et al.  Designing Hyperchaotic Systems With Any Desired Number of Positive Lyapunov Exponents via A Simple Model , 2014, IEEE Transactions on Circuits and Systems I: Regular Papers.

[15]  Changpin Li,et al.  Chaos in Chen's system with a fractional order , 2004 .

[16]  W. Deng,et al.  Chaos synchronization of the fractional Lü system , 2005 .

[17]  Kehui Sun,et al.  High-Order Grid Multiscroll Chaotic Attractors Generated by the Second-Generation Current Conveyor Circuit , 2014, IEEE Transactions on Circuits and Systems II: Express Briefs.

[18]  Weihua Deng,et al.  Generating multi-directional multi-scroll chaotic attractors via a fractional differential hysteresis system , 2007 .

[19]  Ranchao Wu,et al.  Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system , 2014 .

[20]  Chenghua Liang,et al.  A new fractional-order hyperchaotic system and its modified projective synchronization , 2015 .

[21]  Julien Clinton Sprott,et al.  Recent new examples of hidden attractors , 2015 .

[22]  Claude-Henri Lamarque,et al.  Numerical and Experimental Study of Regular and Chaotic Motion of Triple Physical Pendulum , 2008, Int. J. Bifurc. Chaos.

[23]  Amr Elsonbaty,et al.  Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system , 2016 .

[24]  Rongrong Wang,et al.  A new finding of the existence of hidden hyperchaotic attractors with no equilibria , 2014, Math. Comput. Simul..

[25]  Fei Xu,et al.  Integer and Fractional Order Multiwing Chaotic Attractors via the Chen System and the Lü System with Switching Controls , 2014, Int. J. Bifurc. Chaos.

[26]  Ranchao Wu,et al.  Design and implementation of grid multi-scroll fractional-order chaotic attractors. , 2016, Chaos.

[27]  Giuseppe Grassi,et al.  Decomposition Method for Studying Smooth Chua's equation with Application to hyperchaotic multiscroll attractors , 2007, Int. J. Bifurc. Chaos.

[28]  Guanghui Sun,et al.  Generating Multi-Scroll Chaotic Attractors via Switched Fractional Systems , 2011 .

[29]  Xinghuo Yu,et al.  Chaos control : theory and applications , 2003 .

[30]  Weihua Deng,et al.  Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control. , 2006, Chaos.

[31]  Amin Zarei,et al.  Complex dynamics in a 5-D hyper-chaotic attractor with four-wing, one equilibrium and multiple chaotic attractors , 2015 .

[32]  Juebang Yu,et al.  Chaos in the fractional order periodically forced complex Duffing’s oscillators , 2005 .

[33]  Yuan Lin,et al.  Generation and implementation of grid multiscroll hyperchaotic attractors using CCII , 2016 .

[34]  Xinghuo Yu,et al.  Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method , 2004, Autom..

[35]  Xing-yuan Wang,et al.  Dynamic analysis of the fractional-order Liu system and its synchronization. , 2007, Chaos.

[36]  Jan Awrejcewicz,et al.  Routes to chaos in continuous mechanical systems. Part 3: The Lyapunov exponents, hyper, hyper-hyper and spatial–temporal chaos , 2012 .

[37]  E. O. Ochola,et al.  A hyperchaotic system without equilibrium , 2012 .

[38]  G. Leonov,et al.  Localization of hidden Chuaʼs attractors , 2011 .

[39]  Varactor-tuned superconducting filter with constant absolute bandwidth at VHF-band , 2015 .

[40]  O. Rössler.  An equation for hyperchaos , 1979 .

[41]  Jan Awrejcewicz,et al.  Deterministic Chaos in One Dimensional Continuous Systems , 2016 .

[42]  Lei Chen,et al.  Circuit implementation and model of a new multi‐scroll chaotic system , 2014, Int. J. Circuit Theory Appl..

[43]  Yigang He,et al.  A Systematic Design Method for 3-D Grid Multiscroll Chaotic Attractors and Its Circuit Implementation Employing CFOAs , 2015, Int. J. Bifurc. Chaos.

[44]  Michael V. Basin,et al.  A family of hyperchaotic multi-scroll attractors in Rn , 2014, Appl. Math. Comput..

[45]  Jan Awrejcewicz,et al.  Application of the generalized Prony spectrum for extraction of information hidden in chaotic trajectories of triple pendulum , 2014 .