Abstract
In this work, a novel hyperchaotic system is introduced. The system consists of four coupled continuous-time ordinary differential equations with three quadratic nonlinearities. Based on the center manifold and local bifurcation theorems, the existence of pitchfork bifurcation is proved at the origin equilibrium point of the proposed system. Also, the existence of Hopf bifurcation near all the equilibrium points of the system is shown. Moreover, stability analysis of the resulting periodic solutions is analyzed using Kuznetsov’s theory which determines the analytical conditions for the occurrence of supercritical (subcritical) Hopf bifurcation’s type. Numerical verifications such as Lyapunov exponents’ spectrum, Lyapunov dimension, bifurcation diagrams and the continuation software MATCONT are used to show the rich dynamics of the proposed system and to confirm the analytical results. Finally, the hyperchaotic behaviors in this system are suppressed to its three equilibrium points using a novel control method based on Lyapunov stability approach.
Similar content being viewed by others
References
Rössler OE (1979) Continuous chaos—four prototype equations. Ann N Y Acad Sci 316:376–392
Rössler OE (1979) An equation for hyperchaos. Phys Lett A 71:155–157
Matsumoto T, Chua LO, Kobayashi K (1986) Hyperchaos: laboratory experiment and numerical confirmation. IEEE Trans Circuits Syst 33:1143–1147
Kapitaniak T, Chua LO, Zhong G-Q (1994) Experimental hyperchaos in coupled Chua’s circuits. IEEE Trans Circuits Syst I(41):499–503
Kapitaniak T, Chua LO (1994) Hyperchaotic attractors of unidirectionally-coupled Chua’s circuit. Int J Bifurcat Chaos 4:477–482
Khan A, Tyagi A (2017) Analysis and hyper-chaos control of a new 4-D hyper-chaotic system by using optimal and adaptive control design. Int J Dyn Control 5:1147–1155
Khan A, Bhat MA (2017) Hyper-chaotic analysis and adaptive multi-switching synchronization of a novel asymmetric non-linear dynamical system. Int J Dyn Control 5:1211–1221
Khan A, Kumar S (2017) T–S fuzzy observed based design and synchronization of chaotic and hyper-chaotic dynamical systems. Int J Dyn Control. https://doi.org/10.1007/s40435-017-0358-y
Chen A, Lu J-A, Lü J, Yu S (2006) Generating hyperchaotic Lü attractor via state feedback control. Phys A 364:103–110
Ahmad WM (2006) A simple multi-scroll hyperchaotic system. Chaos Solitons Fractals 27:1213–1219
Kengne J, Tsotsop MF, Negou AN, Kenne G (2017) On the dynamics of single amplifier biquad based inductor-free hyperchaotic oscillators: a case study. Int J Dyn Control 5:421–435
Kengne J, Tsotsop MF, Mbe ESK, Fotsin HB, Kenne G (2017) On coexisting bifurcations and hyperchaos in a class of diode-based oscillators: a case study. Int J Dyn Control 5:530–541
Vincent UE, Nbendjo BRN, Ajayi AA, Njah AN, McClintock PVE (2015) Hyperchaos and bifurcations in a driven Van der Pol–Duffing oscillator circuit. Int J Dyn Control 3:363–370
Mahmoud GM, Al-Kashif MA, Farghaly AA (2008) Chaotic and hyperchaotic attractors of a complex nonlinear system. J Phys A Math Theor 41:055104
Mahmoud GM, Mahmoud EE, Ahmed ME (2009) On the hyperchaotic complex Lü system. Nonlinear Dyn 58:725–738
Matouk AE (2009) Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system. Phys Lett A 373:2166–2173
Lan Y, Li Q (2010) Chaos synchronization of a new hyperchaotic system. Appl Math Comput 217:2125–2132
Mahmoud GM, Mahmoud EE (2010) Synchronization and control of hyperchaotic complex Lorenz system. Nonlinear Dyn 80:2286–2296
Chen Z, Yang Y, Qi G, Yuan Z (2007) A novel hyperchaos system only with one equilibrium. Phys Lett A 360:696–701
Chen G (2011) Controlling chaotic and hyperchaotic systems via a simple adaptive feedback controller. Comput Math Appl 61:2031–2034
Hegazi AS, Matouk AE (2011) Dynamical behaviors and synchronization in the fractional order hyperchaotic Chen system. Appl Math Lett 24:1938–1944
Torkamani S, Butcher E (2013) Delay, state, and parameter estimation in chaotic and hyperchaotic delayed systems with uncertainty and time-varying delay. Int J Dyn Control 1:135–163
Abedini M, Gomroki M, Salarieh H, Meghdari A (2014) Identification of 4D Lü hyper-chaotic system using identical systems synchronization and fractional adaptation law. Appl Math Model 38:4652–4661
Matouk AE, Elsadany AA (2014) Achieving synchronization between the fractional-order hyperchaotic Novel and Chen systems via a new nonlinear control technique. Appl Math Lett 29:30–35
El-Sayed AMA, Nour HM, Elsaid A, Matouk AE, Elsonbaty A (2014) Circuit realization, bifurcations, chaos and hyperchaos in a new 4D system. Appl Math Comput 239:333–345
Thamilmaran K, Lakshmanan M, Venkatesan A (2004) Hyperchaos in a modified canonical Chua’s circuit. Int J Bifurcat Chaos 14:221–243
Gao TG, Chen ZQ, Chen G (2006) A hyper-chaos generated from Chen’s system. Int J Mod Phys C 17:471–478
Matouk AE (2015) On the periodic orbits bifurcating from a fold Hopf bifurcation in two hyperchaotic systems. Optik 126:4890–4895
Zhang L (2017) A novel 4-D butterfly hyperchaotic system. Optik 131:215–220
Gao T, Chen Z (2008) A new image encryption algorithm based on hyper-chaos. Phys Lett A 372:394–400
Zhu C (2012) A novel image encryption scheme based on improved hyperchaotic sequences. Opt Commun 285:29–37
Garcia-Martinez M, Čelikovsky S (2015) Hyperchaotic encryption based on multi-scroll piecewise linear systems. Appl Math Comput 270:413–424
El-Sayed AMA, Elsonbaty A, Elsadany AA, Matouk AE (2016) Dynamical analysis and circuit simulation of a new fractional-order hyperchaotic system and its discretization. Int J Bifurcat Chaos 26:1650222
Lin J (2015) Oppositional backtracking search optimization algorithm for parameter identification of hyperchaotic systems. Nonlinear Dyn 80:209–219
Smaoui N, Karouma A, Zribi M (2011) Secure communications based on the synchronization of the hyperchaotic Chen and the unified chaotic systems. Commun Nonlinear Sci Numer Simul 16:3279–3293
Hassan MF (2014) A new approach for secure communication using constrained hyperchaotic systems. Appl Math Comput 246:711–730
He J, Cai J, Lin J (2016) Synchronization of hyperchaotic systems with multiple unknown parameters and its application in secure communication. Optik 127:2502–2508
Fang J, Deng W, Wu Y, Ding G (2014) A novel hyperchaotic system and its circuit implementation. Optik 125:6305–6311
El-Sayed AMA, Nour HM, Elsaid A, Matouk AE, Elsonbaty A (2016) Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system. Appl Math Model 40:3516–3534
Vicente R, Daudén J, Colet P, Toral R (2005) Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to delayed feedback loop. IEEE J Quantum Electron 41:541–548
Pu X, Tian X-J, Zhai H-Y, Qiao L, Liu C-Y, Cui Y-Q (2013) Simulation study on hyperchaos analysis of reforming system based on single-ring erbium-doped fiber laser. J China Univ Posts Telecommun 20:117–121
Haken H (1983) At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point. Phys Lett A 94:71–72
Elabbasy EM, Agiza HN, El-Dessoky MM (2006) Adaptive synchronization of a hyperchaotic system with uncertain parameter. Chaos Solitons Fractals 30:1133–1142
Stenflo L (1996) Generalized Lorenz equations for acoustic-gravity waves in the atmosphere. Phys Scr 53:83–84
Singh S (2016) Single input sliding mode control for hyperchaotic Lu system with parameter uncertainty. Int J Dyn Control 4:504–514
Tripathi P, Aneja N, Sharma BK (2018) Stability of dynamical behavior of a new hyper chaotic system in certain range and its hybrid projective synchronization behavior. Int J Dyn Control. https://doi.org/10.1007/s40435-018-0424-0
Singh JP, Roy BK (2018) Multistability and hidden chaotic attractors in a new simple 4-D chaotic system with chaotic 2-torus behaviour. Int J Dyn Control. https://doi.org/10.1007/s40435-017-0392-9
Jafari S, Sprott JC, Molaie M (2016) A simple chaotic flow with a plane of equilibria. Int J Bifurcat Chaos 26:1650098–1650104
Sprott JC (2011) A proposed standard for the publication of new chaotic systems. Int J Bifurcat Chaos 21:2391–2394
Hsü ID, Kazarinoff ND (1977) Existence and stability of periodic solutions of a third-order nonlinear autonomous system simulating immune response in animals. Proc R Soc Edinburgh Sect A 77:163–175
Matouk AE (2008) Dynamical analysis feedback control and synchronization of Liu dynamical system. Nonlinear Anal Theor Methods Appl 69:3213–3224
Matouk AE, Agiza HN (2008) Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor. J Math Anal Appl 341:259–269
Matouk AE, Elsadany AA (2016) Dynamical analysis, stabilization and discretization of a chaotic fractional-order GLV model. Nonlinear Dyn 85:1597–1612
Wu R, Fang T (2015) Stability and Hopf bifurcation of a Lorenz-like system. Appl Math Comput 262:335–343
Elsadany AA, Matouk AE, Abdelwahab AG, Abdallah HS (2018) Dynamical analysis, linear feedback control and synchronization of a generalized Lotka–Volterra system. Int J Dyn Control 6:328–338
Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, New York
Kuznetsov YA (1998) Elements of applied bifurcation theory, 2nd edn. Springer, New York
Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys D 16:285–287
Kaplan J, Yorke J (1979) Chaotic behavior of multidimensional difference equations. Lecture notes in mathematics. Springer, p 730
Acknowledgements
This work is supported by Deanship of Scientific Research at Majmaah University. The author thanks the anonymous reviewers for providing some helpful comments which improve the style, readability and clarity of this work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Matouk, A.E. Dynamics and control in a novel hyperchaotic system. Int. J. Dynam. Control 7, 241–255 (2019). https://doi.org/10.1007/s40435-018-0439-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40435-018-0439-6