Chaotic and Hyperchaotic Complex Jerk Equations

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1 International Jornal of Modern Nonlinear Theory and Application Pblished Online March 0 ( Chaotic and Hyperchaotic Complex Jerk Eqations Gamal M Mahmod Mansor E Ahmed Department of Mathematics Faclty of Science Assit University Assit Egypt gmahmod@anedeg Received Janary 7 0; revised Febrary 9 0; accepted March 0 ABSTRACT The aim of this paper is to introdce and investigate chaotic and hyperchaotic complex jerk eqations The jerk eqations describe varios phenomena in engineering and physics for example electrical circits laser physics mechanical oscillators damped harmonic oscillators and biological systems Properties of these systems are stdied and their Lyapnov exponents are calclated The dynamics of these systems is rich in wide range of systems parameters The control of chaotic attractors of the complex jerk eqation is investigated The Lyapnov exponents are calclated to show that the chaotic behavior is converted to reglar behavior Keywords: Hyperchaotic; Chaotic; Attractors; Lyapnov Exponents; Jerk Fnction; Control; Complex Introdction Chaos and hyperchaos can occr in systems of atonomos ordinary differential eqations (ODE s) with at least three variables and two qadratic nonlinearities [-4] The Poincaré-Bendixson theorem shows that chaos does not exist in a two-dimensional atonomos system In 994 Sprott [] fond nmerically forteen chaotic systems with six terms and one qadratic nonlinearities In 996 Gottlieb [6] showed that some of these systems cold be written as a single third-order ODE By jerk fnction he means a fnction j sch that the third-order ODE can be written in the form j where j is the time derivative of the acceleration and the eqation is called a jerk eqation The jerk eqations describe varios phenomena in engineering and physics for example electrical circits laser physics mechanical oscillators damped harmonic oscillators and biological systems In the literatre some jerk eqations are introdced and stdied [7-3] Kocić et al [4] considered and stdied two modifications of a 3-dimensional dynamic flow known as jerk dynamical systems of Sprott [3] In this paper we define a complex jerk eqation as an atonomos third-order complex differential eqation of the form: z j zzzzzz () where z is a complex variable the overdot represent the time derivative and j is the complex jerk fnction (time derivative of acceleration z ) and an overbar denotes complex conjgate variables In recent years we introdced and stdied several complex nonlinear systems which appear in many important applications [-] As an example of Eqation () we propose a chaotic complex jerk eqation as: z zz zz z 0 () where α β and η are positive parameters ν is a negative parameter z is complex variable dots represent derivatives with respect to time and an overbar denotes complex conjgate variables We sggest the eqation: z zzz z 0 0 (3) which is an example of a hyperchaotic complex jerk eqation The corresponding real form of (3) (ie z is a real variable) was introdced in [9] and has only chaotic behavior The organization of rest of the paper is as follows: In Section symmetry invariance fixed points of () and stability analysis of the trivial fixed point are discssed Lyapnov exponents are calclated in Section 3 and sed to classify the attractors of () It is clear that or Eqation () has chaotic periodic qasi-periodic attracttors and soltions that approach fixed points In Section 4 we stdy the basic properties of the hyperchaotic complex jerk Eqation (3) Nmerically the range of pa- rameters vales of the system at which hyperchaotic attractors exist is calclated in Section Section 6 contains the control of chaotic attractors of Eqation () by adding a complex periodic forcing The last section contains or conclding remarks Basic Properties of Chaotic Jerk Eqation () In this section we stdy the basic properties of or new Eqation () Eqation () can eqivalently be written as three first-order ordinary differential eqations as: Copyright 0 SciRes

2 G M MAHMOUD ET AL 7 z x x y y yxzz z where z i x 3 i4 and y i 6 are complex variables i j j 6 are real variables dots represent derivatives with respect to time and an overbar denotes complex conjgate variables The real version of (4) is: () The basic dynamical properties of system () are: Symmetry and Invariance From () we note that this system is invariant nder the transformation Therefore if is a soltion of () then is also a soltion of the same system (4) where rcos rsin for 0 π To stdy the stability of E 0 the Jacobian matrix of Eqation () at E 0 is: J E The characteristic polynomial is: 3 0 (8) According to the Roth-Hrwitz condition the real parts of the roots of (8) are negative if and only if Since 0 then E0 is nstable The stability analysis of E can be similarly stdied as we did for the trivial fixed point E 0 Dissipation The divergence of () is: 6 j j j Therefore the system () is dissipative for the case: 0 3 Eqilibria and Their Stability The fixed points of system () can be fond by solving the following eqations: (6) Therefor system (4) has trivial fixed point E The projection in the plane of the non trivial fixed points is a circle: r (7) whose center is at the origin and radis is r The non trivial fixed point can be written in the form E = L l j 3 Lyapnov Exponents and Attractors of Eqation () This section is devoted to calclate Lyapnov exponents and sed their signs to classify attractors of Eqation () Based on these exponents we compte parameters vales of or Eqation () at which chaotic periodic and qasi-periodic attractors and attractors that approach fixed point exist 3 Lyapnov Exponents System () in vector notation can be written as: U t h U t ; (9) where U t t t 3 t 4 t t 6 t t is the state space vector h h h h3 h4 h h6 a set of parameters and t is denoting transpose The eqations for small deviations U from the trajectory Ut are: U t Llj Ut; U l j 346 (0) hl where Ll j= is the Jacobian matrix of the form: j t Copyright 0 SciRes

3 8 G M MAHMOUD ET AL The Lyapnov exponents l of the system is defined by: l l lim log () t t l t 0 To find l Eqations (9) and (0) mst be nmerically solved simltaneosly Rnge Ktta method of order 4 is sed to calclate l For the ch oice 4 and and the initial conditions t and 6 0 We calclate the Lyapnov exponents which are: This means th at or system (4) for this choice of α β μ and η is a chaotic system since one of its Lyapnov exponents is positive and dissipative system becase their sm is negative 3 Attractors of Eqation () 3 Fix 4 and Vary In Figres (a) and (b) we plotted the corresponding Lyapnov exponents λ l l 6 of system () sing the initial conditions t and 6 0 It is clear that from Figre (a) when and 634 the new system () has chaotic attractors It has also periodic attractors for 34 4 while it has qasi-periodic attractors when lies in the interval Soltions of system () that approach nontrivial fixed points are exist for 4 47 As is shown in Figre (b) the vales of 4 and 6 are negative 3 Fix and Vary From Figre (c) one can conclde that () has chaotic attractors for and and periodic attractors for β lies in and Soltions of system () that approach nontrivial fixed points are exist for Fix 4 and Vary As we did before we plot only λ λ and λ 3 in Figre (d) and we see that () has chaotic attractors for and In between above vales of or system has periodic attractors as one sees from Figre (d) 34 Fix 4 and Vary As is shown in Figre (e) the chaotic attractors exist for 0004 Using the same choice of the parameters and initial conditions as in Figre the chaotic attractors of () are plotted in Figre in spaces respectively and Some Properties of Eqation (3) This section deals with the basic dynamical properties of a hyperchaotic jerk Eqation (3) As we did in Section Eqation (3) can be written as a system of three firstorder ordinary differential eqations sch as: z x x y y yzx z () where z i x 3 i 4 and y i6 are complex variables i j j 6 are real variables The real version of (3) is: (3) If is a soltion of (3) then is also a soltion From (3) if 0 then (3) is dissipative System (3) has only trivial fixed point E To stdy the stability of E 0 we calclate the Jacobian matrix of system (3) at E to 0 get: J E Its characteristic polynomial is: 3 P 0 (4) An elementary stdy proves that this polynomial has only one real root r which is therefore nega- 3 tive Since the characteristic eqation is a cbic eqation with real coefficients we will have withot loss of generality P rc c where c is a complex nmber After expanding the above eqation and comparing the coefficients with those of the original characteristic eqation we come p with the relation Rec r We conclde that the real part of c is positive and that th e fixed point is nstable see Ref [9] Lyapnov Exponents and Attractors of Hyperchaotic Eqation (3) In this section we calclate the Lyapnov exponents and attractors of Eqation (3) Copyright 0 SciRes

4 G M MAHMOUD ET AL 9 (a) (b) (c) (d) Copyright 0 SciRes

5 0 G M MAHMOUD ET AL Figre Lyapnov exponents of () with the initial conditions t0 0 d (e) 0 = an 6 0 (a) and 3 verss ; (b) 4 and 6 verss ; (c) and 3 verss ; (d) and 3 verss ; (e) and 3 verss Figre A chaotic attractor of () for 4 and at the same initial conditions as in Figre (a) in space; (b) in space; (c) in space; (d) in space 4 Copyright 0 SciRes

6 G M MAHMOUD ET AL For the choice 03 and the initial conditions t hyperchaotic attractors while it has qasi-periodic attract tors when α lies in the intervals and we calclate the Lyapnov and 0840 exponents which are: The hyperchaotic attractors of (3) sing the same ini and This tial conditions as in Figre 3(a) and for 03 are means that or Eqation (3) for this choice of is hy- plotted in Figres 3(b) and (c) in 3 and per chaotic eqation since two of the Lyapnov expo- 3 spaces respectively nents and are positive and dissipative eqation since the sm of its Lyapnov exponents is negative 6 Control of Chaotic Attractors of System (4) In Figre 3(a) we plotted the corresponding Lyapnov exponents λ l l 6 of Eqation (3) sing the ini- This section is devoted to stdy the control of chaotic tial conditions t attractors of system (4) based on the addition of complex and It is clear that from periodic forcing kexpi t to its first and second eqations so system (4) becom es: Eqation (3) has z xkexp it x y y yxz z z () Figre 3(a) when and Figre 3 Lyapnov exponents and nmerical calclations of the hyperchaotic attractors of (3) sing the initial conditions t and (a) 3 and 4 verss ; (b) A hyperchaotic attractors of (3) in 3 space; (c) A hyperchaotic attractors of (3) in 3 space Copyright 0 SciRes

7 G M MAHMOUD ET AL The real version of () sing 7 t reads: 3 kcos 6 4 ksin (6) What we wold like to see is whether by a sitable selection of vales of k and ω one can control the chaotic soltions of system (4) by converting them from chaotic to periodic with freqency ω For the choice of and 0 and th l con s t 0 4 k e initia dition we have the Lyapnov exponents (for more de- tails abot the calclations of Lyapnov exponents see Ref [6] This means that the hyperchaotic attractor of () is converted to periodic behavior (see Figres 4(a) and (c) before control and Figres 4(b) and (d) after control) 7 Conclsions In this paper we proposed both chaotic and hyperchaotic complex jerk eqations and investigated their dynamics The stability analysis of the trivial fixed points of these compl ex eqations are stdied Thes e eqations appeared in several important applications of p hysics engineering and b iology Figre 4 A chaotic attractor of system () for k 0 w 0 and 7 0 with the same initial conditions as in Figre and (a) efore c l; (b) space before control; (d) 3 4 space after control 4 plane b ontro 4 plane after control; (c) 3 4 Copyright 0 SciRes

8 G M MAHMOUD ET AL 3 Both of or exam ples () and (3) are symmetric and dissipative nder the condition 0 The chaotic Eqa- tion () has both isolated and non-isolated fixed points The projection of non-isolat ed fixed po int in space is a circle with center 00 The dynamic of system () is very complicated as shown in Figre since it has soltions approach to fixed points periodic soltions qasi-periodic soltions and chaotic behavior The hyperchaotic Eqation (3) has only one fixed point The vales of the parameter at which (3) has hyperchaotic attractors is calclated The control of chaotic attractors of Eqation () i s stdied by adding a complex periodic forcing and the reslts are shown in Figre 4 Other examples of Eqation () can be similarly stdied and in- vestigated as we did for Eqations () and (3) REFERENCES [] E N Lorenz Deterministic Nonperiodic Flow Jornal of Atmospheric Sciences Vol 0 No 963 pp 30-4 doi:07/0-0469(963)00<030:dnf>0co; [] O E Rössler An Eqation for Continos chaos Physics Letters A Vol 7 No 976 pp doi:006/ (76)900-8 [3] O E Rossler An Eqation for Hyperchaos Physics Letters A Vol 7 No pp -7 doi:006/ (79)900-6 [4] J C Sprott Simple Chaotic Systems and Circits American Jornal of Physicals Vol 68 No pp doi:09/938 [] J C Sprott Some Simple Chaotic Flows Physical Review E Vol 0 No 994 pp R647-R60 doi:003/physreve0r647 [6] H P W Gottlieb Qestion #38 What Is the Simplest Jerk Fnction That Gives Chaos? American Jornal of Physicals Vol 64 No 996 p doi:09/876 [7] S J Linz Nonlinear Dynamical Models and Jerk Motion American Jornal of Physicals Vol 6 No pp 3-6 doi:09/894 [8] J C Sprott Some Simple Chaotic Jerk Fnctions American Jornal of Physicals Vol 6 No pp doi:09/88 [9] J-M Malasoma What Is the Simplest Dissipative Chaotic Jerk Eqation Which Is Parity Invariant Physics Letters A Vol 64 No 000 pp doi:006/s (99)0089- [0] F Zhang J Heidel and R L Borne Determining Nonchaotic Parameter Regions in Some Simple Jerk Fnc- tions Chaos Solito ns & Fractals Vol 36 No pp doi:006/jchaos [] R Eichhorn S J Linz and P Hänggi Simple Polynomial Classes of Chaotic Jerk Dynamics Chaos Solitons & Fractals Vol 3 No 00 pp - doi:006/s (00)0037-x [] S J Linz Non-Chaos Criteria for Certain Jerk Dynamics Physics Letters A Vol 7 No pp 04-0 doi:006/s (00) [3] J C Sprott Chaos and Time-Series Analysis Oxford University Press New York 003 [4] L M Kocić and S Gegovska-Zajkova On a Jerk Dy- and Robotics Vol 8 namical System Atomatic Control 009 pp 3-44 [] G M Mahmod and T Bontis The Dynamics of Systems of Complex Nonlinear Oscillators International Jornal of Bifrcation and Chaos Vol 4 No 004 pp doi:04/s [6] G M Mahmod M E Ahmed and E E Mahmod Analysis of Hyperchaotic Complex Lorenz Systems International Jornal of Modern Physics C Vol 9 No pp doi:04/s [7] G M Mahmod M A Al-Kashif and A A Farghaly Chaotic and Hyperchaotic Attractors of a Complex Nonlinear System Jornal of Physicss A: Mathematical and Theoretical Vol 4 No 008 pp -04 doi:0088/7-83/4//004 [8] G M Mahmod E E Mahmod and M E Ahmed On the Hyperchaotic Complex Lü System Nonlinear Dynamics Vol 8 No pp doi:0007/s [9] G M Mahmod T Bontis M A Al-Kashif and S A Aly Dynamical properties and synchronization of complex non-linear eqations for detned laser J Dynamical Systems: An International Jornal Vol 4 No 009 pp doi:0080/ [0] G M Mahmod and E E Mahmod Synchronization and Control of Hyperchaotic Complex Lorenz System Mathematics and Compters in Simlation Vol 80 No 00 pp doi:006/jmatcom00030 [] G M Mahmod and E E Mahmod Complete Synchronization of Chaotic Complex Nonlinear Systems with Uncertain Parameters Nonlinear Dynamics Vol 6 No 4 00 pp doi:0007/s y Copyright 0 SciRes