Z. Elhadj. J. C. Sprott UDC

Transcription

1 UDC ABOUT THE BOUNDEDNESS OF 3D CONTINUOUS-TIME QUADRATIC SYSTEMS ПРО ОБМЕЖЕНIСТЬ 3D КВАДРАТИЧНИХ СИСТЕМ З НЕПЕРЕРВНИМ ЧАСОМ Z. Elhaj Univ. f Tébessa 100), Algeria zeraoulia@mail.univ-tebessa.z zelhaj1@yahoo.fr J. C. Sprott Univ. Wisconsin Maison, WI 53706, USA sprott@physics.wisc.eu In this paper, we generalize all the existing results in the current literature for the upper boun of a general 3-D quaratic continuous-time system. In particular, we fin large regions in the bifurcation parameters space of this system it is boune. Наведено узагальнення всiх вiдомих результатiв про верхню для загальної 3D квадратичної системи з неперервним часом. Зокрема, знайдено великi областi у просторi бiфуркацiйних параметрiв системи, де система є обмеженою. 1. Introuction. Chaos in 3-D quaratic continuous-time autonomous systems was iscovere in 1963 by E. Lorenz [1], he propose a simple mathematical moel of a weather system which was mae up of three linke nonlinear ifferential equations. Some surprising results clearly showe the complex behavior from supposely simple equations. Also, it has been note that the behavior of this system of equations was sensitively epenent on the initial conitions of the moel, i.e., it implie that if there were any errors in observing the initial state of the system which is inevitable in any real system, therefore, the preiction of a future state of the system was impossible. Later, Ruelle in 1979, when computer simulation came along, the first fractal shape ientifie took the form of a butterfly; it arose from graphing the changes in weather systems moelle by Lorenz. The Lorenz s attractor shows just how an why weather prognostication is so involve an notoriously wrong because of the butterfly effect. This amusing name reflects the possibility that a butterfly in the Amazon might, in principal, ultimately alter the weather in Kansas. A new chaotic attractor of three-imensional system is coine by Chen an Ueta [], in the pursuit of anti controlling chaos for Lorenz moel [1]. This new chaotic moel reassembles the Lorenz an Rôssler systems [1 3]. However the Chen moel appears to be more complex an more sophisticate []. They both are threeimensional autonomous with only two quaratic terms, but it is topologically not equivalent to the Lorenz equation. Many other systems of three smooth autonomous orinary ifferential equations with two quaratic nonlinear terms have been foun in aition of the Chen system, c Z. Elhaj, J. C. Sprott, ISSN Нелiнiйнi коливання, 010, т. 13, N 4

2 Z. ELHADJ, J. C. SPROTT aiming to generate chaotic attractors; these inclue the Rôssler system [3], recently, Liu & Chen [4] foun another relatively simple 3-D smooth autonomous chaotic system with only quaratic nonlinearities; an some other systems capture by Sprott via computer foun in [5]. Note that the fining of simple 3-D quaratic chaotic systems is quite a very interesting fiel in the stuy of ynamical systems, since many technological applications such as communication, encryption, information storage, uses these chaotic attractors [6]. The bouneness of these systems was the subject of many works. Boune chaotic systems an the estimate of their bouns is important in chaos control, chaos synchronization, an their applications. The estimation of the upper boun of a chaotic system is quite ifficult to achieve technically. In this work, we generalize all the relevant results of the literature an escribe some of these bouns using multivariable function analysis. The most general 3-D quaratic continuous-time autonomous system is given by x = a 0 + a 1 x + a y + a 3 z + a 4 x + a 5 y + a 6 z + a 7 xy + a 8 xz + a 9 yz, y = b 0 + b 1 x + b y + b 3 z + b 4 x + b 5 y + b 6 z + b 7 xy + b 8 xz + b 9 yz, 1) z = c 0 + c 1 x + c y + c 3 z + c 4 x + c 5 y + c 6 z + c 7 xy + c 8 xz + c 9 yz, a i, b i, c i ) 0 i 9 R 30 are the bifurcation parameters. Several researchers have efine an stuie quaratic 3-D chaotic systems as escribe in the references. The generalize Lorenz-like canonical form introuce in [7 1] gives a unique an unifie classification for a large class of 3-D quaratic chaotic systems. This system contains all the well known quaratic systems given in [1 4, 7, 13]. In chaos control, chaos synchronization, an their applications, the estimation of an upper boun of the system uner consieration is an important task. For example, in [14] the bouneness of the Lorenz system [1] was investigate an in [9] the bouneness of the Chen system [] was investigate. Recently, a better upper boun for the Lorenz system for all positive values of its parameters was erive in [15], an it is the best result in the current literature because the estimation overcomes some problems relate to the existence of singularities arising in the value of the upper boun given in [14]. In this paper, we generalize all these results concerning an upper boun for the general 3-D quaratic continuous-time autonomous system. In particular, we fin large regions in the bifurcation parameter space of this system it is boune. The metho is base on multivariable function analysis.. Estimate of the boun for the general system. To estimate the boun for the general system 1), we consier the function V x, y, z) efine by V x, y, z) = x α) + y β) + z γ) ) α, β, γ) R 3 is any set of real constants for which the erivative of ) along the solutions of 1) is given by V t = ωx α 1) ϕy β 1 ) z γ 1 ) + 3) ISSN Нелiнiйнi коливання, 010, т. 13, N 4

3 ABOUT THE BOUNDEDNESS OF 3D CONTINUOUS-TIME QUADRATIC SYSTEMS 3 = ωα 1 + ϕβ 1 + γ 1 βb 0 γc 0 αa 0, ω = αa 4 a 1 + βb 4 + γc 4, ϕ = αa 5 b + βb 5 + γc 5, = αa 6 c 3 + βb 6 + γc 6, α 1 = a 0 αa 1 βb 1 γc 1, if ω 0, ω 4) β 1 = b 0 αa βb γc, if ϕ 0, ϕ γ 1 = c 0 αa 3 βb 3 γc 3, if 0. Note that if ω = 0 or ϕ = 0 or = 0, then there is no nee to calculate α 1, β 1 an γ 1, respectively, an a conition relating a i, b i, c i ) 0 i 9 R 30 to α, β, γ) R 3 is obtaine. If not, we have the formulas given by the last three equalities of 4). The form of the function V t in 3) is possible if the following conitions on the coefficients a i, b i, c i ) 0 i 9 R 30 hol: a 4 = 0, b 4 = a 7, b 5 = 0, b 7 = a 5, c 4 = a 8, c 5 = b 9, c 6 = 0, c 7 = a 9 b 8, c 8 = a 6, c 9 = b 6, b 1 = αa 7 βa 5 a + γc 7, 5) c 1 = a 3 γa 6 + αa 8 + βb 8, c = αa 9 b 3 γb 6 + βb 9, i.e., the system 1) becomes x = a 0 + a 1 x + a y + a 3 z + a 5 y + a 6 z + a 7 xy + a 8 xz + a 9 yz, y = b 0 + b 1 x + b y + b 3 z a 7 x + b 6 z a 5 xy + b 8 xz + b 9 yz, 6) z = c 0 + c 1 x + c y + c 3 z a 8 x b 9 y a 9 + b 8 )xy a 6 xz b 6 yz, with the formulas for b 1, c 1, an c given by the last three equations of 5). To prove the bouneness of system 6), we assume that it is boune an then we fin its ISSN Нелiнiйнi коливання, 010, т. 13, N 4

4 4 Z. ELHADJ, J. C. SPROTT boun, i.e., assume that ω, ϕ,, an are strictly positive, i.e., ωα 1 + ϕβ 1 + γ 1 βb 0 γc 0 αa 0 > 0, a 1 < αa 4 + βb 4 + γc 4, b < αa 5 + βb 5 + γc 5, 7) c 3 < αa 6 + βb 6 + γc 6. Then if system 6) is boune, the function 3) has a maximum value, an the maximum point x 0, y 0, z 0 ) satisfies x 0 α 1 ) Now consier the ellipsoi Γ = { an efine the function ω x, y, z) R 3 : x α 1) ω + y 0 β 1 ) ϕ + y β 1) ϕ + z 0 γ 1 ) F x, y, z) = Gx, y, z) + λhx, y, z), Gx, y, z) = x + y + z, + z γ 1) = 1. = 1, ω, ϕ,, > 0 }, 8) Hx, y, z) = x α 1) ω + y β 1) ϕ + z γ 1) 1 λ R is a finite parameter. Then we have max x,y,z) Γ G = max x,y,z) Γ F an F x, y, z) y F x, y, z) z F x, y, z) x = 1 ϕλβ 1 ϕλ + ) y) = 1 λγ 1 λ + )z) = 1 ωλα 1 ωλ + )x), an the following cases accoring to the value of the parameter λ with respect to the values ω, ϕ, an λ if ω, ϕ, > 0. Otherwise, a similar stuy can be one. i) If λ ω, λ ϕ, an λ, then x 0, y 0, z 0 ) = ) ωλα1 + ωλ, ϕλβ 1 + ϕλ, λγ 1 + λ ISSN Нелiнiйнi коливання, 010, т. 13, N 4

5 ABOUT THE BOUNDEDNESS OF 3D CONTINUOUS-TIME QUADRATIC SYSTEMS 5 an max x,y,z) Γ G = ξ 1 ξ 1 = ω λ α 1 + ωλ) + ϕ λ β 1 + ϕλ) + λ γ 1 + λ). 9) In this case, there exists a parameterize family in λ) of bouns given by 9) of system 6). ii) If λ = ω, ω 0, ϕ 0, 0, ω ϕ, ω ), λ ϕ, λ, then x 0, y 0, z 0 ) = ± ω 1 ξ ) ) + α 1, β 1ϕ ξ 3 ω ϕ, γ 1 ω 10) with the conition ξ = ϕβ 1 + γ 1 )ω ) 3 ω, ξ 3 = ω) ω ϕ) 4 ω 4 ξ 3 ξ. This confirms that the value x 0 in 10) is well efine an the conitions ω 0, ϕ 0, 0, ω ϕ, an ω are formulate as follows: a 1 αa 4 + βb 4 + γc 4, b βb 5 + γc 5 + αa 5, c 3 βb 6 + γc 6 + αa 6, b a 1 a 5 a 4 ) α + b 5 b 4 ) β + c 5 c 4 ) γ, 11) In this case, we have max G = x,y,z) Γ c 3 a 1 a 6 a 4 ) α + b 6 b 4 ) β + c 6 c 4 ) γ. 1 ξ ) ) + α 1 + β 1 ϕ ω ξ 3 ω ϕ) + γ 1 ω ). iii) If λ ω, λ = ϕ ω 0, ϕ 0, 0, ω ϕ, ϕ ), λ, then x 0, y 0, z 0 ) = α 1ω ϕ ω, ± ϕ 1 ξ ) 4 + β 1, ξ 5 ) γ 1 ϕ 1) ISSN Нелiнiйнi коливання, 010, т. 13, N 4

6 6 Z. ELHADJ, J. C. SPROTT ξ 4 = ωϕα 1 ωϕγ 1 ωϕ α 1 ω α 1 + ω γ 1 + ϕ γ 1) ϕ, ξ 5 = ϕ) ϕ ω) with the conition ξ 5 ξ 4. This confirms that the value y 0 in 1) is well efine an the conitions ω 0, ϕ 0, 0, ω ϕ, an ϕ are formulate by the first four equations of 11) an In this case, we have max G = x,y,z) Γ c 3 b a 6 a 5 ) α + b 6 b 5 ) β + c 6 c 5 ) γ. 13) 1 ξ ) ) 4 + β 1 + α 1 ω ϕ ξ 5 ϕ ω) + γ 1 ϕ ). iv) If λ ω, λ ϕ, λ = ω 0, ϕ 0, 0, ω, ϕ), then x 0, y 0, z 0 ) = α 1 ω ω, β 1ϕ ϕ, ± 1 ξ ) ) 6 + γ 1 ξ 7 14) ξ 6 = ωϕ α 1 ωϕβ 1 ωϕα 1 + ω α 1 + ω ϕβ 1 + ϕ β 1), ξ 7 = ϕ) ω) with the conition ξ 7 ξ 6. This confirms that the value z 0 in 14) is well efine an the conitions ω 0, ϕ 0, 0, ω, an ϕ are formulate by the first four equations of 11) an 13), respectively. The other possible cases are treate using the same logic. Theorem 1. Assume that conitions 4), 5), an 7) hol. Then the general 3-D quaratic continuous-time system 1) is boune, i.e., it is containe in the ellipsoi 8). Similar results can be foun using the cases iscusse above. 3. Example. Consier the Lorenz system given by ẋ = a y x), ẏ = cx y xz, ż = xy bz, ISSN Нелiнiйнi коливання, 010, т. 13, N 4

7 ABOUT THE BOUNDEDNESS OF 3D CONTINUOUS-TIME QUADRATIC SYSTEMS 7 i.e., a i = 0, i = 0, 3, 4, 5, 6, 7, 8, 9, a 1 = a, a = a, b i = 0, i = 0, 3, 4, 5, 6, 7, 9, b 1 = c, b = 1, b 8 = 1, c i = 0, i = 0, 1,, 4, 5, 6, 8, 9, c 3 = b, an c 7 = 1. We choose α = β = 0 an γ = a + c as in [15]. Thus V x, y, z) = x + y + z a + c)) ) a + c an = b, ω = a, ϕ = 1, = b, α 1 = 0, β 1 = 0, an γ 1 = a + c V. Then we have = ax t y b z a + c ) ) a + c + b, which is the same as in [15]. Also, it is easy to verify that conitions 5) an 7) hol for this case. The ellipsoi Γ is given by Γ = { x, y, z) R 3 : b a x a+c ) + y b a+c ) z a+c } ) + ) = 1, a, b, c > 0, which is also the same as in [15]. Finally, we have the result shown in [15] that confirms that if a > 0, b > 0, an c > 0, then the Lorenz } system [1] is containe in the sphere Ω = = {x, y, z) R 3 /x + y + z a c) = R, a+c R = a + c) b, if a 1, b, 4 b 1) a + c), if a > b, b <, a + c) b, if a < 1, b a. 4a b a) 4. Conclusion. Using multivariable function analysis, we generalize all the results about fining an upper boun for the general 3-D quaratic continuous-time autonomous system. In particular, we fin large regions in the bifurcation parameters space of this system it is boune. 1. Lorenz E. N. Deterministic nonperioic flow // J. Atmos. Sci P Chen G., Ueta T. Yet another chaotic attractor // Int. J. Bifur. Chaos P Rössler O. E. An equation for continuous chaos // Phys. Lett. A P Lu J. H., Chen G. A new chaotic attractor coine // Int. J. Bifur. Chaos P Sprott J. C. Some simple chaotic flows // Phys. Rev. E P Chen G. Controlling chaos an bifurcations in engineering systems. Boca Raton, FL: CRC Press, Yang Q., Chen G., Zhou T. A unifie Lorenz-type system an its canonical form // Int. J. Bifur. Chaos P Qi G., Chen G., Du S. Analysis of a new chaotic system // Physica A P Zhou T., Tang Y., Chen G. Complex ynamical behaviors of the chaotic Chen s system // Int. J. Bifur. Chaos P Zhou T., Chen G., Yang Q. Constructing a new chaotic system base on the Sh ilnikov criterion // Chaos, Solitons an Fractals P ISSN Нелiнiйнi коливання, 010, т. 13, N 4

8 8 Z. ELHADJ, J. C. SPROTT 11. Zhou T., Chen G. Classification of chaos in 3-D autonomous quaratic systems-i. Basic framework an methos // Int. J. Bifur. Chaos P Celikovsky S., Chen G. On a generalize Lorenz canonical form of chaotic systems // Int. J. Bifur. Chaos P Lu J. H., Chen G., Cheng D., Celikovsky S. Brige the gap between the Lorenz system an the Chen system // Int. J. Bifur. Chaos P Leonov G., Bunin A., Koksch N. Attractor localization of the Lorenz system // Z. a Angew. Math. un Mech P Li D., Lu J. A., Wu X., Chen G. Estimating the bouns for the Lorenz family of chaotic systems // Chaos, Solitons an Fractals P Receive , after revision ISSN Нелiнiйнi коливання, 010, т. 13, N 4