Chaotic dynamics of the Fractional Order Nonlinear Bloch System

EJTP 8, No. 25 (2011) 233–244

Electronic Journal of Theoretical Physics

Chaotic dynamics of the Fractional Order Nonlinear Bloch System Nasr-eddine Hamri∗1 and Tarek Houmor†2 1

Institute of Sciences & Technologie, University Center of Mila, Algeria 2 Department of Mathematics, University of Constantine, Algeria

Received 6 July 2010, Accepted 10 February 2011, Published 25 May 2011 Abstract: The dynamic behaviors in the fractional-order nonlinear Bloch equations were numerically studied. Basic properties of the system have been analyzed by means of Lyapunov exponents and bifurcation diagrams. The derivative order range used was relatively broad. Regular motions (including period-3 motion) and chaotic motions were examined. The chaotic motion identified was validated by the positive Lyapunov exponent. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Dynamical Systems; Fractional Calculus; Bifurcation PACS (2010): 05.45.Pq; 05.45.Xt; 05.45.-a; 05.45.-a; 02.30.Oz

1.

Introduction

The dynamics of fractional-order systems have attracted increasing attentions in recent years. It has been shown that the fractional-order generalizations of many well-known systems can also behave chaotically: [12] and [7] for the fractional Chua system, [3] and [17] for the fractional Duffing one, [8] and [9] for the fractional Chen one and so on. A review of the literature indicates that phase portrait, bifurcation diagram, Poincaré maps and Lyapunov exponent always be the efficient ways to research the integer and fractional-order systems. Many investigations about fractional-order system have been done. Many researches to nonlinear fractional-order equations are based on the frequencydomain method whereas there are some disadvantages of using this method were found in [19]. On the other hand, the dynamic properties in fractional systems are still needing pay more attentions, some basic properties such as fixed point, limit cycles and chaos have been investigated. Bifurcation, Poincaré map and Lyapunov exponent are all good ∗ †

[email protected] tarek [email protected]

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Electronic Journal of Theoretical Physics 8, No. 25 (2011) 233–244

tools. In this paper, the dynamics of the fractional-order nonlinear Bloch system is studied. Nonlinear Bloch system is a system of three nonlinear ordinary differential equations, which define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. In the paper, we will focus on the dynamical behavior of fractional-order nonlinear Bloch system. Experience of dynamical behavior will be considered. Bifurcation of the parameter-dependent system which provides a summary of essential dynamics is investigated. Period-3 windows, coexisting limit cycles and chaotic zones are found. The occurrence and the nature of chaotic attractors are verified by evaluating the largest Lyapunov exponents. This paper is organized as follows. Fractional derivative definitions, a method for solving fractional-order differential equation and some stability results are introduced in Section 2. In Section 3, fractional-order nonlinear Bloch system is presented based on the integer-order system. Bifurcation and the largest Lyapunov exponents of the fractional-order nonlinear Bloch system are studied in Section 4. Finally, some concluding remarks are given in Section 5.

2.

Fractional Calculus Fundamentals

2.1 Definitions of Fractional Derivatives The idea of fractional calculus has been known since the development of the regular calculus, with the first reference probably being associated with Leibniz and L’Hospital in 1695 where half-order derivative was mentioned. Fractional calculus is a generalization of integration and differentiation to non-integer order fundamental operator a Dtr , where a and t are the limits of the operation. The continuous integro-differential operator is defined as ⎧ ⎪ dr ⎪ :r>0 ⎪ ⎪ ⎨ dt r (1) a Dt = 1 :r=0 ⎪ ⎪ ⎪ ⎪ ⎩ t (dτ )−r : r < 0 a The three definitions used for the general fractional differintegral are the GrunwaldLetnikov (GL) definition, the Riemann-Liouville (RL), and the Caputo definition [11, 13]. These definitions are equivalent for a wide class of functions [13]. The GL is given as ⎛ ⎞ ] [ t−a h ⎜r⎟ r −r (2) (−1)j ⎝ ⎠ f (t − jh), a Dt f (t) = lim h h→0 j=0 j where [.] means the integer part. The RL definition is given as f (τ ) 1 dn t r dτ, a Dt f (t) = n Γ(n − r) dt a (t − τ )r−n+1

(3)


235

for (n − 1 < r < n) and where Γ(.) is the Gamma function.The Caputo’s definition can be written as t 1 f (n) (τ ) r D f (t) = dτ, (4) a t Γ(n − r) a (t − τ )r−n+1 for(n − 1 < r < n). The initial conditions for the fractional order differential equations with the Caputo’s derivatives are the same form as for the integer-order differential equations. These integer-order derivatives have a physical meaning and can be measured. On the other hand, when the Riemann-Liouville derivative (3) is considered, it is necessary to specify the values at t0 = 0 in terms of fractional integrals and their derivatives. These data do not have a physical meaning and, consequently, are not measurable. It can be concluded that the Caputo derivative (4) is a better choice for our study, since the initial values required by the Caputo definition coincide with identifiable physical states in our system.

2.2 Numerical Methods for Calculation of The Fractional Order Derivatives Since the analysis of fractional-order systems is not sufficient yet, a suitable numerical method needs to be selected. Among the literature of fractional-order field, two approximation methods have been proposed in order to obtain response of a fractional order system, one of which is the Adams-Bashforth-Moulton predictor-corrector scheme [5, 6], while the other one is the frequency domain approximation [4]. Due to the specificity of the error estimation bound, simulation results obtained by the former method are more reliable than those of the latter [20]. As a result, the former method is used throughout this paper because of its efficiency and reliability. The method is based on the fact that fractional differential equation D∗α y(t) = f (t, y(t)), 0 ≤ t ≤ T, (k)

y (k) (0) = y0 , k = 0, 1, ..., m − 1

(5)

is equivalent to the Volterra integral equation [α]−1

y(t) =

y

(k)

k=0

tkn+1 1 (0) + k! Γ(α)

t 0

(t − s)α−1 f (s, y(s))ds

(6)

Set h = (T /N ), tn = nh, n = 0, 1, ...N ∈ Z + . Then (6) can be discretized as follows: [α]−1

yh (tn+1 ) =

k=0

y (k) (0)

tkn+1 hα + f (tn+1 , yhp (tn+1 ) + k! Γ(α + 2) hα aj,n+1 f (tj , yh (tj )) + Γ(α + 2) j=0 n

(7)


236

where

aj,n+1

⎧ ⎪ ⎪ j = 0, nα+1 − (n − α)(n + 1)α , ⎪ ⎪ ⎨ = (n − j + 2)α + 1 + (n − j)α + 1 − 2(n − j + 1)α + 1, 1 ≤ j ≤ n, ⎪ ⎪ ⎪ ⎪ ⎩ 1, j =n+1 [α]−1

yhp (tn+1 )

=

y

(k)

k=0

tk 1 (0) n+1 + bj,n+1 f (tj , yh (tj )), k! Γ(α) j=0 n

hα ((n + 1 − j)α − (n − j)α ). α The error estimate is maxj=0,1,...N |y(tj ) − yh (tj )| = O(hp ), in which p = min(2, 1 + α). bj,n+1 =

2.3 Stability of the Fractional-Order Systems Recent stability analysis of fractional-order systems, shows that fractional-order differential equations are, at least, as stable as their integer order counterpart, because systems with memory are typically more stable than their memory-less counterpart [2]. We have the following result: Theorem 1. [10] The following autonomous system xα = Ax, dtα

x(0) = x0 ,

(8)

with 0 < α < 1, x ∈ Rn and A ∈ Rn×n , is asymptotically stable if and only if |arg(λ)| > απ/2 is satisfied for all eigenvalues (λ) of matrix A. Also, this system is stable if and only if |arg(λ)| ≥ απ/2 is satisfied for all eigenvalues (λ) of matrix A with those critical eigenvalues satisfying |arg(λ)| = απ/2 having geometric multiplicity of one. The geometric multiplicity of an eigenvalue λ of the matrix A is the dimension of the subspace of vectors v for which Av = λv. Now, Consider the following commensurate fractional-order system: Dq x = f (x),

(9)

where 0 < q < 1 and x ∈ Rn . The equilibrium points of this system are calculated by solving equation f (x) = 0. These points are locally asymptotically stable if all the evaluated at the equilibrium points satisfy eigenvalues of the Jacobian matrix A = ∂f ∂x the following condition [2, 10]: |Arg(eig(A))| > q

π 2

(10)

Fig.1 shows stable and unstable regions in this case. In a 3-D nonlinear dynamical system, a saddle point is an equilibrium point on which


237

the equivalent linearized model has at least one eigenvalue in the stable region and one eigenvalue in the unstable region. In the same system, a saddle point is called saddle point of index 1 if one of the eigenvalues is unstable and other eigenvalues are stable. Also, a saddle point of index 2 is a saddle point with one stable eigenvalue and two unstable eigenvalues. In chaotic systems, it is proved that scrolls are generated only around the saddle points of index 2. Moreover, saddle points of index 1 are responsible only for connecting scrolls [18]. Assume that 3-D chaotic system x˙ = f (x) displays one scroll attractor. Hence, this system has a saddle point of index 2 encircled by a one-scroll attractor. Suppose λ = α ± jβ are unstable eigenvalues for this saddle point of index 2. A necessary condition for fractional system Dq x = f (x) to remain chaotic is keeping the eigenvalue λ in the unstable region. This means π |β| |β| 2 −1 tan q > ⇒ q > tan 2 α π α

(11)

jω stable stable unstable q π /2

σ

− q π /2 stable

unstable stable

Fig. 1 Stability region of linear fractional-order system with order q.

3.

Integer-Order Nonlinear Bloch System

The dynamics of an ensemble of spins usually described by the nonlinear Bloch equation is very important for the understanding of the underlying physical process of nuclear magnetic resonance. The basic process can be viewed as the combination of a precession about a magnetic field and of a relaxation process, which gives rise to the damping of the transverse component of the magnetization with a different time constant. The basic model is derived from a magnetization M precessing in the magnetic induction field B0 in the presence of a constant radiofrequency field B1 with intensity B1 = ωγ1 and frequency ωrf . The following modified nonlinear Bloch equation govern the evolution of


238

the magnetization, ⎧ x ⎪ ⎪ x ˙ = δy + γz(xsin(c) − ycos(c)) − ⎪ ⎪ Γ2 ⎪ ⎨ y y˙ = −δx − z + γz(xcos(c) + ysin(c)) − ⎪ Γ2 ⎪ ⎪ ⎪ z − 1 ⎪ ⎩ z˙ = y − γsin(c)(x2 + y 2 ) − Γ1

(12)

where the variables are properly scaled [1]. It is easy to visualize that fixed the point (x0 , y0 , z0 ) of the above system is given as x0 = f (z0 , γ, c, δ, Γ2 ), y0 = f (z0 , γ, c, δ, Γ2 ) where z0 is given by 2sin(c) 1 2γsin(c) Γ1 2 3 2 2 z0 γ z0 − γ + 2δcos(c) + γ z0 + +δ + + 2γδcos(c) + Γ2 Γ22 Γ2 Γ2 1 2 − + δ = 0, Γ22 a cubic equation. In particular for real root one can always get the restriction on the parameters. The jacobian matrix of system (12), evaluated at the equilibrium (x0 , y0 , z0 ) ⎛ ⎞ 1 γ (sin(c)x0 − cos(c)y0 ) ⎟ ⎜γsin(c)z0 − Γ2 δ − γcos(c)z0 ⎜ ⎟ 1 ⎜ ⎟ J = ⎜−δ + γcos(c)z0 γsin(c)z0 − −1 + γ (cos(c)x0 + sin(c)y0 )⎟ ⎜ ⎟ Γ2 ⎝ ⎠ 1 − −2γsin(c)x0 1 − 2γsin(c)y0 Γ1 Previous works shows that the system (12) possess chaotic attractors for two different sets of parameter values, the first set of parameters is: γ = 10, δ = 1.26, c = 0.7764, Γ1 = 0.5, Γ2 = 0.25 and the second set: γ = 35, δ = −1.26, c = 0.173, Γ1 = 5, Γ2 = 2.5 The form of attractors is given in Fig.2.

0.4 0.4

0.2 0.2

y(t)

0

y(t)

−0.2

0 −0.2

−0.4

−0.4

−0.6 1

0.2

1

0.5

x(t)

0.8 0.6

0

0.4 −0.5

0.2

z(t)

x(t)

0 −0.2 −0.4

−0.2

−0.1

0

0.1

0.2

0.3

z(t)

Fig. 2 Chaotic attractors of system (12) for the parameters: (a) γ = 10, δ = 1.26, c = 0.7764, Γ1 = 0.5, Γ2 = 0.25 . (b) γ = 35, δ = −1.26, c = 0.173, Γ1 = 5, Γ2 = 2.5 .


4.

239

Fractional-Order Nonlinear Bloch System

Here we consider the fractional system. The standard derivative is replaced by a fractional derivative as follows ⎧ q x dx ⎪ ⎪ = δy + γz(xsin(c) − ycos(c)) − ⎪ ⎪ Γ2 ⎪ ⎨ dt dq y y (13) = −δx − z + γz(xcos(c) + ysin(c)) − ⎪ dt Γ2 ⎪ q ⎪ ⎪ dz z−1 ⎪ ⎩ = y − γsin(c)(x2 + y 2 ) − dt Γ1 Our study will be for the two sets of parameters cited above, and using the fractional order q as a bifurcation parameter.

4.1 1st set of Parameters System parameters are specified as: γ = 10, δ = 1.26, c = 0.7764, Γ1 = 0.5, Γ2 = 0.25, for these system parameters, nonlinear Bloch system has one equilibrium and his corresponding eigenvalues are: E = (0.13985, 0.06727, 0.94926) : λ1 = −1.8116, λ2,3 = 2.5574 ± 5.5218j. Hence, the fixed point E is a saddle point of index 2. According to (11), for q > 0.72, the fractional order nonlinear Bloch system with this set of parameters has the necessary condition for remaining chaotic. Applying the predictor-corrector scheme described in subsection 2.2 and using phase diagrams, and the largest Lyapunov exponents, we find that chaos indeed exists in the fractional order system (13) with order less than 3. The system is calculated numerically with q ∈ [0.7, 1], and the increment of q equals to 0.001. Bifurcation diagram is shown in Fig.3. With growth of values of parameter q in the system (13), a cascade of period doubling bifurcations of an original cycle is observed. So for the value q = 0.86 a cycle of the period 2 is born, for the value q = 0.93 a cycle of the period 4 is born, for the value q = 0.94 a cycle of the period 8 is born, etc. Some cycles of the Feigenbaum cascade and a singular Feigenbaum attractor for the value of the parameter q = 0.947, as a result of the period doubling bifurcations, are shown in Fig.4. The cascade of period doubling bifurcation is followed by the subharmonic cascade of bifurcations characterized by the birth of limit cycles of any period in compliance with the scenario established by Sharkovskii [15]. So further increase in the value of the parameter q leads to realization of the Sharkovskii complete subharmonic cascade of bifurcations of stable cycles in accordance with the Sharkovskii order: 1 2 22 23 ... 22 .7 22 .5 22 .3 ... 2.7 2.5 2.3 ... 7 5 3.

(14)


240

0.15 0.1 0.05

x

0 −0.05 −0.1 −0.15 −0.2 −0.25 0.7

0.75

0.8

0.85

0.9

q

0.95

1

Fig. 3 Bifurcation diagram with parameter q increasing form 0.7 to 1. q=0.85

0.3

q=0.86

0.4

0.2

q=0.93

0.4

0.2

0.2

0

y(t)

y(t)

y(t)

0.1 0 −0.2

−0.1 −0.2 0.6

−0.4 0.6 0.4

x(t)

0.7 −0.2

0

0.6

z(t)

q=0.94

0

z(t)

0.5

−0.2

x(t)

0.4

0.5

0.6

0.7

0.8

0.9

z(t)

q=0.947

0.4 0.2

y(t)

0.2

y(t)

0.2

0.7 −0.2

0.4

0.4

0.9 0.8

0.2

0.8

0

x(t)

−0.4 0.6 0.4

1 0.9

0.2

0 −0.2

0 −0.2

0 −0.2

−0.4 0.6

−0.4 0.6 0.4

1 0.8

0.2

0.6

0

x(t)

0.4

1

−0.2

0.2

z(t)

0.8

0.2

0.6

0

0.4

x(t)

0.4 −0.2

0.2

z(t)

Fig. 4 Projections of original cycle, period two cycle, period four cycle, period eight cycle and Feigenbaum attractor in the fractional order nonlinear Bloch system.

The ordering n k in (14) means that the existence of a cycle of period k implies the existence of all cycles of period n. So, if the system (13) has a stable limit cycle of period three then it has also all unstable cycles of all periods in accordance with the Sharkovskii order (14). The Sharkovskii complete subharmonic cascade of bifurcations of stable cycles is proved by existence of a limit cycle of period 6 for the parameter value q = 0.948, a limit cycle of period 5 for q = 0.955 and a limit cycle of period 3 lying in the interval [0.965, 0.979] which with further increase of the parameter q goes through a cascade of period doubling bifurcations. Thus, for q = 0.98 we observe a doubled cycle of period 3. The subharmonic cascade also terminates with the formation of an irregular attractor. Some cycles of this cascade and a subharmonic singular attractor are shown in Fig.5. To demonstrate the chaotic dynamics, the largest Lyapunov exponent should be the first thing to be considered, because any system containing at least one positive Lyapunov exponent is defined to be chaotic [16]. Measuring the largest Lyapunov exponent (LLE) is always an important problem whatever in a fractional order system or in an integral-order


q=0.948

0.4

q=0.955

0.4

0.2

241

q=0.97

0.4 0.2

0.2 0

−0.2

−0.2

−0.4 0.6

−0.4 0.6 0.4

−0.6 0.6 0.4

1

0.2

x(t)

z(t)

0.2

q=0.98

0.4

0.4

z(t)

0.2

z(t)

q=0.99

0.2 0

y(t)

0

y(t)

0.6

0 −0.2

0.4

0.2

1

x(t)

0.4 −0.2

0.8

0.2

0.6

0

0.4 −0.2

0.4

1 0.8

0.2

0.6

0

−0.2 −0.4

0.8

0.2

x(t)

y(t)

y(t)

y(t)

0 0

−0.2 −0.4

−0.2 −0.4

−0.6 0.6

−0.6 0.6 0.4

0.4

1 0.8

0.2 0

x(t)

0.4 −0.2

0.2

1 0.8

0.2

0.6

0.6

0

z(t)

x(t)

0.4 −0.2

0.2

z(t)

Fig. 5 Projections of period six cycle, period five cycle, period three cycle, doubled period three cycle and more complex subharmonic singular attractor in the fractional order nonlinear Bloch system.

system. Wolf and Jacobian algorithms are the most popular algorithm in calculating the largest Lyapunov exponent of integer-order system. However, Jacobian algorithm is not applicable for calculating LLE of a fractional order system, since the Jacobian matrix of fractional order system is hard to be obtained. As to Wolf algorithm [21] which is relatively difficult to implement. Therefore, in this paper, the small data sets algorithm developed by Michael T. Rosenstein etc [14] is chosen to calculating LLE of the Fractional order nonlinear Bloch system, the diagram is plotted in Fig.6. 0.03

Maximal Lyapunov Exponents

0.025 0.02 0.015 0.01 0.005 0 −0.005 −0.01 −0.015 −0.02

0.7

0.75

0.8

0.85

q

0.9

0.95

1

Fig. 6 Maximal Lyapunov Exponents versus q form 0.7 to 1 step with 0.01.

4.2 2nd set of Parameters System parameters are specified as: γ = 35.0, δ = −1.26, c = 0.173, Γ1 = 5.0, Γ2 = 2.5, for these system parameters, nonlinear Bloch system has one equilibrium and his corresponding eigenvalues are:


242

E = (0.02730, 0.00429, 0.99847) : λ1 = −0.19971, λ2,3 = 5.6155 ± 35.685j. Hence, the fixed point E is a saddle point of index 2. The necessary condition to remain chaotic for the fractional order nonlinear Bloch system with this set of parameters is q > 0.90. At q ≈ 0.90 a Hopf bifurcation gives birth to an orbitally stable limit cycle. For certain parameter values, this limit cycle co-exists with another limit cycle with different period, each with its basin of attraction. Fig.7 shows the bifurcation diagram against the parameter q and the coexisting limit cycles for different initial condition

0.05 0

q=0.975

0.4

−0.05 −0.1

0.2

x

y(t)

−0.15 −0.2

0

−0.2

−0.25

−0.4 0.3

−0.3

−0.4 −0.2

0.2

−0.35

0

0.1

−0.4 0.88

0.9

0.92

0.94

0.96

q

0.98

0.2

0

z(t)

1

x(t)

0.4

−0.1

Fig. 7 (a) bifurcation diagram vs q, (b)coexisting limit cycles. Initial conditions: (0.1, 0.1, 0.1) for the thick line and (0.01, 0.01, 0.01) for the thin line.

Fixing the initial conditions at (0.1, 0.1, 0.1) and increasing the parameter q, the initial period one limit cycle will disappear suddenly and is replaced by a period four limit cycle at q = 0.99, the two limit cycles goes through a cascade of period doubling bifurcations which terminates with the formation of irregular attractors as shown in Fig.8.

0.2

0

−0.1

−0.2

−0.2 0.2

−0.4 0.4 0.1

−0.4 0.4

0.2

0.6

x(t)

−0.2

x(t)

z(t)

q=0.9923

0.4

−0.1

0.3

−0.1

q=0.995

0.4

q=0.994

0.2

0 −0.4

z(t)

0.3

0.1

−0.2

0 −0.4

0.2

0

0.1

−0.2

0 −0.2

0.2

0.3 0.2

0

0.2

−0.1

0

−0.2

0.4

0

q=0.9915

0.2

y(t)

0

q=0.99

0.2

y(t)

y(t)

0.4

0.4

q=0.98

0.1

0.2

0.2

0

y(t)

y(t)

y(t)

0.1 0

0

−0.1 −0.2

−0.2 −0.2

−0.4 0.4

−0.3 0.2 0

x(t)

−0.2 −0.4

−0.2

−0.1

0

0.1

z(t)

0.2

0.3

0.2

−0.4 0.4 0.1

x(t)

0

−0.1

−0.2 −0.05

0

0.05

0.1

z(t)

0.15

0.2

0.2 0

x(t)

−0.2 −0.4

−0.2

−0.1

0

0.1

0.2

0.3

z(t)

Fig. 8 Projections of the period doubling bifurcations and irregular attractors.

The largest Lyapunov exponents are calculated numerically with q ∈ [0.85, 1] for an increment of 0.01 which are plotted in Fig.9.


243

0.04

Maximal Lyapunov Exponents

0.03

0.02

0.01

0

−0.01

−0.02

−0.03

−0.04 0.84

0.86

0.88

0.9

0.92

q

0.94

0.96

0.98

1

Fig. 9 Maximal Lyapunov Exponents versus q form 0.85 to 1 step with 0.01 .

Conclusion In this paper, we have studied the dynamics of the fractional-order nonlinear Bloch system by means of the bifurcation diagram and largest Lyapunov exponents. A numerical algorithm is used to analyze the fractional-order system. In this study the fractional order is the explore direction. Through these, Period-doubling and subharmonic cascade routes to chaos were found in the fractional-order nonlinear Bloch equations. Especially, a period-3 window is presented in bifurcation diagram. Moreover, coexisting limit cycles were also found. We calculate the largest Lyapunov exponent by using the small data sets instead of wolf algorithm, which was used frequently in preview research. The results show the validity of the algorithm.

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