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Chaos in fractional order financial delay system
Computers and Mathematics with Applications (
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Chaos in fractional order financial delay system Sachin Bhalekar a,∗ , Varsha Daftardar-Gejji b a
Department of Mathematics, Shivaji University, Kolhapur - 416004, India
b
Department of Mathematics, Savitribai Phule Pune University, Pune - 411007, India
article
info
Article history: Available online xxxx Keywords: LIOUVILLE–CAPUTO fractional derivative Financial system Delay differential equations Predictor–corrector algorithm Chaos
abstract Chaotic dynamics of the fractional order financial system involving time delay is studied. The numerical simulations are done using the new predictor–corrector method proposed by Daftardar-Gejji et al. and by modified fractional Adams method (FAM). The results obtained using new predictor–corrector method are matching with FAM. It is observed that the new predictor–corrector method is more time efficient than the FAM. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Fractional calculus deals with derivatives and integrations of fractional order [1–4]. This subject has history of 300 years, however there is growing interest in its applications in recent past. Phenomenon such as sub/super-diffusion, rheological properties of rocks, electrocardiogram, theory of fractals are appropriately modelled using fractional derivatives as they can formulate ‘‘memory effects’’. Srivastava [5–7] used the theory of fractional derivatives to study different properties of special functions. Recently methods of fractional calculus have been employed to study complex dynamics of financial and economic systems. The financial variables such as stock market prices, interest rates, foreign exchange rates have long memory and hence financial systems are studied using fractional nonlinear models [8]. In recent times it is believed that the ideas from nonlinear dynamics might prove to be useful in the study of economics as nonlinear evolutions take place in many aspects of financial markets. In particular there exists considerable empirical evidence for chaos in financial markets [9]. Hence scientists have investigating financial models which simultaneously possess memory and chaos. In the present paper the delayed version of these models is considered and the time evolution is observed. Insertion of delay in the model improves its dynamics and admits a better description of the real life phenomena. DDEs are proved useful in control systems [10], chaos [11–14], lasers, traffic models [15], NMR [16,17], population dynamics [18], chemical kinetics [19] etc. Some phenomena from economics cannot be described without lags ([20–22] and references cited therein). Stability of fractional order delayed nonlinear systems is discussed in [23,24]. As the initial condition of DDE one has to provide history of the system over the delay interval [−τ , 0]. This is the motivation behind the infinite dimensional nature of the delay systems. Because of infinite dimensionality the DDEs are difficult to analyse analytically [25] and hence the numerical solutions play an important role. Bhalekar and Daftardar-Gejji [26] have extended the fractional Adams–Bashforth–Moulton predictor–corrector scheme [27,28] to solve fractional order DDEs. Further, Daftardar-Gejji et al. proposed a new predictor–corrector method [29,30] to solve such nonlinear equations.
∗
Corresponding author. Fax: +91 231 2691533. E-mail addresses: [email protected], [email protected] (S. Bhalekar), [email protected], [email protected] (V. Daftardar-Gejji). http://dx.doi.org/10.1016/j.camwa.2016.03.009 0898-1221/© 2016 Elsevier Ltd. All rights reserved.
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S. Bhalekar, V. Daftardar-Gejji / Computers and Mathematics with Applications (
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In this paper, the fractional order financial system [31] is generalized to involve delay in each term. The stability conditions and chaos in the proposed system are discussed. The system is simulated using both the methods and the results are compared. 2. Preliminaries 2.1. Fractional calculus We present in this section some basic definitions and properties [1–3,32,33]. Let J = [0, T ], T > 0, C (J , R) denotes the space of all continuous functions from J into R and C n (J , R) denotes the space of all real valued functions defined on J which have continuous nth order derivative. Definition 2.1. The fractional integral of order α ≥ 0 of the function f ∈ C (J , R) is defined as Itα f (t ) =
1
Γ (α)
t
0
f (s) ds, (t − s)1−α
0 < t < T.
Definition 2.2. Liouville–Caputo fractional derivative of order α > 0 of the function f ∈ C n (J , R) is defined as Dαt f (t ) = I n−α Dn f (t ) =
t
1
Γ (n − α)
0
f (n) (s)
(t − s)α−n+1
ds,
n − 1 < α ≤ n, n ∈ N .
Note that m−1
µ µ
It Dt f (t ) = f (t ) −
dk f k=0
µ It t ν =
dt k
tk
(0) , k!
(2.1)
Γ (ν + 1) t µ+ν . Γ (µ + ν + 1)
(2.2)
2.2. Predictor–corrector scheme for fractional delay differential equations In this section, we present the modified Adams–Bashforth–Moulton predictor–corrector scheme described in [26] to solve delay differential equations of fractional order (FDDE). Consider the following FDDE Dαt y(t ) = f (t , y(t ), y(t − τ )) , y(t ) = g (t ),
t ∈ [0, T ] , 0 < α ≤ 1
(2.3)
t ∈ [−τ , 0] .
(2.4)
Consider a uniform grid {tn = nh : n = −k, −k + 1, . . . , −1, 0, 1, . . . , N } where k and N are integers such that h = T /N and h = τ /k. The corrector formula is given by yh (tn+1 ) = g (0) +
hα
Γ (α + 2)
f (tn+1 , yh (tn+1 ) , yh (tn+1−k )) +
hα
n
Γ (α + 2)
j =0
aj,n+1 f tj , yh tj , yh tj−k
,
(2.5)
where aj,n+1 are given by
aj,n+1
nα+1 − (n − α)(n + 1)α , = (n − j + 2)α+1 + (n − j)α+1 − 2(n − j + 1)α+1 , 1,
if j = 0, if 1 ≤ j ≤ n, if j = n + 1.
(2.6)
The unknown term yh (tn+1 ) on the right hand side is replaced by an approximation called predictor which is described as yPh (tn+1 ) = g (0) +
1
n
Γ (α)
j=0
bj,n+1 f tj , yh tj , yh tj−k
,
(2.7)
where bj,n+1 =
hα
α
((n + 1 − j)α − (n − j)α ) .
(2.8)
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Fig. 1a. α = 1, τ = 0.03.
2.3. New predictor–corrector method Daftardar-Gejji et al. used an iterative method [34] to improve the numerical method discussed in preceding section. The new predictor–corrector formula [29,30] is as below: p
yn+1 =
⌈α⌉− 1
φk
k=0
p
+
hα
n
Γ (α + 2)
j=0
aj,n+1 f1 (tj , yj , y(tj − τ )),
p
Γ (α + 2)
ycn+1 = yn+1 + p
k!
hα
p
zn+1 =
tnk+1
f1 (tn+1 , yn+1 , y(tn+1 − τ )), hα
Γ (α + 2)
p
(2.9)
(2.10)
p
f1 (tn+1 , yn+1 + zn+1 , y(tn+1 − τ )).
(2.11)
p
Here yn+1 and zn+1 are called as predictors and ycn+1 is the corrector, and yj denotes the approximate value of the solution. 3. Fractional order financial delay system In [8] Chen proposed the financial system to fractional order Dαt x(t ) = z (t ) + (y(t ) − a) x(t ), α
(3.1)
Dt y(t ) = 1 − by(t ) − x(t ) , 2
(3.2)
α
Dt z (t ) = −x(t ) − cz (t ),
(3.3)
where a is the saving amount, b is the cost per investment and c is the elasticity of demand of commercial markets. The system (3.1)–(3.3) with delay in some state variables is discussed in [31]. In the present paper we consider a generalization of the system (3.1)–(3.3) to include time-delay in each state variable, where a = 3.0 is the saving amount, b = 0.1 is the cost per investment and c = 1.0 is the elasticity of demand of commercial markets: Dαt x(t ) = z (t − τ ) + (y (t − τ ) − a) x (t − τ ) , α
Dt y(t ) = 1 − by (t − τ ) − x (t − τ ) , 2
(3.5)
α
Dt z (t ) = −x (t − τ ) − cz (t − τ ) , x(t ) = 2.0,
y(t ) = 3.0,
z (t ) = 2.0,
(3.4)
(3.6) for t ∈ [−τ , 0] .
(3.7)
4. Numerical simulations We take a = 3.0, b = 0.1 and c = 1.0 and consider various values of α and τ . We use the new predictor–corrector method and fractional Adams method (FAM) for the simulations of 0 < α < 1.
• For α = 1, the system (3.1)–(3.3) (i.e. τ = 0) shows chaotic behaviour. It is observed that the system (3.4)–(3.7) remains chaotic for 0 ≤ τ < 0.05. Fig. 1a shows yz-phase portrait for the system with τ = 0.03. Two cycle is observed for τ = 0.05 (cf. Fig. 1b). The system shows periodic behaviour for τ ≥ 0.05. Cycles are shown in (Figs. 1c, 1d) for τ = 0.06. We have used Software Mathematica command NDSolve for these simulations.
• For 0.85 ≤ α ≤ 1, τ = 0 the system (3.4)–(3.7) shows chaotic behaviour [8]. We fix α = 0.96 and draw solutions/phase portraits for different values of τ . It is observed that the system remains chaotic for 0 ≤ τ < 0.07 and tend to periodic
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Fig. 1b. α = 1, τ = 0.05.
Fig. 1c. α = 1, τ = 0.06.
Fig. 1d. α = 1, τ = 0.06.
behaviour for the higher values of τ . Chaotic phase portraits obtained by using new predictor–corrector method and FAM for the case τ = 0.06 are shown in Figs. 2a and 2b respectively. The periodic oscillations at τ = 0.07 are shown in Fig. 2c (new predictor–corrector method) and 2d (FAM). • For α = 0.93 and for 0 < τ < 0.11 the system shows chaotic behaviour. We plot chaotic phase portraits for τ = 0.10 using new predictor–corrector method and FAM in Figs. 3a and 3b respectively. The periodic limit cycles for τ = 0.11 are shown in Figs. 3c and 3d by using these methods. • The bifurcation value of τ at which the chaotic trajectories turns to periodic for α = 0.88 is 0.17. The change is shown in Figs. (4a)–(4d).
S. Bhalekar, V. Daftardar-Gejji / Computers and Mathematics with Applications (
Fig. 2a. α = 0.96, τ = 0.06. New method, CPU time = 3467.14062.
Fig. 2b. α = 0.96, τ = 0.06. FAM, CPU time = 5720.4375.
Fig. 2c. α = 0.96, τ = 0.07. New method, CPU time = 3464.078125.
Fig. 2d. α = 0.96, τ = 0.07. FAM, CPU time = 5837.1875.
Fig. 3a. α = 0.93, τ = 0.10. New method, CPU time = 3773.75.
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Fig. 3b. α = 0.93, τ = 0.10. FAM, CPU time = 6311.703125.
Fig. 3c. α = 0.93, τ = 0.11. New method, CPU time = 3765.89062.
Fig. 3d. α = 0.93, τ = 0.11. FAM, CPU time = 6636.203125.
Fig. 4a. α = 0.88, τ = 0.16. New method, CPU time = 3797.296875.
Fig. 4b. α = 0.88, τ = 0.16. FAM, CPU time = 6322.734375.
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Fig. 4c. α = 0.88, τ = 0.17. New method, CPU time = 3965.921875.
Fig. 4d. α = 0.88, τ = 0.17. FAM, CPU time = 6634.96875.
5. Conclusions Chaotic dynamics of the fractional order financial delay system is studied. Chaotic system switches to regular behaviour for suitable values of delay. For integer order derivative case system is chaotic for 0 ≤ τ < 0.05. Periodic behaviour is observed for τ ≥ 0.05. For fractional order case α = 0.96 the chaotic system tend to regular behaviour for τ > 0.07. For α = 0.93 and α = 0.88 system attends order at τ = 0.11 and τ = 0.17 respectively. It is observed that the CPU time used by new predictor–corrector method is very less as compared with the fractional Adams method. Acknowledgments S. Bhalekar acknowledges the National Board for Higher Mathematics, Mumbai, India for the Research Grant (Ref. 2/48(6)/2013/NBHM(R.P.)/R&DII/689). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993. L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci. 2003 (54) (2003) 3413–3442. H.M. Srivastava, A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients, J. Math. Anal. Appl. 171 (1) (1992) 1–13. H.M. Srivastava, Applications of fractional calculus to parabolic starlike and uniformly convex functions, Comput. Math. Appl. 39 (3–4) (2000) 57–69. H.M. Srivastava, Some fractional differintegral equations, J. Math. Anal. Appl. 106 (2) (1985) 360–366. W.C. Chen, Nonlinear dynamics and chaos in a fractional-order financial system, Chaos Solitons Fractals 36 (2008) 1305–1314. B. LeBaron, Chaos and nonlinear forecastability in economics and finance, Philos. Trans. Roy. Soc. London Ser. A 348 (1994) 397–404. E. Fridman, L. Fridman, E. Shustin, Steady modes in relay control systems with time delay and periodic disturbances, J. Dyn. Syst. Meas. Contr. 122 (2000) 732–737. S. Bhalekar, V. Daftardar-Gejji, Fractional ordered Liu system with time-delay, Commun. Nonlinear Sci. Numer. Simul. 15 (8) (2010) 2178–2191. V. Daftardar-Gejji, S. Bhalekar, P. Gade, Dynamics of fractional ordered Chen system with delay, Pramana-J. Phys. 79 (1) (2012) 61–69. S. Bhalekar, Dynamical analysis of fractional order Ucar prototype delayed system, Signal Image Video Process. 6 (3) (2012) 513–519. S. Bhalekar, On the Ucar prototype model with incommensurate delays, Signal Image Video Process. 8 (4) (2014) 635–639. L.C. Davis, Modification of the optimal velocity traffic model to include delay due to driver reaction time, Physica A 319 (2002) 557–567. S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, R. Magin, Fractional Bloch equation with delay, Comput. Math. Appl. 61 (5) (2011) 1355–1365. S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, R. Magin, Generalized fractional order Bloch equation with extended delay, Internat. J. Bifur. Chaos 22 (4) (2012) 1250071. Y. Kuang, Delay Differential Equations with Applications in Population Biology, Academic Press, Boston, San Diego, New York, 1993. I. Epstein, Y. Luo, Differential delay equations in chemical kinetics. Nonlinear models: the cross-shaped phase diagram and the Oregonator, J. Chem. Phys. 95 (1991) 244–254. P. Asea, P. Zak, Time–to–build and cycles, J. Econom. Dynam. Control 23 (1999) 1155–1175. L. De Cesare, M. Sportelli, A dynamic IS–LM model with delayed taxation revenues, Chaos Solitons Fractals 25 (2005) 233–244. L. Fanti, P. Manfredi, Chaotic business cycles and fiscal policy: An IS–LM model with distributed tax collection lags, Chaos Solitons Fractals 32 (2007) 736–744. S. Bhalekar, A necessary condition for the existence of chaos in fractional order delay differential equations, Int. J. Math. Sci. 7 (3) (2013) 342–346. S. Bhalekar, Stability analysis of a class of fractional delay differential equations, Pramana-J. Phys. 81 (2) (2013) 215–224. J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Vol. 99, Springer-Verlag, Berlin, 1993.
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[26] S. Bhalekar, V. Daftardar-Gejji, A predictor–corrector scheme for solving nonlinear delay differential equations of fractional order, J. Fract. Calc. Appl. 1 (5) (2011) 1–8. [27] K. Diethelm, N.J. Ford, A.D. Freed, A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam. 29 (2002) 3–22. [28] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal. 5 (1997) 1–6. [29] V. Daftardar-Gejji, Y. Sukale, S. Bhalekar, A new predictor–corrector method for fractional differential equations, Appl. Math. Comput. 244 (2014) 158–182. [30] V. Daftardar-Gejji, Y. Sukale, S. Bhalekar, Solving fractional delay differential equations: A new approach, Fract. Calc. Appl. Anal. 18 (2) (2015) 400–418. [31] W. Zhen, H. Xia, S. Guodong, Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay, Comput. Math. Appl. 62 (2011) 1531–1539. [32] Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnam. 24 (1999) 207–233. [33] R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific, 2014. [34] V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2006) 753–763.
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Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Chaos in fractional order financial delay system Sachin Bhalekar a,∗ , Varsha Daftardar-Gejji b a
Department of Mathematics, Shivaji University, Kolhapur - 416004, India
b
Department of Mathematics, Savitribai Phule Pune University, Pune - 411007, India
article
info
Article history: Available online xxxx Keywords: LIOUVILLE–CAPUTO fractional derivative Financial system Delay differential equations Predictor–corrector algorithm Chaos
abstract Chaotic dynamics of the fractional order financial system involving time delay is studied. The numerical simulations are done using the new predictor–corrector method proposed by Daftardar-Gejji et al. and by modified fractional Adams method (FAM). The results obtained using new predictor–corrector method are matching with FAM. It is observed that the new predictor–corrector method is more time efficient than the FAM. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Fractional calculus deals with derivatives and integrations of fractional order [1–4]. This subject has history of 300 years, however there is growing interest in its applications in recent past. Phenomenon such as sub/super-diffusion, rheological properties of rocks, electrocardiogram, theory of fractals are appropriately modelled using fractional derivatives as they can formulate ‘‘memory effects’’. Srivastava [5–7] used the theory of fractional derivatives to study different properties of special functions. Recently methods of fractional calculus have been employed to study complex dynamics of financial and economic systems. The financial variables such as stock market prices, interest rates, foreign exchange rates have long memory and hence financial systems are studied using fractional nonlinear models [8]. In recent times it is believed that the ideas from nonlinear dynamics might prove to be useful in the study of economics as nonlinear evolutions take place in many aspects of financial markets. In particular there exists considerable empirical evidence for chaos in financial markets [9]. Hence scientists have investigating financial models which simultaneously possess memory and chaos. In the present paper the delayed version of these models is considered and the time evolution is observed. Insertion of delay in the model improves its dynamics and admits a better description of the real life phenomena. DDEs are proved useful in control systems [10], chaos [11–14], lasers, traffic models [15], NMR [16,17], population dynamics [18], chemical kinetics [19] etc. Some phenomena from economics cannot be described without lags ([20–22] and references cited therein). Stability of fractional order delayed nonlinear systems is discussed in [23,24]. As the initial condition of DDE one has to provide history of the system over the delay interval [−τ , 0]. This is the motivation behind the infinite dimensional nature of the delay systems. Because of infinite dimensionality the DDEs are difficult to analyse analytically [25] and hence the numerical solutions play an important role. Bhalekar and Daftardar-Gejji [26] have extended the fractional Adams–Bashforth–Moulton predictor–corrector scheme [27,28] to solve fractional order DDEs. Further, Daftardar-Gejji et al. proposed a new predictor–corrector method [29,30] to solve such nonlinear equations.
∗
Corresponding author. Fax: +91 231 2691533. E-mail addresses: [email protected], [email protected] (S. Bhalekar), [email protected], [email protected] (V. Daftardar-Gejji). http://dx.doi.org/10.1016/j.camwa.2016.03.009 0898-1221/© 2016 Elsevier Ltd. All rights reserved.
2
S. Bhalekar, V. Daftardar-Gejji / Computers and Mathematics with Applications (
)
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In this paper, the fractional order financial system [31] is generalized to involve delay in each term. The stability conditions and chaos in the proposed system are discussed. The system is simulated using both the methods and the results are compared. 2. Preliminaries 2.1. Fractional calculus We present in this section some basic definitions and properties [1–3,32,33]. Let J = [0, T ], T > 0, C (J , R) denotes the space of all continuous functions from J into R and C n (J , R) denotes the space of all real valued functions defined on J which have continuous nth order derivative. Definition 2.1. The fractional integral of order α ≥ 0 of the function f ∈ C (J , R) is defined as Itα f (t ) =
1
Γ (α)
t
0
f (s) ds, (t − s)1−α
0 < t < T.
Definition 2.2. Liouville–Caputo fractional derivative of order α > 0 of the function f ∈ C n (J , R) is defined as Dαt f (t ) = I n−α Dn f (t ) =
t
1
Γ (n − α)
0
f (n) (s)
(t − s)α−n+1
ds,
n − 1 < α ≤ n, n ∈ N .
Note that m−1
µ µ
It Dt f (t ) = f (t ) −
dk f k=0
µ It t ν =
dt k
tk
(0) , k!
(2.1)
Γ (ν + 1) t µ+ν . Γ (µ + ν + 1)
(2.2)
2.2. Predictor–corrector scheme for fractional delay differential equations In this section, we present the modified Adams–Bashforth–Moulton predictor–corrector scheme described in [26] to solve delay differential equations of fractional order (FDDE). Consider the following FDDE Dαt y(t ) = f (t , y(t ), y(t − τ )) , y(t ) = g (t ),
t ∈ [0, T ] , 0 < α ≤ 1
(2.3)
t ∈ [−τ , 0] .
(2.4)
Consider a uniform grid {tn = nh : n = −k, −k + 1, . . . , −1, 0, 1, . . . , N } where k and N are integers such that h = T /N and h = τ /k. The corrector formula is given by yh (tn+1 ) = g (0) +
hα
Γ (α + 2)
f (tn+1 , yh (tn+1 ) , yh (tn+1−k )) +
hα
n
Γ (α + 2)
j =0
aj,n+1 f tj , yh tj , yh tj−k
,
(2.5)
where aj,n+1 are given by
aj,n+1
nα+1 − (n − α)(n + 1)α , = (n − j + 2)α+1 + (n − j)α+1 − 2(n − j + 1)α+1 , 1,
if j = 0, if 1 ≤ j ≤ n, if j = n + 1.
(2.6)
The unknown term yh (tn+1 ) on the right hand side is replaced by an approximation called predictor which is described as yPh (tn+1 ) = g (0) +
1
n
Γ (α)
j=0
bj,n+1 f tj , yh tj , yh tj−k
,
(2.7)
where bj,n+1 =
hα
α
((n + 1 − j)α − (n − j)α ) .
(2.8)
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Fig. 1a. α = 1, τ = 0.03.
2.3. New predictor–corrector method Daftardar-Gejji et al. used an iterative method [34] to improve the numerical method discussed in preceding section. The new predictor–corrector formula [29,30] is as below: p
yn+1 =
⌈α⌉− 1
φk
k=0
p
+
hα
n
Γ (α + 2)
j=0
aj,n+1 f1 (tj , yj , y(tj − τ )),
p
Γ (α + 2)
ycn+1 = yn+1 + p
k!
hα
p
zn+1 =
tnk+1
f1 (tn+1 , yn+1 , y(tn+1 − τ )), hα
Γ (α + 2)
p
(2.9)
(2.10)
p
f1 (tn+1 , yn+1 + zn+1 , y(tn+1 − τ )).
(2.11)
p
Here yn+1 and zn+1 are called as predictors and ycn+1 is the corrector, and yj denotes the approximate value of the solution. 3. Fractional order financial delay system In [8] Chen proposed the financial system to fractional order Dαt x(t ) = z (t ) + (y(t ) − a) x(t ), α
(3.1)
Dt y(t ) = 1 − by(t ) − x(t ) , 2
(3.2)
α
Dt z (t ) = −x(t ) − cz (t ),
(3.3)
where a is the saving amount, b is the cost per investment and c is the elasticity of demand of commercial markets. The system (3.1)–(3.3) with delay in some state variables is discussed in [31]. In the present paper we consider a generalization of the system (3.1)–(3.3) to include time-delay in each state variable, where a = 3.0 is the saving amount, b = 0.1 is the cost per investment and c = 1.0 is the elasticity of demand of commercial markets: Dαt x(t ) = z (t − τ ) + (y (t − τ ) − a) x (t − τ ) , α
Dt y(t ) = 1 − by (t − τ ) − x (t − τ ) , 2
(3.5)
α
Dt z (t ) = −x (t − τ ) − cz (t − τ ) , x(t ) = 2.0,
y(t ) = 3.0,
z (t ) = 2.0,
(3.4)
(3.6) for t ∈ [−τ , 0] .
(3.7)
4. Numerical simulations We take a = 3.0, b = 0.1 and c = 1.0 and consider various values of α and τ . We use the new predictor–corrector method and fractional Adams method (FAM) for the simulations of 0 < α < 1.
• For α = 1, the system (3.1)–(3.3) (i.e. τ = 0) shows chaotic behaviour. It is observed that the system (3.4)–(3.7) remains chaotic for 0 ≤ τ < 0.05. Fig. 1a shows yz-phase portrait for the system with τ = 0.03. Two cycle is observed for τ = 0.05 (cf. Fig. 1b). The system shows periodic behaviour for τ ≥ 0.05. Cycles are shown in (Figs. 1c, 1d) for τ = 0.06. We have used Software Mathematica command NDSolve for these simulations.
• For 0.85 ≤ α ≤ 1, τ = 0 the system (3.4)–(3.7) shows chaotic behaviour [8]. We fix α = 0.96 and draw solutions/phase portraits for different values of τ . It is observed that the system remains chaotic for 0 ≤ τ < 0.07 and tend to periodic
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Fig. 1b. α = 1, τ = 0.05.
Fig. 1c. α = 1, τ = 0.06.
Fig. 1d. α = 1, τ = 0.06.
behaviour for the higher values of τ . Chaotic phase portraits obtained by using new predictor–corrector method and FAM for the case τ = 0.06 are shown in Figs. 2a and 2b respectively. The periodic oscillations at τ = 0.07 are shown in Fig. 2c (new predictor–corrector method) and 2d (FAM). • For α = 0.93 and for 0 < τ < 0.11 the system shows chaotic behaviour. We plot chaotic phase portraits for τ = 0.10 using new predictor–corrector method and FAM in Figs. 3a and 3b respectively. The periodic limit cycles for τ = 0.11 are shown in Figs. 3c and 3d by using these methods. • The bifurcation value of τ at which the chaotic trajectories turns to periodic for α = 0.88 is 0.17. The change is shown in Figs. (4a)–(4d).
S. Bhalekar, V. Daftardar-Gejji / Computers and Mathematics with Applications (
Fig. 2a. α = 0.96, τ = 0.06. New method, CPU time = 3467.14062.
Fig. 2b. α = 0.96, τ = 0.06. FAM, CPU time = 5720.4375.
Fig. 2c. α = 0.96, τ = 0.07. New method, CPU time = 3464.078125.
Fig. 2d. α = 0.96, τ = 0.07. FAM, CPU time = 5837.1875.
Fig. 3a. α = 0.93, τ = 0.10. New method, CPU time = 3773.75.
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Fig. 3b. α = 0.93, τ = 0.10. FAM, CPU time = 6311.703125.
Fig. 3c. α = 0.93, τ = 0.11. New method, CPU time = 3765.89062.
Fig. 3d. α = 0.93, τ = 0.11. FAM, CPU time = 6636.203125.
Fig. 4a. α = 0.88, τ = 0.16. New method, CPU time = 3797.296875.
Fig. 4b. α = 0.88, τ = 0.16. FAM, CPU time = 6322.734375.
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Fig. 4c. α = 0.88, τ = 0.17. New method, CPU time = 3965.921875.
Fig. 4d. α = 0.88, τ = 0.17. FAM, CPU time = 6634.96875.
5. Conclusions Chaotic dynamics of the fractional order financial delay system is studied. Chaotic system switches to regular behaviour for suitable values of delay. For integer order derivative case system is chaotic for 0 ≤ τ < 0.05. Periodic behaviour is observed for τ ≥ 0.05. For fractional order case α = 0.96 the chaotic system tend to regular behaviour for τ > 0.07. For α = 0.93 and α = 0.88 system attends order at τ = 0.11 and τ = 0.17 respectively. It is observed that the CPU time used by new predictor–corrector method is very less as compared with the fractional Adams method. Acknowledgments S. Bhalekar acknowledges the National Board for Higher Mathematics, Mumbai, India for the Research Grant (Ref. 2/48(6)/2013/NBHM(R.P.)/R&DII/689). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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