Skip to main content

A new chaotic behavior of a general model of the Henon map

Abstract

In this paper we are concerned with a general form of the Henon map as a retarded functional equation. The existence of a unique solution is proved. The continuous dependence of the solution and the local stability of fixed points are investigated. Chaos, bifurcation and chaotic attractor of the resulting system are discussed. In addition, we compare our results with the discrete dynamical system of the Henon map.

1 Introduction

Discontinuous (sectionally continuous) dynamical systems have been defined as a problem of retarded functional equation and studied in [110]. The generalized time-delayed Henon map was introduced in [11, 12]. In this work we study the discontinuous (sectionally continuous) dynamical system of the Henon map as a problem of retarded functional equation with two different delays

x(t)=1+βx(t r 1 )α x 2 (t r 2 ), r 1 , r 2 >0,t(0,T],
(1.1)

with

x(t)= x 0 ,t0,
(1.2)

where α>0 and |β|<1.

The existence of a unique continuous dependence solution is proved. The local stability of fixed points is studied. The chaos, bifurcation and chaotic attractor are discussed. Comparison with the corresponding discrete dynamical system of the Henon map

x n + 1 =1+β x n 1 α x n 2 ,n=1,2,3,4,,
(1.3)

is given.

Let f:[0,T]× R n R and r 1 , r 2 ,, r k R + .

Consider the problem of retarded functional equation

x(t)=f ( x ( t r 1 ) , x ( t r 2 ) , , x ( t r k ) ) ,t(0,T],
(1.4)

with the initial condition

x(τ)=ϕ(τ),τ0.
(1.5)

If T is a positive integer, r k =k, ϕ(0)= x 0 , and t=n=1,2,3, , then problem (1.4)-(1.5) will be the discrete dynamical system

x n =f( x n 1 , x n 2 ,, x n k ),n=1,2,3,,T,
(1.6)
x(0)= x 0 .
(1.7)

This shows that discrete dynamical system (1.6)-(1.7) is a special case of the problem of retarded functional equation (1.4)-(1.5).

Consider also the singularly perturbed differential difference equation [13]

ϵ x (t)=x(t)+f ( x ( t 1 ) ) ,
(1.8)

and the singularly perturbed delay differential equation [14]

ϵ x (t)= a 0 x(t)+ j = 1 k x(t m j ),
(1.9)

m j 0, m j Z, j=1,,k.

The limiting cases as ϵ0 of (1.8) and (1.7) are special cases of retarded functional equation (1.4)-(1.5).

Let t(0,r], then tr(r,0], the solution of (1.4)-(1.5) (by the method of steps as in [1316]) is given by

x(t)= x 1 (t)=f ( ϕ ( 0 ) ) ,t(0,r].

For t(r,2r], then tr(0,r], the solution of (1.4)-(1.5) is given by

x(t)= x 2 (t)=f ( x 1 ( t ) ) =f ( f ( ϕ ( 0 ) ) ) = f 2 ( ϕ ( 0 ) ) ,t(r,2r].

Repeating the process we can easily deduce the solution of (1.4)-(1.5) which is given by

x(t)= x n (t)= f n ( ϕ ( 0 ) ) ,t((n1)r,nr],

which is continuous on each subinterval ((k1)r,kr), k=1,2,3,,n, but

lim t k r + x ( k + 1 ) r (t)= f k + 1 ( ϕ ( 0 ) ) x k r ,

which implies that the solution of problem (1.4)-(1.5) is discontinuous (sectionally continuous) on (0,T].

Now we have the following definitions.

Definition 1 The discontinuous (sectionally continuous) dynamical system is the problem of retarded functional equation (1.4)-(1.5).

Definition 2 The fixed points of discontinuous (sectionally continuous) dynamical system (1.4)-(1.5) are the solution of the equation

x(t)=f(t,x,x,,x).
(1.10)

Remark 1 We should notice that the difference equations representing the Henon map in its different cases,

x n + 1 = 1 + β x n 1 α x n 1 2 , n = 1 , 2 , 3 , 4 , , x n + 1 = 1 + β x n α x n 1 2 , n = 1 , 2 , 3 , 4 , , x n + 1 = 1 + β x n α x n 2 , n = 1 , 2 , 3 , 4 , ,

are just special cases of our problem (1.1)-(1.2).

2 Existence and uniqueness

Now consider the discontinuous (sectionally continuous) dynamical system of the Henon map (1.1)-(1.2). The existence of a unique solution as well as the continuous dependence of the solution on the initial data are proved. We study also the continuous dependence of the solution on the parameter α.

Let L 1 = L 1 [0,T], T<, be the class of Lebesgue integrable functions on [0,T] with the norm

f= 0 T |f(t)|dt,f L 1 .

Let D={xR:|x|<k,k= 1 1 | β | } L 1 [0,T].

Definition 3 By a solution of problem (1.1)-(1.2) we mean that the function x L 1 satisfies problem (1.1)-(1.2).

Theorem 1 The sufficient condition for the existence of a unique solution of problem (1.1)-(1.2) is |β|+2αk<1.

Proof Define the operator F: L 1 L 1 on D by

Fx(t)=1+βx(t r 1 )α x 2 (t r 2 ),

then

|Fx(t)|1+β|x|<1+|β|kk.

This proves that F:DD.

Now, for x,yD, we have

|FxFy||β||x(t r 1 )y(t r 1 )|+α| x 2 (t r 2 ) y 2 (t r 2 )|.

Thus we can get

FxFy|β|xy+2αkxy.

If M=|β|+2αk<1, then

FxFyMxy.

So, problem (1.1)-(1.2) has, on D, a unique solution x L 1 . □

Continuous dependence on the initial conditions

Theorem 2 The solution of discontinuous (sectionally continuous) dynamical system (1.1)-(1.2) is continuously dependent on the initial data.

Proof Let x(t) and x (t) be the solutions of dynamical system (1.1)-(1.2) and the dynamical system of equation (1.1) with the initial data

x(0)= x 0 .
(2.1)

Then

|x(t) x (t)||β||x(t r 1 ) x (t r 1 )|+α| x 2 (t r 2 ) x 2 (t r 2 )|,

and we can get

x ( t ) x ( t ) ( r 1 | β | + r 2 α ) | x 0 x 0 |+ ( | β | + 2 α k ) x x .

This implies that

x ( t ) x ( t ) r 1 β + r 2 α 1 | β | 2 α k | x 0 x 0 |.

That is,

|x x |δ x ( t ) x ( t ) ϵ= r 1 β + r 2 α 1 | β | 2 α k δ.

 □

Continuous dependence on the parameter α

Theorem 3 The solution of discontinuous (sectionally continuous) dynamical system (1.1)-(1.2) is continuously dependent on the parameter α.

Proof Let x(t) and x (t) be the solutions of dynamical system (1.1)-(1.2) and the dynamical system

x(t)=1+βx(t r 1 ) α x 2 (t r 2 ),
(2.2)

with the initial data (1.2), then

|x(t) x (t)||α x 2 (t r 2 ) α x 2 (t r 2 )|,

which gives

x ( t ) x ( t ) |β| x x +2α x x +k|α α | x .

This implies that

x ( t ) x ( t ) x 1 | β | 2 α k |α α |.

That is,

|α α |δ x ( t ) x ( t ) ϵ= x 1 β 2 α k δ.

 □

3 Fixed points and stability

Exactly like its discrete counter part, dynamical system (1.1)-(1.2) has two fixed points which are the solutions of the equation

x=1+βxα x 2 .

So, we have

( x fix ) 1 = ( β 1 ) + ( 1 β ) 2 + 4 α 2 α , ( x fix ) 2 = ( β 1 ) ( 1 β ) 2 + 4 α 2 α .

Obviously, they exist only for ( 1 β ) 2 +4α0 [15]. To determine the stability of a fixed point, consider a small perturbation from the fixed point by letting

x(t)= x fix + ϵ 0 λ t .

Substituting in (2.2) we get

x fix + ϵ 0 λ t =1+β ( x fix + ϵ 0 λ t r 1 ) α ( x fix + ϵ 0 λ t r 2 ) 2 ,

which implies that the fixed points are asymptotically stable if all roots of the equation

1=β λ r 1 2α x fix λ r 2
(3.1)

satisfy |λ|<1, where x(t)= x fix +ϵ λ t . Here we study three cases:

  • r 1 = r 2 then ( x fix ) 1 is stable if ( 1 β ) 2 +4α<4.

  • r 2 =2 r 1 then ( x fix ) 1 is stable if 2β+4α<3.

  • r 1 =2 r 2 then ( x fix ) 1 is stable if α< 3 4 ( 1 β ) 2 .

In all the simulations, r 1 and r 2 are rationally dependent.

Figure 1 illustrates the trajectories of (1.3), while Figure 2 illustrates the trajectories of (1.1).

Figure 1
figure1

Trajectories of ( 1.3 ) with α=1.2 and β=0.4 .

Figure 2
figure2

Trajectories of ( 1.1 ) with α=1.2 , β=0.4 , and r 1 = r 2 =1 .

4 Bifurcation and chaos

In this section we show, by numerical experiments illustrated by bifurcation diagrams, that the dynamical behavior of discontinuous (sectionally continuous) dynamical system (1.1)-(1.2) is completely affected by the change in both r and T [17]. We consider three cases for different delays r 1 and r 2 as follows.

  • Case 1: r 1 > r 2 .

Let β=0.3 be fixed and vary α from 0 to 1.4 with step size 0.001 and the initial condition ( x 0 , y 0 )=(0.3,0).

Take r 1 =2 and r 2 =1 and t[0,150] in (1.1)-(1.2) (Figure 3).

Figure 3
figure3

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 =2 and r 2 =1 and t[0,150] .

Take r 1 =0.50 and r 2 =0.25 and t[0,38] in (1.1)-(1.2) (Figure 4).

Figure 4
figure4

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 =0.50 and r 2 =0.25 and t[0,38] .

Take r 1 =0.3 and r 2 =0.1 and t[0,20] in (1.1)-(1.2) (Figure 5).

Figure 5
figure5

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 =0.3 and r 2 =0.1 and t[0,20] .

Take r 1 =0.25 and r 2 =0.15 and t[0,15] in (1.1)-(1.2) (Figure 6).

Figure 6
figure6

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 =0.25 and r 2 =0.15 and t[0,15] .

We see clearly in Figure 3 the bifurcation from a stable fixed point to a stable orbit of period two at α=0.4, and then the bifurcation from period two to period four at α=0.9. The further period doubling occurs at decreasing increments in α, and the orbit becomes chaotic for α1.1.

  • Case 2: r 1 = r 2 .

Take r 1 = r 2 =1 and t[0,100] in (1.1)-(1.2) (Figure 7).

Figure 7
figure7

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 = r 2 =1 and t[0,100] .

Take r 1 = r 2 =2 and t[0,200] in (1.1)-(1.2) (Figure 8).

Figure 8
figure8

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 = r 2 =2 and t[0,200] .

Take r 1 = r 2 =0.1 and t[0,10] in (1.1)-(1.2) (Figure 9).

Figure 9
figure9

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 = r 2 =0.1 and t[0,10] .

Take r 1 = r 2 =0.2 and t[0,15] in (1.1)-(1.2) (Figure 10).

Figure 10
figure10

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 = r 2 =0.2 and t[0,15] .

  • Case 3: r 1 < r 2 .

Take r 1 =1 and r 2 =2 and t[0,200] in (1.1)-(1.2) (Figure 11).

Figure 11
figure11

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 =1 and r 2 =2 and t[0,200] .

Take r 1 =0.1 and r 2 =0.2 and t[0,20] in (1.1)-(1.2) (Figure 12).

Figure 12
figure12

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 =0.1 and r 2 =0.2 and t[0,20] .

Take r 1 =0.25 and r 2 =0.75 and t[0,30] in (1.1)-(1.2) (Figure 13).

Figure 13
figure13

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 =0.25 and r 2 =0.75 and t[0,30] .

Take r 1 =0.15 and r 2 =0.25 and t[0,50] in (1.1)-(1.2) (Figure 14).

Figure 14
figure14

Bifurcation of ( 1.1 )-( 1.2 ) when r 1 =0.15 and r 2 =0.25 and t[0,50] .

5 Chaotic attractor

In this section we are interested in studying the chaotic attractor for three different cases.

  • Case 1: r 2 > r 1 .

Here we rewrite system (1.1)-(1.2) as follows:

x(t)=1+βx(t r 2 )αy(t r 2 ),
(5.1)
y(t)= x 2 ( t ( r 1 r 2 ) ) .
(5.2)

It is worth here to mention what we get when we plot the chaotic attractor for system (5.1)-(5.2) in this case. Figure 15 shows the chaotic attractor when r 1 =2 and r 2 =1, while Figure 16 shows the chaotic attractor of the same when r 1 =0.25 and r 2 =0.15.

Figure 15
figure15

Chaotic attractor of ( 5.1 )-( 5.2 ).

Figure 16
figure16

Chaotic attractor of ( 5.1 )-( 5.2 ).

  • Case 2: r 1 = r 2 =r.

Here system (1.1)-(1.2) is rewritten as

x(t)=1α x 2 (tr)+βx(tr)
(5.3)

with

x(t)= x 0 ,t0.
(5.4)

In this case, the chaotic attractor for r 1 = r 2 =1 and r 1 = r 2 =0.1 looks like in Figures 17 and 18.

Figure 17
figure17

Chaotic attractor of ( 5.3 )-( 1.2 ) with r 1 = r 2 =1 , t[0,75] .

Figure 18
figure18

Chaotic attractor of ( 5.3 )-( 1.2 ) with r 1 = r 2 =0.1 , t[0,7.5] .

  • Case 3: r 1 > r 2 .

Here we also rewrite system (1.1)-(1.2) as follows:

x(t)=1α x 2 (t r 1 )+y(t r 1 ),
(5.5)
y(t)=βx ( t ( r 2 r 1 ) ) .
(5.6)

Here we show the chaotic attractor for system (5.5)-(5.6). Figure 19 shows the chaotic attractor when r 1 =1 and r 2 =2, while Figure 20 shows the chaotic attractor of the same system but with r 1 =0.15 and r 2 =0.25.

Figure 19
figure19

Chaotic attractor of ( 5.5 )-( 5.6 ).

Figure 20
figure20

Chaotic attractor of ( 5.5 )-( 5.6 ).

Since the Lyapunov exponent is a good indicator for the existence of chaos [1821], we compute the Lyapunov characteristic exponents (LCEs) via the Householder QR-based methods described in [22]. LCEs play a key role in the study of nonlinear dynamical systems, and they are a measure of sensitivity of solutions of a given dynamical system to small changes in the initial conditions. One feature of chaos is sensitive dependence on initial conditions; for a chaotic dynamical system, at least one LCE must be positive. Since for non-chaotic systems all LCEs are non-positive, the presence of a positive LCE has often been used to help determine if a system is chaotic or not. Figure 21 shows the LCEs for system (1.1)-(1.2) in the case r 1 > r 2 for β=0.3 with the initial conditions ( x 0 , y 0 )=(0,0). With these parameter values, we find that LCE1=0.3228 and LCE2=1.2461. While Figure 22 shows the LCEs for the same system in the case r 1 < r 2 for β=0.5 with the same initial conditions, we find that LCE1=0.1318 and LCE2=0.8153. Finally, Figure 23 shows the LCEs for system (1.1)-(1.2) in the case r 1 = r 2 for parameter values β=0.6 with the same initial conditions. We find that LCE1=0.3228 and LCE2=1.2460.

Figure 21
figure21

LCEs of ( 5.1 )-( 5.2 ) when r 1 > r 2 .

Figure 22
figure22

LCEs of ( 5.1 )-( 5.2 ) when r 1 < r 2 .

Figure 23
figure23

LCEs of ( 5.1 )-( 5.2 ) when r 1 = r 2 .

6 Conclusion

The discontinuous (sectionally continuous) dynamical system of the Henon map describes dynamical properties for different values of the parameters r 1 , r 2 R + when the time t[0,T] is continuous. Indeed, the stability of fixed points depends on the values of delay parameters r 1 and r 2 as we have seen. The bifurcation diagrams, as well, depend on the values of delay parameters r 1 and r 2 and the time interval [0,T]. We have also noticed that the chaotic attractor of the discontinuous (sectionally continuous) Henon system in its different versions is also affected by the change in r 1 , r 2 and the time interval [0,T]. On the other hand, from Figures 3-4, 7-8, and 11-12 it looks like there is a scale that gives identical chaotic behavior. To summarize, our analytical result (3.1) agrees with the numerical simulations.

References

  1. 1.

    El-Sayed AMA, Nasr ME: Existence of uniformly stable solutions of nonautonomous discontinuous dynamical systems. J. Egypt. Math. Soc. 2011, 19(1–2):91–94. 10.1016/j.joems.2011.09.006

    MathSciNet  Article  Google Scholar 

  2. 2.

    El-Sayed AMA, Nasr ME: On some dynamical properties of discontinuous dynamical systems. Am. Acad. Sch. Res. J. 2012, 2(1):28–32.

    Google Scholar 

  3. 3.

    El-Sayed AMA, Nasr ME: Dynamic properties of the predator-prey discontinuous dynamical system. Z. Naturforsch. A 2012, 67a: 57–60.

    Google Scholar 

  4. 4.

    El-Sayed, AMA, Nasr, ME: On some dynamical properties of the discontinuous dynamical system presents the logistic equation with different delays. i-manag. J. Math. 1(1) (2012)

  5. 5.

    El-Sayed, AMA, Nasr, ME: Some dynamic properties of a discontinuous dynamical system. Alex. J. Math. 3(1) (2012)

  6. 6.

    El-Sayed AMA, Nasr ME: Discontinuous dynamical systems and fractional-orders difference equations. J. Fract. Calc. Appl. 2013, 4(12):1–9.

    Google Scholar 

  7. 7.

    El-Sayed AMA, Salman SM: Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments. Malaya J. Mat. 2012, 1(1):15–19.

    Google Scholar 

  8. 8.

    El-Sayed AMA, Salman SM: Chaos and bifurcation of discontinuous logistic dynamical system with piecewise constant arguments. Malaya J. Mat. 2013, 3(1):14–20.

    Google Scholar 

  9. 9.

    El-Sayed AMA, El-Raheem ZF, Salman SM: On some dynamics of Duffing dynamical system generated by a semi-discretization process with two different delays. Math. Sci. Lett. 2014, 3(2):89–96. 10.12785/msl/030204

    Article  Google Scholar 

  10. 10.

    El-Sayed AMA, Salman SM: Discontinuous dynamical systems generated by a semi-discretization process. Electron. J. Math. Anal. Appl. 2013, 1(1):47–54.

    Google Scholar 

  11. 11.

    Bilal S, Ramaswamy R: The generalized time-delayed Henon map: bifurcations and dynamics. Int. J. Bifurc. Chaos 2013., 23(3): Article ID 1350045

    Google Scholar 

  12. 12.

    Sprott JC: High-dimensional dynamics in the delayed Henon map. Electron. J. Theor. Phys. 2006, 3(12):19–35.

    Google Scholar 

  13. 13.

    Bohai NA: Continuous solutions of systems of nonlinear difference equations with continuous arguments and their properties. Nonlinear Oscil. 2007, 10(2):169–175. 10.1007/s11072-007-0013-1

    MathSciNet  Article  Google Scholar 

  14. 14.

    da Cruze JH, Táboas PZ: Periodic solutions and stability for a singularly perturbed linear delay differential equation. Nonlinear Anal. 2007, 67: 1657–1667. 10.1016/j.na.2006.08.004

    MathSciNet  Article  Google Scholar 

  15. 15.

    Elaydi SN Undergraduate Texts in Mathematics. In An Introduction to Difference Equations. 3rd edition. Springer, New York; 2005.

    Google Scholar 

  16. 16.

    Hale J: Theory of Functional Differential Equations. Springer, New York; 1977.

    Google Scholar 

  17. 17.

    Kuznetsov YA: Elements of Applied Bifurcation Theory. 3rd edition. Springer, Berlin; 2004.

    Google Scholar 

  18. 18.

    Jing ZJ, Yang J: Bifurcation and chaos in discrete-time predator-prey system. Chaos Solitons Fractals 2006, 27: 259–277. 10.1016/j.chaos.2005.03.040

    MathSciNet  Article  Google Scholar 

  19. 19.

    Liu X, Xiao D: Complex dynamic behaviors of a discrete-time predator-prey system. Chaos Solitons Fractals 2007, 32: 80–94. 10.1016/j.chaos.2005.10.081

    MathSciNet  Article  Google Scholar 

  20. 20.

    Wu G-C, Baleanu D: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 2014, 75(1–2):283–287. 10.1007/s11071-013-1065-7

    MathSciNet  Article  Google Scholar 

  21. 21.

    Wu G-C, Baleanu D, Zeng S-D: Discrete chaos in fractional sine and standard maps. Phys. Lett. A 2014. 10.1016/j.physleta.2013.12.010

    Google Scholar 

  22. 22.

    Udwadia FE, von Bremen H: A note on the computation of the largest p -Lyapunov characteristic exponents for nonlinear dynamical systems. J. Appl. Math. Comput. 2000, 114: 205–214. 10.1016/S0096-3003(99)00113-7

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees of this manuscript for their valuable comments and suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ahmed M. A. El-Sayed.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

Authors’ original file for figure 1

Authors’ original file for figure 2

Authors’ original file for figure 3

Authors’ original file for figure 4

Authors’ original file for figure 5

Authors’ original file for figure 6

Authors’ original file for figure 7

Authors’ original file for figure 8

Authors’ original file for figure 9

Authors’ original file for figure 10

Authors’ original file for figure 11

Authors’ original file for figure 12

Authors’ original file for figure 13

Authors’ original file for figure 14

Authors’ original file for figure 15

Authors’ original file for figure 16

Authors’ original file for figure 17

Authors’ original file for figure 18

Authors’ original file for figure 19

Authors’ original file for figure 20

Authors’ original file for figure 21

Authors’ original file for figure 22

Authors’ original file for figure 23

Authors’ original file for figure 24

Authors’ original file for figure 25

Authors’ original file for figure 26

Authors’ original file for figure 27

Authors’ original file for figure 28

Authors’ original file for figure 29

Authors’ original file for figure 30

Authors’ original file for figure 31

Authors’ original file for figure 32

Authors’ original file for figure 33

Authors’ original file for figure 34

Authors’ original file for figure 35

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

El-Sayed, A.M.A., El-Raheem, Z.F.E. & Salman, S.M. A new chaotic behavior of a general model of the Henon map. Adv Differ Equ 2014, 107 (2014). https://doi.org/10.1186/1687-1847-2014-107

Download citation

Keywords

  • retarded functional equation
  • Henon map
  • fixed points
  • existence
  • uniqueness
  • bifurcation
  • chaos
  • chaotic attractor
\