Physics Letters A 373 (2009) 2922–2926

Contents lists available at ScienceDirect

Physics Letters A

www.elsevier.com/locate/pla

Existence of chaos in a piecewise smooth two-dimensional contractive map

Biswambhar Rakshit a,∗, Soumitro Banerjee b

a Department of Mathematics and Centre for Theoretical Studies, Indian Institute of Technology, Kharagpur-721302, Indiab Department of Physics, Indian Institute of Science Education and Research, Mohanpur-741252, Nadia, West Bengal, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 December 2008Received in revised form 17 May 2009Accepted 10 June 2009Available online 17 June 2009Communicated by A.R. Bishop

PACS:05.45.-a05.45.Ac

Keywords:Piecewise smooth mapGlobally contractive mapBorder collision bifurcationChaos

Piecewise smooth maps occur in a variety of physical systems. We show that in a two-dimensionalcontinuous map a chaotic orbit can exist even when the map is contractive (eigenvalues less than unity inmagnitude) at every point in the phase space. In this Letter we explain this peculiar feature of piecewisesmooth continuous maps.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

In recent years the bifurcation theory for piecewise smoothsystems has attracted significant research interest because suchsystems arise in many physical, engineering and biological sys-tems. Under stroboscopic sampling a piecewise smooth flow yieldsa piecewise smooth map. In smooth maps, a bifurcation occurswhen a real eigenvalue or a complex conjugate pair of eigenval-ues of a fixed point crosses the unit circle. In piecewise smooth(PWS) maps, on the other hand, a bifurcation can also occur whena fixed point (or a periodic orbit) collides with the borderline be-tween the two regions of smooth behavior. This involves an abruptchange in the eigenvalues of the Jacobian matrix. This leads to anew class of bifurcations, called border collision bifurcations [1–5].It has been shown that such bifurcations can lead to atypical tran-sitions like a period one orbit directly bifurcating into a chaoticorbit, or a periodic orbit suddenly vanishing as it hits the border.

The classification of border collision bifurcations for one-dimensional (1-D) PWS map is more or less complete [2,6–9].A detailed study of the border collision bifurcations in piecewisesmooth 2-D maps has been reported in [10,11] by exploiting thenormal form map. Later it was found that many interesting dy-namics [12,13] can occur in the 2-D map when the map is globally

* Corresponding author. Tel.: +91 3222 281645.E-mail address: biswambhar@cts.iitkgp.ernet.in (B. Rakshit).

0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2009.06.025

contractive. Dutta et al. [12] presented a novel border collision bi-furcation in which multiple co-existing periodic attractors are cre-ated simultaneously. Hassouneh et al. [13] demonstrated anotherinteresting bifurcation, known as the “dangerous border collisionbifurcation”.

In our present investigation, we have focused on such 2-D mapswhose fixed points are stable at both sides of the border. Since themap is globally contractive, one does not expect the occurrenceof chaos — for which a stretching behavior is necessary. We havedetected this counterintuitive phenomenon, and have investigatedthe mechanism for the occurrence of chaos in a globally contrac-tive piecewise smooth continuous map.

2. The normal form map

We consider such PWS maps whose leading order Taylor termin the neighborhood of the border is linear. For such maps thenormal form can be expressed as [10]

(x(n+1)

y(n+1)

)=

⎧⎨⎩

JL( x(n)

y(n)

) + Cμ, if x(n) � 0,

JR( x(n)

y(n)

) + Cμ, if x(n) > 0,(1)

where

JL =(

τL 1)

, JR =(

τR 1)

, C =(

1)

,

−δL 0 −δR 0 0

B. Rakshit, S. Banerjee / Physics Letters A 373 (2009) 2922–2926 2923

τL is the trace and δL is the determinant of the Jacobian matrixJL and τR is the trace and δR is the determinant of the Jacobianmatrix JR .

The fixed points of the system (1) in both sides of the boundaryare given by:

L∗ =(

μ

1 + δL − τL,

−δLμ

1 + δL − τL

),

R∗ =(

μ

1 + δR − τR,

−δRμ

1 + δR − τR

). (2)

If the x-component of L∗ is negative, the fixed point exists. Else itdoes not. However, when the x-component of L∗ is positive, itera-tions from initial conditions in the left half are influenced by the“nonexistent” fixed point, which is called a “virtual” fixed point,and is denoted by L̄∗ . Similarly, when the x-component of R∗ ispositive, the fixed point exists; else it is a virtual fixed point de-noted by R̄∗ .

The stability of L∗ and R∗ are determined by the eigenvalues

λL± = 1

2

(τL ±

√τ 2

L − 4δL), λR± = 1

2

(τR ±

√τ 2

R − 4δR).

Let us choose the parameters |δL | < 1 and |δR | < 1, so that thesystem is dissipative. The conditions

−(1 + δL) < τL < (1 + δL), (3a)

−(1 + δR) < τR < (1 + δR), (3b)

then ensure that the map is globally contractive. We have re-stricted our attention to the parameter space range given by (3).Moreover, due to the symmetry of the map any dynamical be-havior observed for μ > 0 will also be observed for μ < 0, if thevalues of τL and τR are interchanged. In our present investigationwe have considered μ > 0.

3. Coexistence of chaotic attractor along with a stable fixed point

We have observed that in the piecewise smooth map (1) achaotic attractor can co-exist with a stable fixed point. The bifur-cation diagram in Fig. 1 shows an example, with parameter valueschosen to illustrate this behavior. Fig. 1 reveals that on the left-hand side of the border only a stable fixed point exists for μ < 0,but for μ > 0 a chaotic orbit co-exists with a stable fixed point andperiod-4 attractor. Thus, numerical observation shows that even ifthe map is globally contractive, a chaotic orbit can come into exis-tence.

Fig. 2 presents the bifurcation diagram with τL as the variableparameter, with μ fixed at a positive value. The other parametersare the same as in Fig. 1. This figure reveals the following.

• There are certain ranges of the parameter τL for which achaotic orbit comes into existence, and before the occurrenceof chaos one of the high-periodic orbits loses stability.

• These high periodic orbits follow a particular type of symbolsequence. They are all maximal orbits [14] of type Ln R , i.e.,one point lies on the right-hand side of the border and allother points lie on the left-hand side of the border.

• The chaotic attractor suddenly disappears through a boundarycrisis.

Let us probe how these high periodic orbits become unstableand what happens to them afterwards. First we analyze the ex-istence and stability of the period-3 LLR orbit. Throughout thisLetter we will name particular types of orbit depending on thepartitions (L or R) in which the points fall. For example, a LLRperiod-3 orbit that has two points in the left-hand side and the

Fig. 1. Bifurcation diagram with τL = 0.90, τR = −1.59, δL = 0.7, and δR = 0.7.

Fig. 2. Bifurcation diagram with μ = 0.5, τR = −1.59, δL = 0.7, and δR = 0.7.

third one in the right-hand side is called an LLR orbit. Supposethat the orbit has the points (xL1 , yL1), (xL2 , yL2), (xR1 , yR1 ). Theexistence conditions of the LLR orbit are given by xL1 < 0, xL2 < 0,xR1 > 0. Since it is not possible to have a period-3 orbit with allthe points in one linear side, the last condition is always satisfied.

So the existence conditions are given by

(1 + τR − δL + τLτR + δLδR + δRτL)μ

1 + δ2L δR + τRδL + δRτL + δLτL − τRτ 2

L

< 0, (4)

(1 + τL − δR + τRτL + δRδL + δLτR)μ

1 + δ2L δR + τRδL + δRτL + δLτL − τRτ 2

L

< 0. (5)

The condition of stability of this orbit is given by eigenvaluesof the composite matrix J L J L J R , which must be inside the unitcircle. If τLLR denotes the trace and δLLR denotes the determinantof the above Jacobian matrix, then the criterion of stability is givenby

−(1 + δLLR) < τLLR < (1 + δLLR). (6)

Using (6) the stability conditions are obtained as

S1 = 1 + δ2L δR + τRδL + δRτL + δLτL − τRτ 2

L > 0, (7)

S2 = 1 + δ2L δR − τRδL − δRτL − δLτL + τRτ 2

L > 0. (8)

The stability condition (7) refers to τLLR < (1 + δLLR) while thecondition (8) refers to −(1 + δLLR) < τLLR . Notice that one of thestability conditions (7) is the same as the denominator of the ex-istence conditions (4) and (5). So this LLR orbit may lose stabilitywhenever the stability condition (8) is not satisfied. This happenswhen one of the eigenvalues crosses the unit circle through −1and the LLR orbit becomes a flip saddle. From Fig. 3 we see thatthere exists a parameter region where a LLR flip saddle exists.

Here we can observe a fundamental difference from one-dimensional non-smooth maps. In a one-dimensional map, if theslope at every point is less than unity, the magnitude of the slopeof the higher iterates of the map will always be less than unity.But in a two-dimensional map the higher iterates can be expansiveeven if the map is globally contractive, since in this case matricesare multiplied to give the Jacobian of the high periodic fixed point.In Fig. 4 we show the evolution of a small circle taken around a

2924 B. Rakshit, S. Banerjee / Physics Letters A 373 (2009) 2922–2926

Fig. 3. The parameter space region where a period-3 LLR orbit exists. Here δL = 0.7,δR = 0.7, and μ = 0.5.

Fig. 4. Evolution of a circle, taken around a point of the period-3 LLR saddle, show-ing an expansive behavior after the third iterate of the map. Here τL = 0.8915,τR = −1.59, δL = 0.7, δR = 0.7, and μ = 0.5.

Fig. 5. Bifurcation diagram with μ = 0.5, τR = −1.59, δL = 0.7, and δR = 0.7.

point of the period-3 LLR saddle. Here we can observe the expan-sive nature of the third iterate of the map.

Let us now analyze what happens when the period-3 LLR orbitbecomes unstable. The bifurcation diagram for the section {0.87 �τL � 0.905, τR = −1.59, δL = 0.7, δR = 0.7,μ = 0.5} is presented inFig. 5. At the point τL = τ ∗

L ≈ 0.8783, the largest eigenvalue of theJacobian matrix J L J L J R crosses the unit circle through −1 andthe period-3 LLR orbit turns into a flip saddle. This leads to theemergence of a period-6 orbit with a symbol sequence given bythe concatenation of LLR with itself. Since the map is linear oneach side of the border at x = 0, the period 6 orbit produced inthe period doubling instantly collides with the border. Thus thereis no parameter range for which the stable period-6 orbit actuallyoccurs [15]. This leads to an abrupt transition to a six band cyclicchaotic attractor (Fig. 6).

The period-3 LLR orbit now becomes a flip saddle, and we alsohave a period-6 saddle orbit. Fig. 7 shows the stable and unstablemanifolds of the period-6 saddle orbit as well as the LLR saddleorbit. The neighborhood of one of the points of the period-3 sad-dle is shown in Fig. 8. Fig. 8(a) shows that the stable and unstablemanifolds of the LLR saddle orbit do not have a homoclinic inter-section. But in Fig. 8(b) we can observe a homoclinic intersectionof the stable and unstable manifolds of the period-6 saddle or-

Fig. 6. Phase portrait showing the six piece chaotic attractor for τL = 0.89, τR =−1.59, δL = 0.7, δR = 0.7, and μ = 0.5.

Fig. 7. The stable (blue) and unstable (red) manifolds of the LLR flip saddle (de-noted by black circles) and period-6 saddle (denoted by green circles) for τL = 0.89,δL = 0.7, τR = −1.59, δR = 0.7, and μ = 0.5. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this Letter.)

bit, and this gives rise to a six-piece cyclic chaotic attractor. In thephase space there are three segments filled with 6-cycles, and theperiod-3 orbit which undergoes the bifurcation is located betweenthem, separating the six segments. The homoclinic period-6 sad-dle orbit created at the bifurcation is located inside the six-piecechaotic attractor.

As the parameter τL is varied, the pieces of the six-piece chaoticattractor merge pairwise to give rise to a three-piece cyclic chaoticattractor (Fig. 9) at τL ≈ 0.8915. As long as the six-piece chaoticattractor exists, the LLR saddle cycle is not included in the chaoticset, and it is not a homoclinic saddle (Fig. 8(a)). The homoclinicbifurcation of the LLR saddle cycle occurs (Fig. 10) at the parame-ter value where the six-piece chaotic attractor merges into a threepiece chaotic attractor.

At τL ≈ 0.9025, a boundary crisis occurs and the chaotic at-tractor disappears. Fig. 11 shows the basins of attraction of thethree coexisting orbits close to the boundary crisis. At this pointthe chaotic orbit comes very close to the fractal basin boundary.When they make contact with the change of parameter, the chaoticorbit disappears.

In Fig. 12 we have shown the parameter region where a chaoticattractor appears through period doubling of the LLR orbit. Atother parameter ranges, a similar phenomenon occurs for the otherhigh-periodic fixed points. In Fig. 13 we have shown the parameterspace regions where a chaotic attractor, created through the insta-bility of the respective high-period orbits, co-exist with a stablefixed point.

Due to the symmetry of the map, for negative values of μ thesame phenomenon will be observed. In this case the symbol se-quence will get reversed, and the LR R orbit will be flip saddleinstead of the LLR orbit.

B. Rakshit, S. Banerjee / Physics Letters A 373 (2009) 2922–2926 2925

Fig. 8. Zoomed portion of Fig. 7: (a) the stable (blue) and unstable (red) manifolds of the period-3 cycle, and (b) those for the period-6 cycle. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this Letter.)

Fig. 9. Phase portrait showing the three-piece chaotic attractor for τL = 0.8915, τR =−1.59, δL = 0.7, δR = 0.7, and μ = 0.5.

Fig. 10. A portion of the stable (blue) and unstable (red) manifolds of LLR flip saddle(denoted by black circle) for τL = 0.8915, δL = 0.7, τR = −1.59, δR = 0.7, and μ =0.5. (For interpretation of the references to colour in this figure legend, the readeris referred to the web version of this Letter.)

From Fig. 14 we observe that the basin of attraction is of a verycomplicated structure. The basin boundary between the period-1attractor and the chaotic attractor is formed by the stable manifoldof the LR R regular saddle while the basin boundary between theperiod-4 attractor and the chaotic attractor is formed by the stablemanifold of the LLR R regular saddle. Both the basin boundariesare fractal due to the presence of homoclinic intersection.

In general we have found that

• For μ > 0, an attractor of type Ln R and a regular saddle or-bit of type Ln−1 R2 come into existence, where n � 2. As wechange the parameter the Ln R attractor undergoes a perioddoubling bifurcation. The resulting period-2n orbit hits theborder instantly and turns into a 2n piece chaotic attractor.

• For μ < 0, the same phenomenon is observed if the values ofτL and τR are interchanged. Only the symbol sequence will getreversed.

Fig. 11. Basins of attraction of the three coexisting attractors at τL = 0.9025, τR =−1.59, δL = 0.7, δR = 0.7 and, μ = 0.5. Aqua: Chaotic, Magenta: period-1, Blue:period-4. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this Letter.)

Fig. 12. The parameter space region where the chaotic attractor appears througha period doubling of the LLR orbit and co-exist with a stable fixed point. HereδL = 0.7, δR = 0.7, and μ = 0.5.

Fig. 13. The regions of the parameter space where a chaotic attractor co-exist witha stable fixed point. Here δL = 0.7, δR = 0.7, and μ = 0.5.

2926 B. Rakshit, S. Banerjee / Physics Letters A 373 (2009) 2922–2926

Fig. 14. Basins of attraction of the three coexisting attractors at τL = 0.9, τR =−1.59, δL = 0.7, δR = 0.7, and μ = 0.5. Magenta: Chaotic, Aqua: period-1, White:period-4. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this Letter.)

4. Conclusion

We have reported the occurrence of chaos in a piecewisesmooth map that is contractive at every point of the phase space.We have explained the mechanism by illustrating the pathway inwhich a period doubling is followed immediately by a border col-lision leading to an homoclinic intersection. We have also shownthat in a 2-D map, a higher iterate fixed point can have eigen-value outside the unit circle even though each of the Jacobianmatrices may have eigenvalues inside the unit circle. Thus a global

contractive map can harbour an expansive behavior of the higheriterates.

Acknowledgements

One of the authors B. Rakshit is thankful to Council of Scientificand Industrial Research, Government of India, for a senior researchfellowship.

References

[1] H.E. Nusse, J.A. Yorke, Physica D 57 (1992) 39.[2] H.E. Nusse, J.A. Yorke, Int. J. Bifur. Chaos 5 (1) (1995) 189.[3] M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk, Piecewise-Smooth

Dynamical Systems: Theory and Applications, Springer-Verlag, Applied Math-ematical Sciences, London, 2008.

[4] Z.T. Zhusubaliyev, E. Mosekilde, Bifurcations and Chaos in Piecewise-SmoothDynamical Systems, World Scientific, Singapore, 2003.

[5] M. di Bernardo, M.I. Feigin, S.J. Hogan, M.E. Homer, Chaos Solitons Frac-tals 10 (11) (1999) 1881.

[6] S. Banerjee, M.S. Karthik, G.H. Yuan, J.A. Yorke, IEEE Tran. Circuits Syst.-I 47 (3)(2000) 389.

[7] Y.L. Maistrenko, V.L. Maistrenko, L.O. Chua, Int. J. Bifur. Chaos 3 (6) (1993) 1557.[8] Y.L. Maistrenko, V.L. Maistrenko, S.I. Vikul, L.O. Chua, Int. J. Bifur. Chaos 5 (3)

(1995) 653.[9] Y.L. Maistrenko, V.L. Maistrenko, S.I. Vikul, L.O. Chua, Chaos Solitons Frac-

tals 9 (1) (1998) 67.[10] S. Banerjee, P. Ranjan, C. Grebogi, IEEE Trans. Circuits Syst.-I 47 (5) (2000) 633.[11] S. Banerjee, C. Grebogi, Phys. Rev. E 59 (4) (1999) 4052.[12] M. Dutta, H.E. Nusse, E. Ott, J.A. Yorke, G.H. Yuan, Phys. Rev. Lett. 83 (1999)

4281.[13] M.A. Hassouneh, E.H. Abed, H.E. Nusse, Phys. Rev. Lett. 92 (7) (2004) 070201.[14] W. Chin, E. Ott, H.E. Nusse, C. Grebogi, Phys. Rev. E 50 (1994) 4427.[15] D.J.W. Simpson, J.D. Meiss, SIAM J. Appl. Dyn. Syst. 7 (2008) 795.