Chaos, Solitons & Fractals 45 (2012) 341–350

Contents lists available at SciVerse ScienceDirect

Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier .com/locate /chaos

Stability, attractors, and bifurcations of the A2 symmetric flow

J.M. González-MirandaDepartamento de Física Fundamental, Universidad de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain

a r t i c l e i n f o

Article history:Received 19 July 2011Accepted 25 December 2011Available online 28 January 2012

0960-0779/$ - see front matter � 2012 Elsevier Ltddoi:10.1016/j.chaos.2011.12.013

E-mail address: jmgonzalezm@ub.edu

a b s t r a c t

The A2 symmetric flow, initially introduced to study effects of symmetry in chaos synchro-nization, displays a variety of attractors and bifurcations much richer than initially though.These are studied in this article by means of two approaches. A linear stability analysis isused to determine fixed points, the nature of its stability, and where oscillatory solutionsare expected. Nonlinear techniques such as bifurcation diagrams, Lyapunov exponentsand phase space plots, are used to find and classify these oscillations and their bifurcations.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Chaotic dynamics is among the most relevant phenom-ena displayed by nonlinear systems. When these systemsevolve in a continuous time, and their dynamics is de-scribed by autonomous ordinary differential equations(ODEs), the simplest models able to display chaos have tobe at least of order three. Because of this, systems of threeordinary differential equations are among the mathemati-cal models most commonly used to study the fundamen-tals of the theory of nonlinear dynamics and chaos. Wellknown examples of such systems are the Lorenz [1], theRössler [2], and the Chua [3] oscillators. A mathematicallyinteresting collection of order three ODEs has been foundby Sprott [4] in a search for the simplest systems able todisplay chaos. For one or other reason, new systems of thistype are being studied since the last years [5].

We will present here a study of the bifurcation diagramof the A2 symmetric flow. This is one of several ODEs ini-tially introduced in Ref. [6] to study the effect of spatialsymmetries of chaotic attractors on chaos synchronization.Further study showed that symmetry [7–9], may play a rel-evant role in the applications of chaos to secure communi-cations [10,11]. Additional developments showed thatsome properties of this system were important to theunderstanding of the conditions of occurrence of general-ized synchronization of chaos [12–14]. This makes the A2

. All rights reserved.

symmetric flow specially interesting and worthy of furtherstudy in the subfield of synchronization and control ofchaos [15–17], which is a very active one in today researchin nonlinear dynamics and chaos, and perhaps in othersubfields [18,19].

This system depends on three parameters. Until now,the research made on it is limited to a single set of valuesof these parameters for which the system displays a spa-tially symmetric chaotic attactor, which has the shape ofa single ring-shaped band. It would be useful for furtherapplications of this system, in chaos synchronization orother subfields, to have a wider view of the different non-linear dynamic behaviors, chaotic or not, that it may dis-play. This might also be useful to better understandprevious results. Because of this, we have made a system-atic study of the dynamics, stability and bifurcation dia-grams displayed by this system, whose main results wereport in the present article. After a linear stability analy-sis, and guided by it, our approach consist in detectingand characterizing a variety of routes to chaos and attrac-tors by means of phase space plots, bifurcation diagramsand Lyapunov exponents. This bears a methodological sim-ilarity with numerical approaches followed in many papersin the literature that study bifurcations in nonlinear maps[20–22] and flows [23–26].

The results obtained are presented in two sections:Section 2, on linear stability analysis, where the fixedpoints of the system are obtained and its stabilitydetermined, and Section 3 on non linear analysis, where

342 J.M. González-Miranda / Chaos, Solitons & Fractals 45 (2012) 341–350

bifurcation diagrams, Lyapunov exponents and phasespace plots are used to characterize chaotic and periodicoscillations. A last section (Section 4) is devoted to discus-sion and summary of these results.

2. Linear stability analysis

We will study the dynamics displayed by the followingsystem of three autonomous ordinary differential equa-tions [6]:

_x ¼ yþ A sinðXyÞ; ð1Þ_y ¼ �y� ðz� RÞx; ð2Þ_z ¼ x2 � z; ð3Þ

with ðx; y; zÞ 2 R3 the system state, and A; X; R 2 R thebifurcation parameters. These equations, which areinvariant under the change (x,y,z) M (�x,�y,z), have beenintroduced to study the effect of symmetry in chaos syn-chronization [6]. In this, and other research that followed[7,10,12], the system parameters were held fixed atA = 3.2, X = 1.4 and R = 5.2. For these values, the systemevolves toward an attractor that is chaotic and has theshape of a symmetric single loop in the plane x–y. Thephase space motion occurs on a simple annular stripcentered in the origin. The system dynamics follows a sym-metric rotation in the x–y plane around the origin, andoscillates up and down along the z-axis in such a way thatmaxima of z occur when x(t) and y(t) are in the second andfourth quadrants of the x � y plane and the minima whenthey are in the first and third quadrant. The value for thelargest Lyapunov reported in Ref [6] was k1 = 0.128, whichindicates a chaotic attractor.

2.1. Fixed points and eigenvalues

The space of parameters for this system (A,X,R) hasdimension 3 and can be identified with R3. However, aninspection of equations Eqs. (1)–(3) shows that they areinvariant under the change (A,X,R) M (�A,�X,R). There-fore we can restrict our study to the semi-space whereA P 0. We call it

Pþ ¼ fðA;X;RÞjA 2 Rþ; X 2 R; R 2 Rg: ð4Þ

A linear stability analysis of this system requires thedetermination of the fixed points xF = (xF,yF,zF). These aregiven by the solutions of the following system of nonlinearalgebraic equations

0 ¼ yF þ A sinðXyFÞ; ð5Þ0 ¼ �yF � ðzF � RÞxF ; ð6Þ0 ¼ x2

F � zF : ð7Þ

To find these fixed points we must solve the first equationfor yF. Because of the trigonometric function, this is a tran-scendental equation, which in most cases can only besolved numerically. Once yF has been determined, we usethe third equation to rewrite the second equation as0 ¼ x3

F � RxF þ yF : This a third degree equation in xF, havingthree roots that can be obtained by radicals. The thirdequation gives zF = x2

F , once xF has been determined. Linear

stability analysis of this system is complicated because ofEq. (5) that may have a large number of roots dependingon the values of A and X. Each of these roots then may re-sult in one or three fixed points, depending on the numberof real roots given by Eq. (6).

The stability of the fixed points is determined by theeigenvalues of the system Jacobian matrix. These are givenby the three roots, Ki=1,2,3(A,X,R;xF), of the polynomialequation

K3 þ 2K2 þ bðA;X;R; xFÞKþ cðA;X;R; xFÞ ¼ 0; ð8Þ

with

bðA;X;R; xFÞ ¼ 1þ x2F � R

� �½1þ AX cosðXyFÞ�; ð9Þ

cðA;X;R; xFÞ ¼ 3x2F � R

� �½1þ AX cosðXyFÞ�: ð10Þ

Once a fixed point, xF, has been determined its stability isobtained by solving Eq. (8), which again is a cubic equation.

2.2. The fundamental equilibria

For all values of the system parameters we always havethe trivial solution yF = 0 of Eq. (5). In this case, Eq. (6) hasthree roots: xF 2 f0;

ffiffiffiRp

;�ffiffiffiRpg. The first one exist for all

ðA;X;RÞ 2 Pþ, gives zF = 0, and then x0 = (0,0,0). The othertwo roots, xF ¼ �

ffiffiffiRp

, are only real and different than zerofor R > 0. Then, if R > 0, we have the additional fixed pointsxð0Þ� ¼ ð�

ffiffiffiRp

;0;RÞ associated to the solution yF = 0 of Eq. (5).We call these three fixed points the fundamental equilibria.

For x0, the characteristic equation becomes

P0ðKÞ ¼ ð1þKÞ½K2 þK� Rð1þ AXÞ� ¼ 0: ð11Þ

Its roots are: K1(x0) = �1, which is independent of systemparameters, and

K2;3ðA;X;R; x0Þ ¼12�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4Rð1þ AXÞ

ph i: ð12Þ

Having in mind that K1 < 0, an inspection of this expres-sion tells us that, regarding stability, we have essentiallytwo cases. (1) x0 is unstable when 4R(1 + AX) > 0, becausein this case K2;3 2 R; K2 > 0 and K3 < 0; therefore, the ori-gin is a saddle with a one-dimensional unstable manifold.(2) x0 is stable when 4R(1 + AX) < 0 because in this caseeither K2;3 2 R and K3 < K2 < 0, when 0 > 4R(1 + AX) P �1,and x0 is a sink, or K2;3 2 C with Re½K2� ¼ Re½K3� < 0, when4R(1 + AX) < �1; therefore, x0 is a spiral sink.

The bifurcation points where the system dynamicschanges stability are given by the condition R(1 + AX) = 0.When this condition is verified there are two eigenvaluesequal to �1 and the other is null. This corresponds to abifurcation in which the one-dimensional unstable mani-fold that occurs for 4R(1 + AX) > 0, becomes stable bymeans of a change of sign of its eigenvalue. From a geomet-rical point of view, the condition R(1 + AX) = 0 in theparameter space, Pþ, defines two significant surfaces: theplane R = 0, and the sheet

A1ðR;XÞ ¼ �1=X; ð13Þ

which is a surface orthogonal to the R = 0 plane and whoseintersection with this plane is a hyperbola. This separatesPþ in four regions of different stability [Fig. 1(a)]. For

-2.0 -1.0 0.0 1.0 2.0

Ω

0.0

1.0

2.0

3.0

4.0

5.0

A

0.0

1.0

2.0

3.0

4.0

5.0

A

(a)

(b)

Stable for R<0Unstable for R>0

Unstable for R<0

Stable for R>0 LΩ

LA

Fig. 1. (a) The stability of x0 in the planes given by the condition R = const.The solid lines are the intersections of the surfaces A1(R,X) (thick line)and A3(R,X) (thin line) with the plane R = const. Two dashed lines (LA andLX) are given as references for the discussion in Section 3. (b) The stabilityof xð0Þ� : these fixed points are stable inside the region limited by thesurfaces A1(R,X) (thick line) and A2(R,X). This last is shown in the two-dimensional plot by means of a set of six dashed contour lines for R > 1(R = 1.1, 1.5, 2.0, 3.0, 5.0, 9.0), and six dotted contourlines for R < 1(R = 0.10, 0.15, 0.20, 0.30, 0.50, 0.90). Arrows signal the direction ofincreasing R in each case. The thin solid line is again A3(R,X).

J.M. González-Miranda / Chaos, Solitons & Fractals 45 (2012) 341–350 343

R > 0, A1(R,X) separates Pþ in two regions: one located atthe concave side of this surface, where x0 is stable, andother located at the convex side, where x0 is unstable.For R < 0 we have the same two regions with the stabilityexchanged.

We consider now the stability of the two additionalfixed points xð0Þ� . The characteristic equation becomes

P�ðKÞ ¼ K3 þ 2K2 þKþ 2Rð1þ AXÞ ¼ 0; ð14Þ

and is the same for the two points; therefore, they sharestability properties.

We can analyze this stability by defining the function

f ðKÞ ¼ K3 þ 2K2 þK; ð15Þ

and the constant

C ¼ 2Rð1þ AXÞ; ð16Þ

so that Eq. (14) is written as P±(K) = f(K) + C = 0.The function f(K) is a cubic, it intersects the K axis in

two points, Ka = 0 and Kc = �1, has a minimum at Kb =�1/3 being f(Kb) = �4/27, and a maximum at Kc beingf(Kc) = 0. Moreover, f(K) ? ±1 when K ? ±1. Therefore,as shown in Fig. 2(a), we have the following scenarios: (i)if C < 0, we have a real positive and two complex conjugateeigenvalues; then, xð0Þ� are unstable, (ii) if 0 < C 6 4/27, wehave three real eigenvalues that are all negativeK3 6 �1 6K2 < �1/3 6K1 < 0, and we can assure thatxð0Þ� are stable fixed points, and (iii) if 4/27 < C, we have areal negative eigenvalue, K1 < �1, and two complex conju-gate eigenvalues, whose real part will determine the fixedpoints stability.

To have information on the real parts of the complexeigenvalues we have resorted to solve solutions of Eq.(14). The results, presented in Fig. 2(b), are in accordancewith those of the above paragraph. We have four casesdepending on the value of C. (1) If C < 0; Re½K2;3� < 0, then,xð0Þ� are spiral-in saddles. (2) If 0 < C 6 4=27; xð0Þ� are sinks.(3) If 4=27 < C < 2; Re½K2;3� < 0, and xð0Þ� are spiral-in sinks.(4) If 2 < C; Re½K2;3� > 0, and xð0Þ� lose their stability becom-ing spiral-out saddles. The bifurcations changing stability,as C increases, are given by the conditions C = 0 and C = 2.That means that xð0Þ� are stable for 0 < R(1 + AX) < 2.

The condition for the first bifurcation is R(1 + AX) = 0that, being R > 0, is meted again by the points in the sheetA1(R,X) = �1/X[Fig. 1(b)]. This surface separates Pþ in tworegions of different stability. We have stable fixed points inthe side where this surface is convex, and unstable pointsat the side where it is concave.

The condition for the second bifurcation, 2 = 2R(1 + AX),defines a second surface in Pþ,

A2ðR;XÞ ¼ ð1� RÞ=RX: ð17Þ

This is significant for R > 0. Its shape can be inferred fromthe contour lines plotted in Fig. 1(b). For R > 1, in the vol-ume limited by the surfaces A1(R,X) and A2(R,X) we havestable xð0Þ� fixed points. Outside that volume, xð0Þ� pointsare unstable. Therefore, above the plane given by the con-dition R = 1, we have a stability region to the left ofFig. 1(b), for X < 0, which becomes narrower as R increases,because A2(R,X) ? A1(R,X) when R ?1. Below the R = 1

plane, the stable region includes positive values of X, andbecomes narrower in the R direction as A and X increase.

2.3. Secondary equilibria

Geometrically speaking, the solutions of Eq. (5) are gi-ven by the intersections of the function g(n) = �Asin(Xn)with the straight line y = n, with n 2 R (see Fig. 3). Thenumber of these solutions is always odd, because of theroot in the origin and the antisymmetry of the functionsg(n) and y = n. When AX are small enough yF = 0 is the onlysolution. For X < 0, there is only the fundamental root forAX P �1, while two new solutions appear for AX < �1 asillustrated in Fig. 3(a). For X > 0, two additional solutionsappear when AX equals the critical value aC = 4.6033 <3p/2. These become four as AX increases as illustrated inFig. 3(b). Of course, in the two cases more solutions ofEq. (5) arise as jAXj continues to increase. Therefore,Eq. (5) has only the solution, yF = 0, when �1 6 AX < aC.Consequently, besides A1(R,X), we have the following rel-evant surface

A3ðR;XÞ ¼ aC=X; ð18Þ

limiting the region where we have only the fundamentalequilibria (this is plotted in Fig. 2). Further increase of jAXjresults in the addition of an even number of solutions ofEq. (5) each time the functions g(n) and y = n becometangent to each other. Then, the number of solutions ofEq. (5) tends to infinity as jAXj goes to infinity.

-1.8 -1.2 -0.6 0.0

Λ

-0.4

-0.2

0.0

0.2

0.4

P

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

C

-2.4

-1.2

0.0

1.2

Λ

(a)

(b)

Fig. 2. (a) Real roots of P±(K) = 0, contributing to define the stability ofthe equilibria xð0Þ� ¼ ð�

ffiffiffiRp

;0;RÞ for C = 0 (thick solid line), C = 4/27 (thinsolid line), C = 2/27 (dashed line), C = �4/27 (dotted line), and C = 8/27(dot-dashed line). (b) Exact solutions of the equation P(K) = 0 as functionsof C. Real parts of eigenvalues are plotted as solid lines and imaginaryparts as dashed lines. The thick lines are for the first (always real)eigenvalue, and the thin lines for the second and third eigenvalues. Fromleft to right the vertical dotted lines, corresponding to C = 0, C = 4/27 andC = 2, separate regions with distinct stability nature.

π/2 -π/4 0 π/4 π/2

-1.0

-0.5

0

0.5

1.0

g/A

ξ A|Ω|=1.4A|Ω|=1.0A|Ω|=0.6

-2π -π 0 π 2πξ|Ω|

-1.0

-0.5

0

0.5

1.0

g/A

ξAΩ=3.6AΩ=4.6AΩ=6.6

(a)

(b)

Fig. 3. Graphical calculation of the roots of Eq. (5) as the intersection ofg(n) = �Asin(Xn) (thick line), with the straight line y = n (thin lines fordifferent values of AX as said in the legends). There is always thefundamental root yF = 0 (filled square). Moreover for jAXj large enoughnew roots appear. (a) The raise of two new roots (filled circles) for X < 0.(b) Emergence of two additional roots (open circles) that immediatelybecome four (filled circles) for X > 0.

344 J.M. González-Miranda / Chaos, Solitons & Fractals 45 (2012) 341–350

If the condition given by �1 6 AX < aC is meted only thefundamental equilibria exist, and the surfaces A1(R,X) andA3(R,X), both orthogonal to the R = 0, tell us where wehave only the fundamental equilibria. Our objective inthe next section will be to gain some insight on the kindof attractors that develop when we have only the funda-mental fixed points, and these are unstable. Because thedivergence of the flow given by Eqs. (1)–(3) is constantand equal to �2 for all ðA;X;RÞ 2 Pþ, we have a contractingflow anywhere the fixed points are unstable, and we ex-pect to have some type of oscillatory solutions.

We have to note that for R < 0, x0 is only unstable wheresecondary equilibria exist. Because of this, in what follows,our study will be restricted to R > 0; that is, to the regionR � Pþ of the parameter space defined as

R ¼ fðA;X;RÞjA; R 2 Rþ; X 2 R; �1 6 AX < aCg: ð19Þ

A case with three unstable fixed points is quite common inthe literature, well known examples are the Lorenz [1], andChua [3] systems. In that sense, we consider it as a regularcase, being non-regular (or special) those with many fixedpoints that are allowed to this system.

3. Nonlinear analysis

As an approach to obtain an insight on the dynamicbehaviors available to the A2 symmetric flow, we have

computed bifurcation diagrams and spectra of Lyapunovexponents, as functions of each one of the system parame-ters A, X and R. Bifurcation diagrams have been con-structed by plotting the maxima, Z(p), of the oscillationsof the variable z(t) obtained from a numerical integrationof Eqs. (1)–(3) for a set of different values of one of thesystem parameters, p 2 {A,X,R}, chosen as bifurcationparameter, while the others are held fixed at the valuessaid above. Lyapunov exponents have been computedthrough standard techniques for systems of differentialequations [27]. Moreover, phase space plots of selected tra-jectories in the system attractors complete this non-linearanalysis. These have been obtained from the integration ofEqs. (1)–(3) by means of a fourth-order Runge–Kutta algo-rithm, and neglecting the dynamics of the first t = 20,000time steps to avoid transitories. Integration steps rangingbetween 0.1 and 0.001 have been tested to ensure struc-tural stability of these solutions. The results presented,unless otherwise stated, have been obtained from the sameinitial condition (x0,y0,z0) = (0.1,0.2,0.3). Additional checksfor structural stability have been performed trying otherinitial condition values. These results are described in thenext subsections.

3.1. The A bifurcation parameter

We studied the dynamics along the vertical line, labeledLA in Fig. 1(a), with X = 1.4, R = 5.2 and A 2 (0,3.2288). Thisis inside the regular region, and is such that the three equi-libria are unstable. We observe in Fig. 4, five qualitatively

Fig. 4. (a) Bifurcation diagram and (b) Lyapunov spectrum for the A2symmetric flow for X = 1.4, R = 5.2 when A is taken as the bifurcationparameter. Vertical dotted lines separate the regions of different dynamicbehavior.

Fig. 5. Projections of periodic trajectories of the A2 symmetric flow ontothe x–y plane for fixed X = 1.4 and R = 5.2. The bifurcation parameter, A,takes increasing values given on top of each plot. In case (a) A = 0.60, thethin line is for initial conditions (0.1,0.2,0.3), and the thick line is for(�0.1,�0.2,0.3).

J.M. González-Miranda / Chaos, Solitons & Fractals 45 (2012) 341–350 345

different dynamic behaviors when A is taken as the bifur-cation parameter. For A 2 (0,1.465) the bifurcation dia-gram is given by two segments that suddenly disappearat A = 1.465, and the Lyapunov spectrum indicates aperiodic behavior. For A 2 [1.465,2.444) the bifurcationdiagram shows a single segment, and the Lyapunov spec-trum a periodic behavior again. From A = 2.444 we observewhat appears to be a period doubling cascade to chaos.This is not the case, as we will see. Properly, we have toconsider three more cases: (i) the dynamics in A 2[2.444,2.915), with a bifurcation diagram given by twolines and a Lyapunov spectrum characteristic of periodicorbits, (ii) the dynamics in A 2 [2.915,3.170), where thebifurcation diagram and the Lyapunov spectrum show aperiod doubling transition to chaos, and (iii) the dynam-ics in A 2 [3.170,3.228), which both, bifurcation diagramand Lyapunov spectrum characterize as chaotic.

This bifurcation diagram and Lyapunov spectrum can bebetter understood by means of plots of trajectories fol-lowed by the system at judiciously chosen values of A.These are presented in Fig. 5. The attractor whose projec-tions are displayed in Fig. 5(a) as thin lines illustrates thedynamics for A 2 (0,1.465). The double line observed isnot a period-2 cycle as one might initially guess, but anasymmetric orbit that has two different maxima for z(t).If the system is started at (x0,y0,z0) = (�0.1,�0.2,0.3) in-stead, we obtain a cycle that is the reflection around thez-axis of the former. That means that for A 2 (0,1.465)the system is periodic and bistable. This bistability is lostat A = 1.465, in a discontinuous transition where thebifurcation diagram abruptly starts to display a single line.This is a signal of a symmetric periodic orbit as the onedisplayed in Fig. 5(b) that occurs in the interval A 2[1.465,2.444). At A = 2.444, we have a continuoustransition where a single symmetric periodic orbit losesits symmetry giving rise to a couple of coexisting periodic

orbits in A 2 [2.444,2.915). One of such orbits is displayedin Fig. 5(c). This look quite alike to Fig. 5(b); however theloss of symmetry is noticeable if one looks to the concavi-ties close to the main diagonal of the frame.

Chaotic dynamics then develops. In the intervalA 2 [2.915,3.170) we have a legitimate cascade of perioddoubling bifurcations to chaos. This corresponds todynamics in one of two coexisting asymmetric, first peri-odic and then chaotic attractors, as illustrated in Fig. 5(d)and (e), respectively. In this region we have chaotic bista-bility in the sense that depending on the initial conditionsthe system can go to an attractor like the one shown inFig. 5(e), or to other (not shown for the sake of clarity)which is the reflection of this around the z-axis. Whenthe two bands in the bifurcation diagram merge, atA = 3.170, bistability is lost and the system dynamics oc-curs in a symmetric attractor like the one studied previ-ously in the literature [6]. This is illustrated in Fig. 5(f).Then, in A 2 [3.170,3.228) we have a single loop symmetricchaos, surrounding the three fixed points.

3.2. The X bifurcation parameter

We consider now the attractors and bifurcations thatdevelop when, with A = 3.2 and R = 5.2 fixed, we changeX in the interval X 2 (�0.2660,1.4385), where only thefundamental equilibria exist and are unstable. This seg-ment is included in the horizontal line labeled LX inFig. 1(a). In this case, we obtain a quite complex bifurcationstructure made by the two chaotic regions at the borders ofthis interval. These are connected by a variety of periodicorbits in the middle. We separate the study of this casein two parts: the left side, X 2 (�0.2660,�0.1400], andthe right side, X 2 (�0.1400,1.4385), of the bifurcationdiagram.

The dynamics in the left side of this interval is shown inFig. 6. The chaotic region occurs for X 2 (�0.2660,�0.177485). It contains some periodic windows, and is fol-lowed by a series of periodic orbits. The type of chaotic

7.0

8.0

9.0

10.0

11.0

12.0

Z

-0.26 -0.24 -0.22 -0.20 -0.18 -0.16 -0.14

Ω

-2.4

-1.8

-1.2

-0.6

0.0

λ

(a)

(b)

Fig. 6. (a) Bifurcation diagram and (b) Lyapunov spectrum for the A2symmetric flow for A = 3.2, R = 5.2, being X the bifurcation parameter.Vertical dotted lines separate the regions of different dynamic behavior.

Fig. 7. Projections of trajectories of the A2 symmetric flow onto the x–yplane for fixed A = 3.2 and R = 5.2. The bifurcation parameter X takesincreasing values given in the labels on top of each plot.

346 J.M. González-Miranda / Chaos, Solitons & Fractals 45 (2012) 341–350

dynamics in the region of negative values of X is displayedin Fig. 7(a)–(c). It is a symmetric attractor, there is no bista-bility, but a new topology is observed here. Instead of a sin-gle band encircling the three fixed points we observe aLorenz-like dynamics with motion in two loops, centeredat xð0Þþ or xð0Þ� , and jumps between these loops when thephase space point approaches x0. As X increases we seehow a chaotic attractor, initially made of a single band[Fig. 7(a)], becomes a double band [Fig. 7(b)], and then afourth band attractor [Fig. 7(c)].

A transition from chaotic to periodic dynamics occurs atX = �0.177485. The emerging periodic attractor is a periodeight symmetric orbit with the same topology then thechaotic attractors, as shown in Fig. 7(d). At first sight thistransition could have been classified as a regular perioddoubling route to chaos. However, this is not the case here,as shown in Fig. 8 where fine details of this transition re-gion are presented. When we cross X = �0.177485 in thedirection of decreasing X, each line in the bifurcation dia-gram, corresponding to a period-8 orbit, splits in two linesthat correspond a chaotic attractor having 16 bands. WhenX is still decreased each of these two bands separate intwo additional chaotic bands, giving a 32 bands chaoticattractor. The chaotic attractors having many bands havea largest Lyapunov exponent smaller than 0.007; thatmeans chaos is very mild in them. Further decrease in X,increases chaos considerably, up to have a largest Lyapu-nov exponents of the order of 0.1. In the process, thesebands get wider, merge with their neighbors until a fullydeveloped single band attractor, like that shown inFig. 7(a) develops.

We show in Fig. 7(e)–(h) example attractors illustrat-ing the periodic orbits that one obtains beyond this tran-sition as X increases. Fig. 7(e) shows an example ofperiod-4 orbit. This appears to result as a period halvingof the period-8 shown in Fig. 7(d), in the same way as

the period-2 attractor in Fig. 7(g) looks like a period halv-ing of that in Fig. 7(c). However, in these cases, the periodhalving process that we observe as X increases is also anatypical one. We illustrate this through an orbit interme-diate between the two above [Fig. 7(f)]. We see that theorbit first becomes asymmetric and then this asymmetricorbit gets symmetric. So we have two bifurcations [Figs. 6and 8(a) and (b)]. One is from a symmetric orbit to abistable asymmetric orbit. This is signaled by two nullLyapunov exponents and a discontinuity in the first deriv-ative in the upper line of the bifurcation diagram. In thesecond bifurcation, a bistable asymmetric attractor be-comes symmetric signaled by two null Lyapunov expo-nent and the merge of couples of lines in the bifurcationdiagram.

In Fig. 7(h), we show the asymmetric period one cyclethat occurs when we approach X = 0 from the negativeside. This results from the period-2 attractor in Fig. 7(g)by means of a symmetry breaking bifurcation as the onejust described, which occurs at X = �0.1523. It is to benoted that the symmetry breaking transitions, [e.g., fromFig. 7(e) to (f), or from Fig. 7(g) to (h)] occur when the sym-metric periodic orbit collides with x0, which is a saddle.This suggest that at the transition point we have an homo-clinic orbit.

11.85

11.90

11.95

12.00

12.05

Z

-0.1776 -0.1772 -0.1768 -0.1764

Ω

-0.008

-0.004

0.000

0.004

0.008

λ

-9.0 -8.0 -7.0y

11.90

11.95

12.00

z

-9.0 -8.0 -7.0y

-9.0 -8.0 -7.0y

(b)

(c) Ω=−0.1870

(a)

(d) Ω=−0.1777 (e) Ω=−0.1774

Fig. 8. Detail of (a) the bifurcation diagram and (b) the two largestLyapunov exponents for the same case shown in Fig. 6. Enhancement ofthe top-left projection of the attractor on the y–z plane for the values of Xindicated on top of each plot.

Fig. 9. (a) Bifurcation diagram and (b) Lyapunov spectrum for the A2symmetric flow for A = 3.2, R = 5.2, being X the bifurcation parameter.Vertical dotted lines separate the regions of different dynamic behavior.

J.M. González-Miranda / Chaos, Solitons & Fractals 45 (2012) 341–350 347

The main features of the dynamics on the right side ofthe X interval, X 2 (�0.1400,1.4385), are summarized inFig. 9. Although the bifurcation diagram and Lyapunovspectra look complex, no really new dynamic behaviors,compared with what has said before, develop here. ForX [ 1.2648 we observe periodic dynamics of differenttypes very similar to those studied until now. These changeby means of symmetry breaking, or by means of perioddoubling bifurcations. Examples of the attractors aroundthe bifurcation changing symmetry that occurs atX = 0.0998 are illustrated in Fig. 7 (i) and (j), and a peri-od-8 orbit resulting from the period doubling cascade thatfollows in Fig. 7(k).

For X P 1.2648 we have a chaotic dynamics similar tothat described in the above subsection on the A bifurcationparameter. We briefly illustrate this with the asymmetricand bistable chaotic attractor that occurs near X = 1.2648[Fig. 7(l)]. This asymmetric chaos becomes symmetric atX = 1.2899 and again asymmetric and bistable atX = 1.24098. We finally note that the abrupt discontinu-ities observed in the bifurcation diagram at X = 0.58 andX = 1.2 are essentially abrupt changes in the size of theattractor.

3.3. The R bifurcation parameter

The dependence on R, shown in Fig. 10, is characterizedby an overall increase of the value of Z(R) with R. We havestudied the bifurcation diagram and Lyapunov spectrumfor A = 3.2, X = 1.4 and R 2 (0.0,12.0). In this interval we

have three unstable fixed points, except for a tiny intervalnear R = 0, in accordance with the results of Section 2[Fig. 2(b)]. We can distinguish a central complex chaoticregion with periodic windows interspersed. This occursfor R 2 (2.27,11.14), and is surrounded by regions ofperiodic dynamics, which present no novelties respectwhat we have seen before. The periodic behavior inR 2 (0.18,1.72) can be of one the two types: the bistableasymmetric one (the bifurcation diagram is given by twolines), or the symmetric one with no bistability (bifurca-tion diagram is given by one line). The same happens inR 2 [11.14,12.00).

A noteworthy feature of this bifurcation diagram is thecoexistence of unlike attractors, mainly in R 2 (1.72,2.56)and in R 2 (5.42,6.21). This coexistence is different fromthe bistability that we have seen before. In the presentcase, the coexisting attractors are not pairs of attractorsthat are identical copies of each other obtained by a sym-metry transformation. Instead, qualitatively differentattractors coexist for the same parameter values. For theinterval (1.72,2.56) this is illustrated in Fig. 11(a) and (b),where we see a inner symmetric cycle, which is the contin-uation of what is obtained for small R. This coexists withother wider symmetric attractor, which is periodic forsmall R, and becomes chaotic as R increases. Being thetwo attractors symmetric, we obtain a class of bistabilitydifferent that the one presented in the above subsections.However, we can have cases where this bistability be-comes tristability. This occurs when one of the two sym-metric coexisting attractors becomes asymmetric andthen bistable. Even fourfold stability can be observed whenthere is the coexistence of two pairs of bistable asymmetricattractors. We can observe this multistability, for example,in R 2 (5.42,6.21).

In the chaotic region we can distinguish a first chaoticinterval R 2 (2.27,3.05) with a small fluctuation of Z(R) thatcorresponds to a symmetric chaotic attractor like the outer

Fig. 10. (a) Bifurcation diagram, and (b) Lyapunov spectrum for the A2symmetric flow for A = 3.2, X = 1.4 and taking R as the bifurcationparameter. Vertical dotted lines separate the regions of different dynamicbehavior.

Fig. 11. Projections of trajectories of the A2 symmetric flow onto the x–yplane for fixed A = 3.2 and X = 1.4. The bifurcation parameter R takesincreasing values as said in the labels on top of each plot. In (a) and (b),the thick inner curves are for initial conditions (0.1,0.2,0.3), the othercurves are for (0.1,0.2,0.9).

348 J.M. González-Miranda / Chaos, Solitons & Fractals 45 (2012) 341–350

one in Fig. 11(b). This is of the type of a single loop sur-rounding the three fixed points seen before. However, ithas different shape, wide, and region of existence. This cha-otic attractor suddenly disappears at R � 3.05, through acrisis [28], becoming an asymmetric bistable periodic cy-cle, as shown in Fig. 11(c). The chaotic and periodic behav-iors inside the interval R 2 [3.05,5.55) are very alike tothose described for Figs. 4 and 9.

Regarding the interval R 2 [5.55,11.14), we first notethat in the left part of this interval, we have a period-3symmetric attractor, as the one shown in Fig. 11(d). ForR < 6.21 this coexist with a symmetric cycle of the typeseen in Fig. 5(b). By increasing R, the period-3 symmetricattractor becomes a chaotic symmetric attractor of thetype shown in Fig. 11(e). This is of the single loop type,but spreads over a wider region of the phase space. More-over, it shows a more complex structure, and has a positiveLyapunov exponent that is roughly twice of that of otherchaotic attractors seen before. The transition from period-icity to chaos is of the type shown in Fig. 4, performed infour steps. First, a symmetry breaking bifurcation trans-forms the period-3 symmetric attractor in an asymmetricbistable one. Second, this turns bistable asymmetricchaotic around R = 6.5. Third, this becomes bistable chaoticasymmetric, and finally chaotic symmetric. Further in-crease of R adds little qualitative new dynamics, only moretransition of the type just described, and more symmetricsingle loop chaotic attractors, like the one in Fig. 11(f),although with its own distinct appearance, phase space re-gion of existence, and having a larger maximal Lyapunovexponent.

We finally note that our study of oscillatory dynamicshas not been restricted to the results presented in this sec-tion. We have computed bifurcation diagrams along otherlines, parallel and diagonal to those shown here. We did

not found more significant new results with respect towhat we have presented. This suggest that, despite oursearch for oscillatory solutions is limited to three orthogo-nal segments, this is enough to obtain the most relevantattractors and bifurcations available to the system in theregular region.

4. Comments and conclusions

We have presented a study of the dynamic behaviorsavailable to the A2 symmetric flow. A linear stability anal-ysis has shown that the dynamics to be expected for thissystem are quite diverse. For one side, we have identifieda set of parameter values, called the regular region, wherethere is a fixed point at the origin and, eventually, twoadditional symmetric ones. In that region we have identi-fied subregions of stability where at least one fixed pointis stable and no oscillations are expected, and subregionswhere all fixed points are unstable and oscillatory dynam-ics are likely. Moreover, we have found that beyond theregular region, the number of fixed points increases inwhat we called the non-regular region, which we left forfuture analysis.

Oscillatory solutions in the regular region display anunusual variety of attractors and bifurcations. These in-clude pairs of non-symmetric periodic and chaotic attrac-tors, which display bistability. There are also symmetricperiodic and chaotic attractors that can be either, singleloops that circle around the three fixed points, or doubleloops with each loop circling around one of the symmetricfixed points. It is to be noted that system having threeunstable fixed points use to follow the typical double-loopLorenz-like dynamics. Attractors, as many shown here,made of a single loop around the three fixed points are rarein the literature. The existence of regions of multistability,where up to four different attractors can coexist alsodeserves to be noted.

J.M. González-Miranda / Chaos, Solitons & Fractals 45 (2012) 341–350 349

The transitions between these different attractors occurthrough bifurcations, which in some cases are quite rareand peculiar. We distinguish three cases that, to theknowledge of the present author, seldom or never beforehave been described in the literature. One is the coexis-tence of two period doubling routes to chaos, when wehave bistable asymmetric attractors, as in SubSection 3.1.Another case is the period doubling through symmetrybreaking, seen in Section 3.2. A third case is the direct jumpfrom a period-8 periodic attractor to a period-16 chaoticone, observed for the X bifurcation parameter, also seenin Section 3.2.

An important result here is the observation of many dif-ferent dynamic behaviors in a single system, even when werestrict ourselves to the regular region. This is unusual,even if we consider that we have studied a model withthree nonlinearities. Such models, usually do not show somuch complexity. For example, recent studies of the Hind-marsh–Rose model of a neuron, showed that it displays avariety of attractors and complex bifurcations [29,30,26],but not to the degree seen here. More illustrative is thecase of the model of NF � jB oscillations [31] that, havingthree nonlinearities, two of them rational functions, onlydisplays periodic oscillations.

We like to note the some of the results presented in thisarticle for the A2 symmetric flow bear a similarity withothers reported in the literature for other flows or maps.In particular, Gonchenko et al. [25], in a study a three-dimensional Hénon map, have reported the existence ofseveral types of attractors, including Lorenz-like attractors,in some domains of the parameter space. Broer et al. [21] ina study of the fattened Arnold map, have found several sce-narios of coexistence of attractors, and the occurrence ofinfinitely many sinks as a parameter is varied. We do nothave found this last phenomenon in our study; however,the emergence of an infinity of fixed points beyond the reg-ular region appears reminiscent of that result by Broeret al. [21]. Scenarios of multistability have also been foundand studied by Vitolo et al. [22] in a three dimensional dis-crete map, obtained from the flow of a Poincaré–Takensnormal form vector field, and aimed to study generic phe-nomena of Hopf-saddle-node bifurcations for fixed points.

Finally, we note that this system has proven to be usefulto study several aspects of chaos synchronization in thepast [6,10,12]. The variety of attractors and bifurcationsseen here, suggest that another possible application isthe mathematical study of attractors and bifurcations. Forexample, the study of the mechanisms behind these attrac-tors and bifurcations could be a possible application of thetechniques for global bifurcations. Some approaches of thatkind that had worked in the past, include the application oflinear stability analysis to fixed points and appropriateinvariant curves as done in Refs. [12,32]. Alternatively,one can resort to the analysis of the system low-dimen-sional dynamics by means of embedded discrete maps asdone in Refs. [24,33]. This approach would ease the appli-cation of analytical techniques, of the kind used in Refs.[20–22]. Moreover, the dynamics when there are morethan three fixed points, appears interesting because sys-tems with many fixed points have received relatively littleattention in the literature. However, according to Thomas

[34], and Sprott and Chlouverakis [35] they may presentinteresting behaviors, such as the so called labyrinth chaos.

Acknowledgment

Research partly supported by DGI.

References

[1] Lorenz E. Deterministic non-periodic flow. J Atmos Sci1963;20:130–41.

[2] Rössler O. An equation for continuous chaos. Phys Lett A1976;57:397–8.

[3] Matsumoto T, Chua LO, Komuro M. The double scroll. IEEE Trans CAS1985;32:798–818.

[4] Sprott JC. Some simple chaotic flows. Phys Rev E 1994;50:R647–50.[5] Sprott JC. Elegant chaos. Singapore: World Scientific Pub. Co.; 2010.[6] González-Miranda JM. Synchronization of symmetric chaotic

systems. Phys Rev E 1996;53:5656–69.[7] González-Miranda JM. Bistable generalized synchronization of

chaotic systems. Comput Phys Commun 1999;121–122:429–31.[8] Guan S, Lai CH, Wei GW. Bistable chaos without symmetry in

generalized synchronization. Phys Rev E 2005;71:036209. 11 pp..[9] Basto M, Semiao V, Calheiros F. Dynamics in spectral solutions of

Burgers equation. J Comput Appl Math 2007;205:296–304.[10] González-Miranda JM. Communications by synchronization of

spatially symmetric chaotic systems. Phys Lett A 1999;251:115–20.[11] Wu L, Zhu S. Multi-channel communication using chaotic

synchronization of multi-mode lasers. Phys Lett A 2003;308:157–61.[12] González-Miranda JM. Generalized synchronization in directionally

coupled systems with identical individual dynamics. Phys Rev E2002;65:047202. 4 pp..

[13] Jovic B, Berber S, Unsworth CP. A novel mathematical analysis forpredicting master–slave synchronization for the simplest quadraticchaotic flow and Ueda chaotic system with application tocommunications. Physica D 2006;216:31–50.

[14] Guan S, Li K, Lai CH. Chaotic synchronization through couplingstrategies. Chaos 2006;16:023107. 9 pp..

[15] González-Miranda JM. Synchronization and control ofchaos. London: Imperial College Press; 2004.

[16] Musielak ZE, Musielak DE. High-dimensional chaos in dissipativeand driven dynamical systems. Int J Bifurcat Chaos Appl Sci Eng2009;19:2823–69.

[17] Banerjee S, editor. Chaos synchronization and cryptography forsecure communications: applications for encryption. Pennsylvania:IGI Global; 2010.

[18] Hogan SJ et al., editors. Nonlinear dynamics and chaos: where do wego from here? UK: Institute of Physics Publishing; 2002.

[19] Wang CW, editor. Nonlinear phenomena research perspectives. NewYork: Nova Science Publishers Inc.; 2007.

[20] Simó C. On the Hénon–Pomeau attractor. J Stat Phys1979;21:465–94.

[21] Broer H, Simó C, Tatjer JC. Towards global models near homoclinictangencies of dissipative diffeomorphisms. Nonlinearity1998;11:667–770.

[22] Vitolo R, Broer H, Simó C. Routes to chaos in the Hopf-saddle-nodebifurcation for fixed points of 3D-diffeomorphisms. Nonlinearity2010;23:1919–47.

[23] Broer H, Simó C, Vitolo R. Bifurcations and strange attractors in theLorenz-84 climate model with seasonal forcing. Nonlinearity2002;15:1205–67.

[24] González-Miranda JM. Observation of a continuous interior crisis inthe Hindmarsh–Rose neuron model. Chaos 2003;13:845–52.

[25] Gonchenko SV, Ovsyannikov II, Simó C, Turaev D. Three-dimensionalHénon-like maps and wild Lorenz-like attractors. Int J Bifurcat ChaosAppl Sci Eng 2005;15:3493–508.

[26] Rech PC. Dynamics of a neuron model in different two-dimensionalparameter-spaces. Phys Lett A 2011;375:1461–4.

[27] Wolf A et al. Determining Lyapunov exponents from a time series.Physica D 1985;16:285–317.

[28] Grebogi C, Ott E, Yorke JA. Crises, sudden changes in chaoticattractors and chaotic transients. Physica D 1983;7:181–200.

[29] González-Miranda JM. Complex bifurcation structures in theHindmarsh–Rose neuron model. Int J Bifurcat Chaos Appl Sci Eng2007;17:3071–83.

350 J.M. González-Miranda / Chaos, Solitons & Fractals 45 (2012) 341–350

[30] Storace M, Linaro D, de Lange E. The Hindmarsh–Rose neuron model:Bifurcation analysis and piecewise-linear approximations. Chaos2008;18:033128. 10 pp..

[31] Krishna S, Jensen MH, Sneppen K. Minimal model of spikyoscillations in NF � jB signaling. PNAS 2006;103:10840–5.

[32] González-Miranda JM. Linear stability analysis of a boundary crisisin a minimal chaotic flow. Physica D 2010;239:322–6.

[33] González-Miranda JM. Highly incoherent phase dynamics in theSprott E chaotic flow. Phys Lett A 2006;352:83–8.

[34] Thomas T. Deterministic chaos seen in terms of feedback circuits:analysis, synthesis, ‘labyrinth chaos’. Int J Bifurcat Chaos Appl SciEng 1999;9:1889–905.

[35] Sprott JC, Chlouverakis KE. Labyrinth chaos. Int J Bifurcat Chaos ApplSci Eng 2007;17:2097–108.