International Symposium on Nonlinear Theory and its ApplicationsXi’an, PRC, October 7–11, 2002

Towards an automatic extraction of bifurcation diagrams

Francois ALARY, Pascal ACCO and Daniele FOURNIER-PRUNARET

LESIA – Laboratoire d’Etude des Systmes Informatiques et Automatiques,INSA – Institut National des Sciences Appliques, DGEI,

135 avenue de Rangueil, 31077 Toulouse, France.http://www.lesia.insa-tlse.frfrancois.alary@insa-tlse.frpascal.acco@insa-tlse.fr

daniele.fournier@insa-tlse.fr

Abstract

Analysis of non-linear dynamical systems is often basedon drawing bifurcation diagram of the parametric plane.A brute force solution commonly used is to scan theparametric plane and plot by colours the order of thehighest attractive cycle found. The scan can be done fordiscrete systems mainly because simulation of continu-ous systems is too long. This paper presents a parallelsoftware, called Ubicus, that uses a cluster of comput-ers to scan the parametric plane of a discrete equation aswell as continuous ones. A simple scan of the parametricplane of the well-known Van Der Pol-Duffing Equation iscompared with theoretical results. Parallelisation allowsto run more simulations and to memorise more singular-ities than classical scans. Ubicus, can build a dynamicaldata-base that will allow in the future, to identify somebifurcation curves. The software uses the free MessagePassing Interface library on Linux systems and its codeis freely distributed by the authors.

Introduction

A useful way to analyse dynamical systems is to scanexistence of attractive cycles over the parametric plane.The result is a coloured parametric plane where coloursrepresent the order of the highest cycle found. For eachpoint of the parametric plane, several different initialconditions are tried until a cycle is found. If all trieslead to an unbounded solution then the point is markedas divergent – white colour. If all tries do not divergeand do not show any cycle of order less than a maximumnumber (usually 15) the point is marked in black colour.In such black regions, chaotic attractive behaviours canbe obtained.

Scanning the parametric plane of a continuous sys-tem with a reasonable accuracy is a very long task fora single computer. The use of parallel computation will

keep simulation time reasonable even if we increase thenumber of tries and memorise all the singularities andthe properties found for each parametric point. A lay-ered parametric scan, with one layer for each colour, willhelp dealing with coexistent cycles. Moreover, the dy-namical data-base can be used to extract automaticallya partial bifurcation diagram.

1. Parallelisation method

The less expensive hardware solution is to use a clusterof personal computers linked by an Ethernet network.In this case, communication between processors is veryslow so data dependencies have to be very low. The timetaken by communications should not reduce the speed upgiven by the number of processors. So the parallelisationmethod which determines directly the performance ofthe system should limit inter-process communication.

We have considered two different parallelisation meth-ods: the owned grains method and the public grainmethod shown in fig. 1. The owned grains method is themore intuitive one. It simply cuts the parametric planeinto pieces, called grains, and allocates a fixed number ofgrains to each processor. The master process sends coor-dinates of the grains to the slaves and then collects oneby one all the results of each slave. Each grain belongsto one processor, the advantage is the minimal quantityof communication but the disadvantage is a low qualityof the load balancing. Each processor is more or lesspowerful and each grain is more or less fast to explore(unbounded domains are very fast to explore and poten-tially chaotic ones are very slow). Consequence is thatmost process are on idle task while the slowest node cal-culates the hardest grain.

To improve the load balancing we choose to use moregrains than processors: this is the public grains method.When a slave process finishes a grain, the master send

GrainsProcessors

Public grains method.

Owned grains method.

P4 finish first and stop P1,P2 and P3.

P3P4

P1P2P3P4

P2P1

G2G1 G3 G4

G5 G6 G7 G8

G9 G10 G11 G12G13 G14 G15 G16

G1 G2 G3 G4

G5 G6 G7 G8

G9 G10 G11 G12

G13 G14 G15 G16

G1 G9 G15

G2 G6 G10 G13 G16

G3 G7 G12 G16

G4 G5 G8 G16G11 G14

P4

P2P1

P3

load−balancing.Tasks−sharing parallelisation and

Figure 1: load balancing

another grain and so on until the last one ( fig. 1).It drastically limits the idle time and improve perfor-mances.

Using the Amdahl law [2] we can determine the ef-fective speedup of the system knowing its parallelisationfactor. Running the same parametric scan for differentnumbers of nodes, we can identify the parallelisation fac-tor of Ubicus.

2. Program validation with Duffing Van Der Polprevious results

In this section, we will use previous results of a verywell-known example of modelisation: the van der Pol-Duffing equation. More precisely, we will see how wellthe bifurcation curves of this equation can match theparametric space structure obtained with Ubicus [3].

d2x

dt2+ ε

dx

dt+ x3 = B cos t + B0 (1)

or : {dxdt = ydydt = −ε y − x3 + B cos t + B0

(2)

2.1. Duffing Van Der Pol equations and bifurca-tion structure

The equation (1) is an order two non-autonomous dif-ferential equation depending on the parameter vectorλ = [B B0 ε]t. The parameter ε will be fixed while(B, B0) will represent the parametric plane. The numer-ical integration of the equation is obtained with a fourth

order Runge-Kutta algorithm, with a constant step ofπ

128 . Then we consider the two-dimensional Poincaremap T defined by:

X = T k(X0, λ) (3)

where X0 = [x0 y0]t is the considered vector of initialconditions for the numerical integration. The singulari-ties admitted by T allow us to characterise the solutionsof (1):

• A fixed point of T corresponds to a solution of (1)with a period τ = 2π. It means that (3) satisfiesX = T (X,λ).

• A k order cycle of T corresponds to a solution of (1)with a period τ = 2kπ. Then (3) satisfies X =T k(X, λ)

Let us introduce some general considerations aboutbifurcation curves. Here is a non-exhaustive list of com-mon bifurcations which can be found in the parameterplane:

• A fold bifurcation curve Λj(k)0

is obtained when oneand only one of the associated eigenvalue with a k-order cycle is S=+1.

• A flip bifurcation curve Λjk is such that one and only

one of the eigenvalue associated with a k-order cycleis S=-1. In these two cases , j is the rotation orderof the cycle.

• A Neimark-Hopf bifurcation curve Γjk corresponds

to two eigenvalues S1,2 = e+−jφ. These curves are

not considered in the text.

In [3],authors deal with bifurcation structures of theVan der Pol-Duffing equation. [3] makes reference tocusp points Cj

q and other structures such as lip and quasi-lip structure (respectively Lk

q and Lkq ).

In order to validate Ubicus, we will consider only bi-furcation curves of type Λj

k involving 1,2 and 4-ordercycles. The parametric plane obtained with Ubicus ispresented in fig. 2.

For each value of the parameters (B,B0), several cycleorders may coexist, so we choose to represent in this pic-ture the cycle of highest order. Also, only singularitiesof order up to 15 are represented for a better vision ofthe parametric plane. Singularities of order 16 or morecorrespond to the black colour, while couples of param-eters (B, B0) which admit no singularity correspond tothe white colour.

However, it is sometimes difficult to know for whichparameter values a bifurcation occurs. Ubicus allows usto select the specific orders we want to see by generating

2 61 3 4 5 7 8 9 10 11 12 1314 15 16+

B1 2 3

3

2

1

4

B0

Figure 2: Duffing parametric sweep done by Ubicus

a layer for each cycle orders. Layers corresponding toorder 1,2 and 4 are represented separately in fig. 4. Theblack layer corresponds to cycle of order higher than 15.Superposition of the layers corresponding to order 1, 2and 4 shown in fig. 3 helps to see the classical period-doubling structure studied in [3].

1 2 4 16+

Figure 3: Bifurcation curves and parametric scan to-gether

3. Improvements and parallelisation factor esti-mation

The purpose of this part is to estimate the time im-

of fixed points in the parametric of order 2 cycles in the parametricplane (blue color) plane (green color)

plane (red color)of order 4 cycles in the parametric of other singularities in the

parametric plane (black color)

Layer 1 shows the localisation Layer 2 shows the localisation

Layer 4 shows the localisation Layer 3 shows the localisation

Figure 4: Presentation of the different layers

provements in function of nodes numbers, i.e., the pro-cessors numbers. In the following part we will use thesimplest theory: the Amdahl Law.

3.1. Definitions and recall of Amdahl law

A classical tool for estimating processing time im-provement in function of the number of nodes is theAmdahl Law [2]. This law is built on the simple hy-pothesis that a program contain a sequential part anda parallel one. Let f be the parallel fraction and s thesequential one, we have by definition:

s + f = 1 (4)

Let T (1) be the processing time with one processor,the Amdahl law gives :

T (1) = T (1)(f + s) = T (1) (f + (1− f)) (5)

If p is the cluster’s processors number, and T (p) theprocessing time with p processors, we have:

T (p) = T (1)(

f

p+ (1− f)

)(6)

Now we can define the processing speedup by :

S(p) =T (1)T (p)

=1

fp + 1− f

(7)

Suppose we have an infinite power of calculation, i.e.,p → ∞, then S(∞) = 1

1−f . This is the Amdahl lawwhich fix the maximal speedup that can be reached withan infinite number of computers. With (6), we can es-timate f by executing several simulations with differentnumbers of processors. Of course, simulations must bedone with the same parameters.

3.2. Identification of Ubicus parallelisation fac-tor

To measure f we have done simulations with 5 proces-sors on a discrete-time system (DPCM system) becausethe execution is faster than in the continuous case (nu-merical integration takes a lot of time). This system ismodelized by :

xn+1 = yn

yn+1 = a yn + b xn + a tanh(p(s− yn))+b tanh(p(s− xn))

(8)

The parametric scan is presented in fig. 5. Results are

a−15

−15 15

15

98764321 5 10 11 12 13 14 15 16+b

Figure 5: Parametric scan of the DPCM system

presented in fig. 6 It represents the restitution time infunction of the number of processors with a simple op-timisation algorithm. We can estimate f and moreover,we can estimate with a reasonable extrapolation the pro-cessing time with a greater number of nodes than 5.

We find the value f=0.98 and, according to the Amdahllaw, it means that the best speedup is 50.

4. Conclusion

A software able to perform parametric scans of dis-crete systems as well as continuous systems in a reason-

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

50

100

150

200

250

300

350

400

450

f=0.98027

T(1)=218.7399

Figure 6: Restitution time of the process with differentnodes number

able amount of time is presented. Ubicus has been vali-dated by a scan of the well-known Van Der Pol-Duffingequation which corresponds to the bifurcation diagramestablished in the literature. Ubicus memorise all thesingularities and their parameters such as points of thecycles. This dynamical data-base will soon allows usto automatically extract bifurcations curves and iden-tify what kind of bifurcation has occurred. Source code,setup instructions can be found on [1]. Up to date in-formations will be found on this site such as developersmanual and other parametric scans of differential equa-tions.

References

[1] P. Acco and F. Alary. Ubicus web page.http://www.lesia.insatlse.fr/˜acco/ubicus/ubicus dyn.html.

[2] G. Amdahl. Validity of the single-processor ap-proach to achieving largescale computing capabili-ties. In AFIPS Press, editor, In Proceedings of the1967 AFIPS Conference, volume 30, pages 483–485,1967.

[3] C. Mira, M. Touzani-Qriouet, and H. Kawakami.Bifurcation structures generated by the nonau-tonomous duffing equation. International Journal OfBifurcation and Chaos, 9(7):1363–1379, 1999.