On the Takens-Bogdanov Bifurcation in the Chua’s Equation

1722 IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.9 SEPTEMBER 1999 PAPER Special Sec ion on Nonlinea Theo y and I s Applica ions On he Takens-Bogdano Bi u ca ion in he Chua’s Equa ion An onio ALGABA†, Emilio FREIRE††, Es anislao GAMERO††, and Alejand o J. RODR´ IGUEZ-LUIS††,Nonmembe s SUMMARY The analysis o he Takens-Bogdano bi u ca- ion o he equilib ium a he o igin in he Chua’s equa ion wi h a cubic nonlinea i y is ca ied ou . The local analysis p o ides, in fi s app oxima ion, diffe en bi u ca ion se s, whe e he p esence o se e al dynamical beha iou s (including pe iodic, homoclinic and he e oclinic o bi s) is p edic ed. The local esul s a e used as a guide o apply he adequa e nume ical me hods o ob ain a global unde s anding o he bi u ca ion se s. The s udy o he no mal o m o he Takens-Bogdano bi u ca ion shows he p esence o a degene a e (codimension- h ee) si ua ion, which is analyzed in bo h homoclinic and he e oclinic cases. key wo ds: bi u ca ions, oscilla ions, Chua ci cui 1. In oduc ion The main objec i e o his wo k is o p o ide a deep un- de s anding o some non i ial dynamical beha iou e- la ed o he Takens-Bogdano bi u ca ion (double-ze o eigen alue o he linea iza ion ma ix) in he Chua’s equa ion wi h a cubic nonlinea i y. This equa ion mod- els an elec onic ci cui , whose mos impo an ea u es a e i s simplici y (only one nonlinea i y, which we ha e aken as an odd cubic polynomial), and he complex beha iou s ha can exhibi . Some o hese beha iou s a e analy ically explained in he s udy we will pe o m. The analysis o his ci cui has been sou ce o a la ge bibliog aphy (see Ma sumo o e al. [9]). The mos widely conside ed case co esponds o a piecewise linea implemen a ion o he nonlinea de ice (see, e.g., Madan [8] and e e ences he ein). Unde his hypo h- esis, he heo e ical analysis o he s a e equa ions can- no p ofi om many esul s o diffe en iable dynamics and, in pa icula , o bi u ca ion heo y (see, o in- s ance, [5]). We conside he e he Chua’s equa ion wi h a cubic nonlinea i y: ˙x=α(y−ax3−cx), ˙y=x−y+z, (1) ˙z=−βy. We a e in e es ed in hose bi u ca ion aspec s ela ed Manusc ip ecei ed Decembe 1, 1998. Manusc ip e ised Ap il 13, 1999. †The au ho is wi h he Depa men o Ma hema ics, Escuela Poli ´ecnica Supe io , Uni e si y o Huel a, Spain. ††The au ho s a e wi h he Depa men o Applied Ma h- ema ics II, Escuela Supe io de Ingenie os, Uni e si y o Se illa, Spain. o he Takens-Bogdano bi u ca ion ha he equilib- ium a he o igin in he abo e equa ion exhibi s. The impo ance o his bi u ca ion lies in he possibili y o finding global effec s (homoclinic and he e oclinic mo- ions) om a local bi u ca ion s udy (see Ma sumo o e al. [9]). Chua’s equa ion (1) has been analyzed, e.g., by Khibnik e al. [7], Huang e al. [6], Pi ka e al. [10]. In he wo las pape s, a new linea e m is included in he las equa ion o (1), in o de o ake in o accoun small esis i e effec s in he induc o . In Khibnik e al. [7], he analysis is done by keeping fixed a,c, so ha he e is no possibili y o a Takens-Bogdano bi u ca- ion. The au ho s ca y ou he analysis o he Hop bi u ca ion, and es ablish i s connec ion wi h a homo- clinic bi u ca ion. This connec ion can be explained wi h he analysis we will pe o m he e (compa e Fig. 10 o [7] wi h Fig. 1 he e). In ou analysis o he Chua’s equa ion (1), we will conside he non i ial cases a=0,α=0. No e he symme y (x, y, z)→(−x, −y, −z) i ex- hibi s. The o igin is always an equilib ium poin , and he linea iza ion ma ix a his poin is:   −αc α 0 1−11 0−β0  .(2) I is a s aigh o wa d compu a ion o show ha , aking c=cc=0,β=βc=α, he linea iza ion ma ix a he o igin has a double ze o eigen alue and a hi d eigen alue −1. Then, we ha e a bidimensional cen e mani old and a one-dimensional s able mani old. To analyze his linea codimension- wo bi u ca ion, we ake cand βas bi u ca ion pa am- e e s, and look o he bi u ca ion beha iou s co e- sponding o pa ame e alues close o he c i ical ones: cc,βc. 2. No mal Fo m Fi s ly, we conside he Chua’s equa ion wi h he pa- ame e e alua ed a hei c i ical alues. In o de o pu he equa ion in an app op ia e o m, we make he linea ans o ma ion: ALGABA e al: BIFURCATIONS IN CHUA’S EQUATION 1723 (a) (c) (b) (d) Fig. 1 Bi u ca ion se o he Chua’s equa ion, o a=1,α=1.3. (a) Nume ical bi u - ca ion se . (b) Quali a i e bi u ca ion se . The configu a ion o equilib ia and pe iodic o bi s in each zone appea s in Fig. 4. (c) Zoom o he neighbou hood o Takens-Bogdano poin . (d) Zoom o he neighbou hood o he degene a e homoclinic connec ion poin .   x y z  =  α0−α 011 −α1α    X Y Z  , b inging he linea iza ion ma ix (2) o Jo dan o m:   01 00 −1  . In he new a iables, a hi d-o de cen e mani old (compu ed by using a ecu si e p ocedu e de eloped in F ei e e al. [2]) is gi en by Z=α4a(X3−3X2Y+6XY2−6Y3). This app oxima ion enables us o ob ain he fi h-o de educed sys em on he cen e mani old. Nex , we will pu he educed sys em in an app o- p ia e o m. Fo ha , we use nea -iden i y ans o - ma ions leading o no mal o m. Using he algo i hm de eloped in Game o e al. [3], we ob ain he ollowing fi h-o de no mal o m o he educed sys em on he cen e mani old: ˙ X=Y, ˙ Y=a3X3+b3X2Y+a5X5+b5X4Y, (3) whe e a3=−aα4,b 3=3aα3(α−1), a5=3a2α8,b 5=−3a2α7(8α−5). We obse e ha he coefficien a3is always nonze o (al hough i may be posi i e o nega i e). Howe e , he coefficien b3 anishes o he alue α= 1, whe e a degene a e Takens-Bogdano bi u ca ion akes place. To know how he pa ame e s c,βaffec o he no mal o m (3), we suspend he sys em (adding he i ial equa ions ˙c=0, ˙ β= 0) and compu e he cen e mani old o he suspended sys em. A e some com- pu a ions, and neglec ing he highe -o de e ms in he pa ame e s, we pu in co espondence he Chua’s equa- ion (1) – aking c≈ccand β≈βc– wi h: ˙ X=Y, ˙ Y=1X+2Y+a3X3 +b3X2Y+a5X5+b5X4Y, ˙ Z=−Z, whe e 1724 IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.9 SEPTEMBER 1999 1=−α2c, 2=−(β−α)+α(α−1)c. I is a s aigh o wa d compu a ion o show ha hese un olding pa ame e s sa is y he ans e sali y condi- ion: ∂(1, 2) ∂(c, β)   c=0,β=α =α2=0, and hen, he change o pa ame e s c, β by 1, 2is a local diffeomo phism. In nex sec ions we will add ess he s udy o bo h, nondegene a e and degene a e cases. 3. Nondegene a e Cases When bo h a3and b3a e nonze o, we a e dealing wi h a nondegene a e Takens-Bogdano bi u ca ion [5]. I s classifica ion depends on he sign o a3, and he coe - ficien s a5,b5do no play any ole in he subsequen analysis. So, in he s udy o he nondegene a e cases i is enough o compu e he hi d-o de no mal o m, i.e., he no mal o m (3) unca ed up o hi d o de (any- way, he whole fi h-o de no mal o m will be equi ed la e in he analysis o he degene a e case). Depending on he sign o a3(which is de e mined by he sign o he pa ame e a), wo diffe en si ua ions a ise: 3.1 Homoclinic Case This case applies when he coefficien a3is nega i e, ha is, o a>0. We can conside b3<0, whose analysis can be ound in Guckenheime and Holmes [5] ( he s udy o b3>0 can be educed o he abo e one by changing he sign o , X, 2). In he bi u ca ion se a ound he o igin, he ol- lowing codimension-one bi u ca ions a e p esen : •A pi ch o k bi u ca ion o he o igin PI. •A Hop bi u ca ion o he o igin H. •A Hop bi u ca ion o he non i ial equilib ia h. •A homoclinic connec ion o he o igin Hm (in ac , a pai o homoclinic connec ions, due o he sym- me y). •A saddle-node bi u ca ion o pe iodic o bi s SN1. The local app oxima ions o hese bi u ca ions appea in he quo ed book, and hey a e no included he e o he sake o b e i y. I is impo an o no ice ha , as we ha e compu ed 1,2 o fi s o de as a unc ion o pa ame e s c,β, we ha e a use ul s a ing poin o he nume ical analysis we will p esen la e . 3.2 He e oclinic Case This si ua ion co esponds o a3>0. I occu s in Chua’s equa ion (1) when a<0. We can es ic o he case b3<0 ( he case b3>0 can be handled by changing he sign o , X, 2). The ollowing bi u ca- ions a e p esen (see Guckenheime and Holmes [5]): •A pi ch o k bi u ca ion o he o igin PI. •A Hop bi u ca ion o he o igin H. •A he e oclinic connec ion be ween non i ial equi- lib ia H . As abo e, we ha e no included he local exp essions o hese bi u ca ions. 4. Degene a e Cases The complexi y o bi u ca ion beha iou s g ows in he degene a e case co esponding o he anishing o b3, which occu s a he c i ical alue αc= 1. In his case, he fi h-o de e ms in (3) a e necessa y o de e mine he local bi u ca ion beha iou . Fo he c i ical alue o α, we find b5=−9a2α7= 0. This degene a e case is a codimension- h ee si ua- ion, and a hi d un olding pa ame e is equi ed. We will ake 3=α−αc=α−1, oge he wi h 1and 2, o desc ibe his bi u ca ion. In he degene a e cases, he knowledge o he whole fi h-o de no mal o m (3) is equi ed. One o he fi h- o de e ms in he no mal o m (3) — he coefficien a5— can be elimina ed by escaling he ime in e ms o he s a e a iables. This ope a ion do no al e he alues o he emaining coefficien s in he fi h-o de no mal o m. As in he nondegene a e cases, wo diffe en si ua- ions a e possible. Each one will be conside ed in nex subsec ions. 4.1 Homoclinic Case This is he case when a3<0. We can assume b5> 0 ( he case b5<0 can be educed o he abo e by changing he sign o , Y, 2, 3). In Rousseau and Li [14] and Rod ´ıguez-Luis e al. [12], i is shown ha , besides he bi u ca ions p esen in he nondegene a e case (see Sec . 3.1), he ollowing ones appea : •A degene a e Hop bi u ca ion o he o igin Hd. F om he e, a saddle-node bi u ca ion o pe iodic o bi s SN2 eme ges. •A degene a e Hop bi u ca ion o he non i ial equilib ia hd. F om he e, a saddle-node bi u ca- ion o pe iodic o bi s sn eme ges. •A degene a e homoclinic connec ion Hmd. F om he e, wo saddle-node bi u ca ions o pe iodic o - bi s sn and SN3 eme ge. •A cusp o saddle-node bi u ca ions o pe iodic o - bi s C1, whe e SN1 and SN2 collapse. ALGABA e al: BIFURCATIONS IN CHUA’S EQUATION 1725 4.2 He e oclinic Case This co esponds o he case a3>0. We can educe ou s udy o he case b5>0, by changing he sign o , Y, 2, 3i necessa y. Beyond he bi u ca ions p esen in he nondegene a e case (see Sec . 3.2), he ollowing bi u ca ions appea (see Rousseau [13]): •A degene a e Hop bi u ca ion Hd. F om he e, a saddle-node bi u ca ion o pe iodic o bi s SN eme ges. •A degene a e he e oclinic connec ion H d, whe e he he e oclinic connec ion changes i s s abili y. He e, he abo e saddle-node bi u ca ion o pe iodic o bi s SN ends. 5. Nume ical S udy Now, we will look o his ich bi u ca ion beha iou p edic ed by he heo y, a ound his degene a e Takens- Bogdano bi u ca ion, in he Chua’s equa ion. Fi s ly, we ha e selec ed a= 1 in o de o p esen bi u ca ion se s co esponding o he homoclinic case. We will o- cus on he degene a e case ha occu s a cc=0,βc=1, αc=1. The nume ical esul s a e p esen ed in Figs. 1 and 3, co esponding o fixed alues o α=1.3 and 0.8, e- spec i ely. They a e loca ed on bo h sides o he c i ical alue αc= 1 whe e he degene acy akes place. The local esul s achie ed in p e ious sec ions ha e been essen ials as a guide in he use o he adequa e nume ical con inua ion me hods (see Doedel e al. [1], Rod ´ıguez-Luis e al. [11]), in o de o ex end globally he local in o ma ion. In Fig. 1 we ha e aken α=1.3. This is he iches si ua ion om he poin o iew o diffe en bi u ca ion beha iou s. In he bi u ca ion se d awn in (a) we ha e conside ed he ange β∈(0,1.4), c∈(−0.8,1). The nume ical esul s o he quo ed pic u e ha e been ob- ained wi h AUTO [1]. Se e al cu es a e so close ha a e almos indis inguishable. We e e o he quali- a i e cu es (Fig. 1(b)) and he wo zooms ((c)–(d)) o cla i y he bi u ca ion se . All he bi u ca ions p e- dic ed by he analysis o Sec . 3.1 a e p esen . Mo e- o e , a cusp o saddle-node bi u ca ions o pe iodic o - bi s C2, whe e SN3 and SN4 collapse, also appea s. The p esence o his cusp may be ela ed o a highe degene acy in he homoclinic Hm. Fu he s udy will be needed o unde s and i . Fo he momen , we show nume ically ha , in- c easing α, he wo cusps C1 and C2 mee s in a beak- o-beak singula i y. This ac is p esen ed in he bi- u ca ion se s o Fig. 2, whe e we only d aw he ou saddle-node bi u ca ions o pe iodic o bi s. No ice ha , o α=1.45, he saddle-node bi u ca ions SN1and SN2collapse a C1, and SN3and SN4a C2. La e , (a) (b) Fig. 2 Beak- o-beak singula i y o a= 1. Pa ial bi u ca ion se including only he saddle-node bi u ca ion o pe iodic o bi s cu es. (a) Si ua ion a α=1.45, be o e he collision o he cusps. (b) Si ua ion a α=1.5, a e he collision. Fig. 3 Nume ical bi u ca ion se o he Chua’s equa ion o a=1,α=0.8. The configu a ion o equilib ia and pe iodic o bi s in each zone appea s in Fig. 4. o α=1.5, a saddle-node cu e (SN1–SN3) joins TB and Hmd, whe eas SN2–SN4a e no he same cu e. In Fig. 3, he si ua ion is simple , and only he bi- u ca ions p edic ed by he local analysis appea . This bi u ca ion se also illus a es he homoclinic nonde- gene a e case, conside ed in Sec . 3.1. 1726 IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.9 SEPTEMBER 1999 Fig. 4 Configu a ion o equilib ia and pe iodic o bi s in he homoclinic case. Fig. 5 Nume ical bi u ca ion se o he Chua’s equa ion o a=−1, α=1.3. The configu a ion o equilib ia and pe iodic o bi s in each zone appea s in Fig. 7. The configu a ions o equilib ia and pe iodic o bi s p esen in each zone o he bi u ca ion se s, o his homoclinic case, a e depic ed schema ically in Fig. 4. Now, we selec a=−1, o deal wi h he he e- oclinic case. As abo e, we ha e aken α=1.3 and α=0.8. Each alue co esponds o a side o he c i - ical alue αc= 1 whe e he degene acy akes place. The nume ical esul s a e p esen ed in Figs. 5 and 6. The configu a ions o equilib ia and pe iodic o bi s a e ske ched in Fig. 7. All he cu es and configu a ions a e explained by he p e ious heo e ical analysis. In pa - Fig. 6 Nume ical bi u ca ion se o he Chua’s equa ion o a=−1, α=0.8. The configu a ion o equilib ia and pe iodic o bi s in each zone appea s in Fig. 7. icula , he configu a ion in he icini y o a he e oclinic nondegene a e Takens-Bogdano bi u ca ion coincides wi h he si ua ion o Fig. 6. The analysis ca ied ou is a s aigh way o de ec global dynamics om a local analysis. Fo ins ance, in Fig. 8 we show wo global connec ions. Namely, a homoclinic o bi loca ed on he cu e Hm o Fig. 1. I co esponds o he pa ame e alues a=1,α=1.3, β=0.892 and c≈−0.435. We ha e ep esen ed i s p ojec ion on o he xy plane. Also, a he e oclinic o bi loca ed on he cu e H o Fig. 5, co esponding o he ALGABA e al: BIFURCATIONS IN CHUA’S EQUATION 1727 Fig. 7 Configu a ion o equilib ia and pe iodic o bi s in he he e oclinic case. (a) (b) Fig. 8 (a) P ojec ion on he xy plane o he homoclinic con- nec ion ha appea s, o a=1,α=1.3, a β=0.892 and c≈−0.435. (b) P ojec ion on he xy plane o a pai o he e o- clinic connec ions ha appea s, o a=−1, α=1.3, a β=1.295 and c≈0.2052. alues a=−1, α=1.3, β=1.295 and c≈0.2052. These global connec ions, ha nea he Takens- Bogdano bi u ca ion a e plana phenomena, de elop a idimensional s uc u e by mo ing, o example, he pa ame e α. In he plana si ua ion, he pe iodic o - bi s ha e a mono onically inc easing pe iod when ap- p oaching homoclinici y. On he o he hand, when he homoclinic connec ion en e s in he Shil’niko egion, a wiggling beha iou appea s, wi h a sequence o saddle- (a) (b) Fig. 9 (a) Bi u ca ion diag am (pe iod e sus β) o a=1,α= 10, c=−0.5. (b) P ojec ion on he xy plane o he homoclinic connec ion ha appea s, o a=1,α= 10, c=−0.5, β≈0.7705. node and pe iod-doubling bi u ca ions o pe iodic o - bi s (see Glendinning and Spa ow [4]). This beha iou is p esen ed in Fig. 9, whe e we ha e included a bi u - ca ion diag am, plo ing he pe iod agains β. Also, a phase po ai o he Shil’niko homoclinic connec- ion is d awn. No ice he saddle- ocus cha ac e o he equilib ium a he o igin. 6. Conclusions The s udy o he Takens-Bogdano bi u ca ion is a pow- 1728 IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.9 SEPTEMBER 1999 e ul me hod ha p o ides aluable in o ma ion abou pe iodic beha iou and global dynamics. Nea he Takens-Bogdano bi u ca ion, he phe- nomena a e plana , bu mo ing he pa ame e s a away, we can expec ha hey de elop a idimensional s uc u e, and hen hey can p o ide a ou e o chao ic dynamics. In his pape , we ha e ca y ou he analysis o he Takens-Bogdano bi u ca ion o he equilib ium a he o igin in he Chua’s equa ion wi h a cubic nonlinea - i y. De i ing he co esponding no mal o m, we pu in e idence he p esence o degene a e cases. Then, we ob ain heo e ically local and global bi u ca ions, ha p o ide in o ma ion abou pe iodic beha iou s and ho- moclinic and he e oclinic mo ions. The comple ion o he bi u ca ion se equi es nume ical me hods. These allow us o de ec he p esence o a cusp o saddle-node bi u ca ion o pe iodic o bi s, and also o a beak- o- beak singula i y. Mo eo e , ou analysis explains he p esence o se e al codimension- wo bi u ca ions de- ec ed nume ically in Khibnik e al. [7]. Namely, a degene a e Hop bi u ca ion o he o igin, a degene a e homoclinic and a cusp o saddle-node o pe iodic o bi s (see Fig. 10 o [7]). Re e ences [1] E. Doedel, X. Wang, and T. Fai g ie e, “AUTO94, so - wa e o con inua ion and bi u ca ion p oblems in o dina y diffe en ial equa ions,” Applied Ma h. Repo , CIT, 1995. [2] E. F ei e, E. Game o, E. Ponce, and L.G. F anquelo, “An algo i hm o symbolic compu a ion o cen e mani olds,” Lec . No . in Comp. Sci., ol.358, pp.218–230, 1989. [3] E. Game o, E. F ei e, and E. Ponce, “No mal o ms o plana sys ems wi h nilpo en linea pa ,” In . Se . Num. Ma h., ol.97, pp.123–127, 1991. [4] P. Glendinning and C. Spa ow, “Local and global beha - io nea homoclinic o bi s,” J. S a is ical Physics, ol.35, pp.645–696, 1984. [5] J. Guckenheime and P. Holmes, “Nonlinea oscilla ions, dynamical sys ems, and bi u ca ions o ec o fields,” Appl. Ma h. Sci. Se ies, ol.42, Sp inge -Ve lag, 1986. [6] A. Huang, L. Pi ka, and C.W. Wu, “Chua’s equa ion wi h cubic nonlinea i y,” In . J. Bi u ca ion and Chaos, ol.6, pp.2175–2222, 1996. [7] A. Khibnik, D. Roose, and L.O. Chua, “On pe iodic o bi s and homoclinic bi u ca ions in Chua’s ci cui wi h a smoo h nonlinea i y,” In . J. Bi u ca ion and Chaos, ol.3, pp.363– 384, 1993. [8] R.A. Madan, Chua’s Ci cui : A Pa adigm o Chaos, Wo ld Scien ific Publ., Singapo e, 1993. [9] T. Ma sumo o, M. Komu o, H. Kokubu, and R. Tokunaga, Bi u ca ions. Sigh s, Sounds, and Ma hema ics, Sp inge - Ve lag, Tokyo, 1993. [10] L. Pi ka, C.W. Wu, and A. Huang, “Lo enz equa ion and Chua’s equa ion,” In . J. Bi u ca ion and Chaos, ol.6, pp.2443–2489, 1996. [11] A.J. Rod ´ıguez-Luis, E. F ei e, and E. Ponce, “A me hod o homoclinic and he e oclinic con inua ion in wo and h ee dimensions,” NATO ASI Se ies C, ol.313, pp.197– 210, 1990. [12] A.J. Rod ´ıguez-Luis, E. F ei e, and E. Ponce, “On a codi- mension 3 bi u ca ion a ising in an au onomous elec onic ci cui ,” In . Se . Num. Ma h., ol.97, pp.301–306, 1991. [13] C. Rousseau, “Uni e sal un olding o a singula i y o a sym- me ic ec o field C∞-equi alen o y∂ ∂x +(±x3±x6y)∂ ∂y ,” Lec . No . in Ma h., ol.1455, pp.334–355, 1990. [14] C. Rousseau and C. Li, “A sys em wi h h ee limi cy- cles appea ing in a Hop bi u ca ion and dying in a homo- clinic bi u ca ion: The cusp o o de 4,” J. Diff. Eq., ol.79, pp.132–167, 1989. An onio Algaba was bo n in Pe˜na oya-Pueblonue o (Spain). He ecei ed he Licenciado en Ciencias Ma em´a icas deg ee in 1979 and he Doc- o en Ciencias Ma em´a icas deg ee in 1996, bo h om he Uni e si y o Se ille. Since Oc obe 1989, he has been wi h he Depa men o Applied Ma hema ics a he Uni e si y o Huel a, whe e he s a ed wi h a esea ch g an in 1989 and became P o eso Ti ula in 1998. His esea ch in- e es lies in he fields o plana ec o fields, bi u ca ions in dynamical sys ems wi h applica ions o elec onic ci cui s. Emilio F ei e was bo n in Reus (Spain). He ecei ed he B.S. deg ee in elec ical enginee ing in 1975 and he Ph.D. deg ee in 1982, bo h om he Uni- e si y o Se ille. Since Oc obe 1980, he has been wi h he Depa men o Applied Ma hema ics a he Uni e si y o Se ille, whe e he became P o esso in 1990. His esea ch in e es lies in he fields o bi- u ca ions in dynamical sys ems (ODEs) wi h applica ions o elec onic ci cui s. Es anislao Game o was bo n in Se ille (Spain). He ecei ed he Li- cenciado en Ciencias Ma em´a icas de- g ee in 1986 and he Doc o en Ciencias Ma em´a icas deg ee in 1990, bo h om he Uni e si y o Se ille. Since Oc obe 1986, he has been wi h he Depa men o Applied Ma hema ics a he Uni e si y o Se ille, whe e he became P o eso Ti ula in 1993. His esea ch in e es lies in he fields o bi u ca ions in low-dimensional dynamical sys ems (ODEs) and hei applica ions (mainly o elec onic sys ems). Alejand o J. Rod ´ıguez-Luis was bo n in Le´on (Spain). He ecei ed he Licenciado en F´ısica deg ee in 1984 and he Doc o en Ciencias F´ısicas deg ee in 1991, bo h om he Uni e si y o Se ille. Since Oc obe 1986, he has been wi h he Depa men o Applied Ma hema ics a he Uni e si y o Se ille, whe e he be- came P o eso Ti ula in 1993. His e- sea ch in e es lies in he fields o bi u ca- ions in low-dimensional dynamical sys- ems (ODEs) and hei applica ions (mainly o elec onic sys- ems). View publica ion s a sView publica ion s a s