We continue the study of the structural stability and the bifurcations of planar bimodal linear dynamical systems (BLDS) (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we enlarge the study of the bifurcation diagram of saddle/spiral BLDS to saddle/source BLDS and in particular we study the position of the homoclinic bifurcation with regard to the new improper node bifurcationPostprint (published version

Ferrer Llop, Josep

Peña Carrera, Marta

Susín Sánchez, Antonio

English

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Bifurcation Diagram for Saddle/SourceBimodal Linear Dynamical SystemsJOSEP FERRERUniversitat Polite`cnica de CatalunyaDepartament de Matema`tiquesDiagonal 647, 08028 BarcelonaSPAINjosep.ferrer@upc.eduMARTA PEN˜AUniversitat Polite`cnica de CatalunyaDepartament de Matema`tiquesDiagonal 647, 08028 BarcelonaSPAINmarta.penya@upc.eduANTONI SUSINUniversitat Polite`cnica de CatalunyaDepartament de Matema`tiquesDiagonal 647, 08028 BarcelonaSPAINtoni.susin@upc.eduAbstract: We continue the study of the structural stability and the bifurcations of planar bimodal linear dynamicalsystems (BLDS) (that is, systems consisting of two linear dynamics acting on each side of a straight line, assumingcontinuity along the separating line). Here, we enlarge the study of the bifurcation diagram of saddle/spiral BLDSto saddle/source BLDS and in particular we study the position of the homoclinic bifurcation with regard to the newimproper node bifurcation.Key–Words: Piecewise linear system, structural stability, bifurcation diagram1 IntroductionPiecewise linear systems constitute a class of non-linear systems which have attracted the interest of re-searchers because of their interesting properties andthe wide range of applications from which they arise.Even the planar continuous BLDS (planar continuousbimodal linear dynamical systems, that is, two pla-nar linear subsystems acting in complementary half-planes, assuming continuity in the separating straightline) have complex dynamic behaviors as well as ap-plications (see, for example, [1], [2], [3] and [5]).Our aim is a full characterization of the planarcontinuous BLDS structurally stable and a system-atic study of the bifurcations between them, both interms of the coefficients of the matrices which definethe system. The structural stability of a system guar-antees that its qualitative behavior is preserved un-der small perturbations of their parameters, whereasqualitative changes occur at the bifurcation points.We point out that both concepts (structural stabilityand bifurcation) depend on the equivalence relationwhich precises the idea that two dynamical systemshave the ”same qualitative behavior”. For example,for the equilibrium point of a single (non degenerate)planar linear system, those having positive trace andpositive determinant form a unique (structurally sta-ble)C0-class (sources) whereas they are partitioned infour C1-classes (spirals and nodes as structurally sta-ble classes; improper nodes and starred nodes as bifur-cations). Here we follow [11], where two planar con-tinuous BLDS are equivalent if there is a homeomor-phism of R2, preserving the separating line, whichmaps the orbits of a system into those of the other oneand it is differentiable when restricted to finite peri-odic orbits (see Definition 3). We maintain also thenomenclature in [11].Till now, several partial studies exist concerningequilibrium points, periodic orbits or homoclinic or-bits (see [8], [9], [10], [12]). Our aim is a full inte-gration of these and other behaviors in a complete bi-furcation diagram, in particular analyzing their persis-tence under small perturbations. For example, in [6]it becomes clear that in general the periodic orbits arestructurally stable and that two bifurcations are possi-ble for disappearing: an ordinary homoclinic bifurca-tion and a special kind of Hopf bifurcation. Indeed, inan ordinary Hopf bifurcation the periodic orbit col-lapses to the equilibrium point inside it. Whereasin our case, the spiral inside the periodic orbit doesnot collapse but change from divergent to convergent,through a continuum of periodic orbits. In addition, in[7] we prove that beyond both bifurcations there is nota zone of structural stability, but a sequence of sad-dle/tangency or tangency/saddle bifurcations whoselimit is the corresponding bifurcation.For this global study, the starting point is the re-duced form of the matrices representing a continu-ous BLDS obtained in [4]. Then, in [6] we havespecialized the general criteria for structural stabilityin [11] and we have pointed out that additional spe-cific studies (concerning periodic orbits, saddle-loop(or homoclinic) orbits, saddle/tangency orbits and tan-gency/saddle orbits) are needed when one of the sub-Josep Ferrer et al.International Journal of Mathematical and Computational Methods http://www.iaras.org/iaras/journals/ijmcmISSN: 2367-895X 345 Volume 1, 2016systems is a spiral.As a first goal, we focus our attention in the sad-dle/spiral case because it is the only one where allthese elements can appear so that, more complex be-haviors and applications are expected. Thus, also in[6], we have studied the periodic orbits and the saddle-loop orbits for the case of a saddle/spiral system. Forthis case, in [12] one specifies the conditions for theexistence of saddle-loop orbits. In general, for pla-nar continuous BLDS, in [8] it is proved that there ex-ists at most one saddle-loop orbit or limit cycle, whichthen must be attracting or repelling. All these previouspartial results are collected in Section 3. As referred,in [7] we complete the bifurcation diagram of a sad-dle/spiral system.Here we enlarge this study to the transformationof the considered spiral into a node, through an im-proper node. More precisely, we consider a saddle(with negative trace T ) as left subsystem and a sourceas right subsystem, that is, having positive both the de-terminant  and the trace  . It is well-known that theright subsystem will be an improper divergent nodefor  = 0  2p, a divergent node for  > 0 anda divergent spiral for  < 0. We present figures ofthese transformation.For the spiral case, Theorem 6 says that an homo-clinic orbit appears for an unique value 0 < H < 0,and that there is a finite periodic hyperbolic orbit foreach 0 <  < H . Here we prove that H ! 0+,when T ! 0  and that H !  0 , when T ! 1. Thus the interval  2]0; H [ of structurally sta-ble systems having a finite periodic orbit increases tothe whole ]0; 0[ when the trace of the spiral in theright decreases to  1. As a consequence, the in-terval ]H ; 0[ containing the tangency/saddle bifurca-tion decreases to zero length.Throughout the paper, R will denote the set ofreal numbers, Mnm(R) the set of matrices hav-ing n rows and m columns and entries in R (in thecase where n = m, we will simply write Mn(R))and Gln(R) the group of non-singular matrices inMn(R). Finally, we will denote by e1; : : : ; en the nat-ural basis of the Euclidean spaceRn.2 Structural stability of planar bi-modal linear systemsLet us consider a bimodal linear dynamical system(BLDS) given by two subsystems each one acting in ahalfspace:_x(t) = A1x(t) +B1 if Cx(t)  0;_x(t) = A2x(t) +B2 if Cx(t)  0;where A1; A2 2 Mn(R); B1; B2 2 Mn1(R);C 2M1n(R). We assume that the dynamics is con-tinuous along the separating hyperplane H = fx 2Rn : Cx = 0g; namely, that both subsystems coin-cide for Cx(t) = 0.By means of a linear change in the state variable x(t),we can consider C = (1 0 : : : 0) 2 M1n(R). HenceH = fx 2 Rn : x1 = 0g and continuity along H isequivalent to:B2 = B1; A2ei = A1ei; 2  i  n:We will write from now on B = B1 = B2.Definition 1 In the above conditions, we say that thetriple of matrices (A1; A2; B) defines a continuous bi-modal linear dynamical system. (BLDS.)The placement of the equilibrium points will playa significative role in the dynamics of a BLDS. So, wedefine:Definition 2 Let us assume that a subsystem of aBLDS has a unique equilibrium point, not lying inthe separating hyperplane. We say that this equilib-rium point is real if it is located in the halfspace cor-responding to the considered subsystem. Otherwise,we say that the equilibrium point is virtual.Our goal is to characterize the planar continuousBLDS which are structurally stable in the sense of[11] in terms of the coefficients A1, A2 and B, andto analyze the bifurcations appearing in the boundaryvalues between them.Definition 3 A triple of matrices (A1; A2; B) defin-ing a continuous BLDS is said to be (regu-larly) structurally stable if it has a neighbor-hood V (A1; A2; B) such that for every triplet(A01; A02; B0) 2 V (A1; A2; B) there is a homeomor-phism of R2 preserving the hyperplane H whichmaps the oriented orbits of (A01; A02; B0) into those of(A1; A2; B) and it is differentiable when restricted tofinite periodic orbits.A natural tool in the study of BLDS is simplify-ing the matrices A1; A2; B by means of changes inthe variables x(t) which preserve the qualitative be-havior of the system (in particular, the condition ofstructurally stability). So, we consider linear changesin the state variables space preserving the hyperplanesx1(t) = k, which will be called admissible basischanges. Thus, they are basis changes given by a ma-trix S 2 Gln(R),S = 1 0U W!;W 2 Gln 1(R); U 2Mn 11(R):Josep Ferrer et al.International Journal of Mathematical and Computational Methods http://www.iaras.org/iaras/journals/ijmcmISSN: 2367-895X 346 Volume 1, 2016See [4] for the resulting reduced forms.Also, translations parallel to the hyperplaneH areallowed.3 PreliminariesBy specializing to BLDS the general necessary andsufficient conditions in [11], in [6] one proves the fol-lowing results.Theorem 4 [6] Let us consider planar continuousBLDS.(1) If such a BLDS is structurally stable, then thetriples of matrices representing it can be reduced(by means of an admissible basis change and atranslation parallel to the separating line) to theform:A1 = T 1 D 0!; A2 =  1  0!;B = 0b!; b 6= 0In particular, the only tangency point is (0; 0).(2) If one of the subsystems is a center, a degeneratenode, an improper node or a starred node, thenthe BLDS is not structurally stable.(3) For the remainder BLDS, if none subsystem is areal spiral then the BLDS is structurally stable.Explicitly (for b > 0; when b < 0, we obtain thesymmetric ones) when:the left subsystem is a real saddle, a virtual nodeor a virtual spiralthe right subsystem is a virtual saddle or a realnode(4) Additional conditions must be verified if one ofthe subsystems is a real spiral (in the right half-plane if b > 0):(4.1) A BLDS real saddle/real spiral is struc-turally stable if and only if:(a) the finite periodic orbits are hyper-bolic(b) there are not saddle-loop orbits(c) there are not finite orbits connecting asaddle and a tangency point(4.2) A BLDS virtual node/real spiral is struc-turally stable if and only if condition (a)holds(4.3) A BLDS virtual spiral/real spiral is struc-turally stable if and only if condition (a)holds and also:(a’) the infinite periodic orbit at infinity ishyperbolicRemark 5 In (1) of the above Theorem one can takeb = 1 (by means of a change of scale and a symme-try, if necessary), but we will consider general b 6= 0because of the homogeneity in the obtained formulas.In [6] one focuses on conditions (a), (b) of case(4.1) for divergent spirals ( > 0). Thus, let us assumea BLDS as in (1) of Theorem 4, verifying:- The left subsystem is a (real) saddle, i.e.: D <0; b > 0. In particular, its equilibrium point is( bD ; T bD ), and the invariant manifold cut theseparating line at (0;  b2 ) and (0;  b1 ), where2 < 0 < 1 are the eigenvalues of A1 (1 +2 = T , 12 = D.)- The right subsystem is a (real) spiral, i.e.:  >0; 2 < 4; b > 0. In particular, its equilibriumpoint is ( b ;  b). We write   i,  > 0 theeigenvalues of A2 (2 =  , 2 + 2 = .)We summarize the results in [6] concerning thiscase (4.1). Moreover, we precise the uniqueness ofthe finite periodic orbit in (2.c) (see [8]).Theorem 6 [6] Let us assume the case (4.1) above,that is:b > 0; D < 0; > 0; 2 < 4in (1) of Theorem 4, and let be2 < 0 < 1 the eigenvalues of A1 i;  > 0; the eigenvalues of A2Then, for  > 0:(1) If T  0, then there are not homoclinic orbitsnor finite periodic orbits.(2) If T < 0, the only homoclinic (i.e., saddle-loop)orbit appears for the value H of  verifyingt =1ln(222121   1 +22   2 +)beingexp(t) sin(t ')+ M= 0; +'  t  32+'whereM > 0 and 0 < ' <  are defined byM cos(') =   2; M sin(') = :Moreover, H < TD .Josep Ferrer et al.International Journal of Mathematical and Computational Methods http://www.iaras.org/iaras/journals/ijmcmISSN: 2367-895X 347 Volume 1, 2016(3) If T < 0, a unique finite periodic orbit existsfor 0 <  < H , being attractive hyperbolic(and transverse to the separating axis). No sad-dle/tangency orbits exist.Corollary 7 [6] The systems in case (4.1) with T < 0and 0 <  < H are structurally stable.The following theorem specifies the value of H .Theorem 8 [12] In the conditions of Theorem 6, His the value of  verifying12ln(222121   1 +22   2 +)  (2 arctan 2 2 arctan   11) = 0:Remark 9 For  < 0 one has symmetric results:(1’) If T  0, there are not homoclinic orbits norfinite periodic orbits.(2’) If T > 0, the only homoclinic orbit appears fora unique value H < 0 of  verifying H > TD .(3’) If T > 0, a unique finite periodic orbit exists foreach H <  < 0, being hyperbolic (and trans-verse to the separating axis). Hence, the systemis structurally stable.4 The bifurcation at an impropernodeLet us consider a BLDS (A1; A2; B) as in (1) of The-orem 4, withT < 0; D < 0;  > 0; > 0; b > 0;that is, the left subsystem is a real saddle and the rightone is a real source. From Theorem 4, we haveProposition 10 In the above conditions, let 0 =2p. Then:(1) For  > 0 one has a structurally stable systemsaddle/node.(2) For  = 0 one has a bifurcation sad-dle/improper node.(3) For  < 0 one has a system saddle/spiral, whichis structurally stable if H <  and no tan-gency/saddle orbit appears.Remark 11 When H <  < 0 as in (3) above, in[7] one proves that tangency/saddle orbits appear justfor a decreasing sequence (1; 2; :::)! +H . Thus, in(3) above the system is structurally stable for all 1 < < 0. In particular, the tangent orbit intersects theseparating line in just one additional point.Example 12 We include figures of the transition  <0;  = 0;  > 0 for T =  1; D =  1; = 5 andb = 1, so that 0 = 2p5 = 4:4721.More precisely, Figure 1 and Figure 2 correspondto  = 0   1. Notice that the tangent orbit (as wellas the ones under the spiral/saddle orbit) intersectsthe separating line in just an additional point. Theremainder orbits intersect twice in the main quadrantof the saddle and a third time under it.Figure 3 corresponds to  = 0. Now the tangentorbit does not intersect the separating line because itcannot cross the new invariant line arising from theequilibrium point of the improper node.In Figure 4, corresponding to  = 0 + 1, this in-variant line splits into two of them, giving an ordinarynode.-5 -4 -3 -2 -1 0 1 2 3 4 5 6-8-7-6-5-4-3-2-101tau= 3.4721Figure 1: Fig. corresponding to Example 12,  < 0Our main goal is to determine the variation of Hwith regard to 0 when 0 > T >  1. In other words,the variation of the interval ]0; H []0; 0[ where thestructurally stable finite periodic orbits appear. Orequivalently, the variation of the interval ]H ; 0[]0; 0[ where the tangency/saddle bifurcations appear.Theorem 13 In the above conditions, we fix D;and b. Then:(1) H ! 0+, when T ! 0 .(2) H !  0 , when T !  1.Josep Ferrer et al.International Journal of Mathematical and Computational Methods http://www.iaras.org/iaras/journals/ijmcmISSN: 2367-895X 348 Volume 1, 2016-50 -40 -30 -20 -10 0 10 20 30-100-90-80-70-60-50-40-30-20-10010tau= 3.4721Figure 2: Fig. corresponding to Example 12,  < 0-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3-5-4-3-2-101tau= 4.4721Figure 3: Fig. corresponding to Example 12,  = 0Proof: From Theorem 6, we know that, for eachvalue of T < 0, H is the unique value of  > 0verifying the condition in (2) of Theorem 6, or equiv-alently, the equation in Theorem 8.(1) Clearly, for T = 0 (that is, 1 =p D;2 = p D), the value  = 0 (that is,  = 0;  =p) verifies the equation in Theorem 8.(2) Now, we consider T !  1. We must provethat, then, the equation in Theorem 8 is verifiedif and only if  ! 2p.In order to that, we writex =12; y =p4  2so thatx < 0; 0  y < 2AsT = 1 + 2; 2 < 0 < 1-2 -1 0 1 2 3 4 5 6 7 8-9-8-7-6-5-4-3-2-101tau= 5.4721Figure 4: Fig. corresponding to Example 12,  > 0when T !  1, necessarily 2 !  1 andx! 0 . We must prove that, then, y ! 0+.With the above notation, we have1 = Dx;  =y2;  =2;  =q4  y2so that the equation in Theorem 8 results12ln(1D2x2D2x2  Dp4  y2x+1 p4  y2x+x2 ) ==p4  y2y(2   arctanp4  y2  xy   arctan 2 Dp4  y2xDxy)As0 q4  y2 < 2pwhen x! 0 , for the left term we haveD2x2  Dp4  y2x+1 p4  y2x+x2 ! 12ln(1D2x2D2x2  Dp4  y2x+1 p4  y2x+x2 )! +1On the other hand, as  2   arctanp4  y2  xy   arctan 2 Dp4  y2xDxy 3the only possibility that the right term convergesto +1 is that y ! 0+, as desired.utJosep Ferrer et al.International Journal of Mathematical and Computational Methods http://www.iaras.org/iaras/journals/ijmcmISSN: 2367-895X 349 Volume 1, 20165 ConclusionWe enlarge the study of the bifurcation diagram ofsaddle/spiral BLDS to saddle/source BLDS. In par-ticular, being  the trace of the source, we precise thevariation of the interval of values 0 <  < H forwhich structurally stable finite periodic orbits appear:it converges to 0 if T ! 0, and to 0 <  < 2pwhenT !  1, where T is the trace of the saddle and isthe determinant of the source (assumed fixed).Acknowledgements: The research is supported byDGICYTMTM2011-23892 (first and second author)and TIN2014-52211-C2-1-R partially supported bythe Spanish Ministerio E&C and FEDER founding(third author).References:[1] J. Artes, J. Llibre, J.C. Medrado and M.A. Teix-eira, Piecewise linear differential systems withtwo real saddles, Math. Comput. Simul. 95,2013, pp. 13–22.[2] K. Camlibel, M. Heemels and H. Schumacher,Stability and controllability of planar bimodallinear complementarity systems, Proc. IEEEConf. Decis. Control, 2003, pp. 1651–1656.[3] M. Di Bernardo, D.J. Pagano and E. Ponce,Nonhyperbolic boundary equilibrium bifurca-tions in planar Filippov systems: a case study ap-proach, Int. J. Bifurcation and Chaos 18, 2008,pp. 1377–1392.[4] J. Ferrer, D. Magret and M. Pen˜a, Bimodalpiecewise linear systems. Reduced forms, Int. J.Bifurcation and Chaos 20, 2010, pp. 2795–2808.[5] J. Ferrer, D. Magret and M. Pen˜a, DifferentiableFamilies of Planar Bimodal Linear Control Sys-tems,Math. Probl. Eng., 292813, 2014, pp. 1–9.[6] J. Ferrer, M. Pen˜a and A. Susin, Structural sta-bility of planar bimodal linear systems, Math.Probl. Eng., 892948, 2014, pp. 1–8.[7] J. Ferrer, M. Pen˜a and A. Susin, Bifurcation di-agram of saddle/spiral bimodal linear systems,submitted to Int. J. Bifurcation and Chaos.[8] E. Freire, E. Ponce, F. Rodrigo and F. Torres,Bifurcation sets of continuous piecewise linearsystems with two zones, Int. J. Bifurcation andChaos 8, 1998, pp. 2073–2097.[9] E. Freire, L. Pizarro and A.J. Rodriguez-Luis,Numerical continuation of homoclinic orbitsto non-hyperbolic equilibria in planar systems,Nonlin. Dyn. 23, 2000, pp. 353–375.[10] J. Llibre, M. Ordonez and E. Ponce, On the ex-istence and uniqueness of limit cycles in pla-nar continuous piecewise linear systems withoutsymmetry, Nonlin. Anal. 14, 2013, pp. 2002–2012.[11] J. Sotomayor and R. Garcia, Structural stabil-ity of piecewise-linear vector fields, J. Diff.Eqs. 192, 2003, pp. 553–565.[12] B. Xu, F. Yang, Y. Tang and M. Lin, Homoclinicbifurcations in planar piecewise-linear systems,Discr. Dyn. Nature and Society, 732321, 2013,pp. 1–9.Josep Ferrer et al.International Journal of Mathematical and Computational Methods http://www.iaras.org/iaras/journals/ijmcmISSN: 2367-895X 350 Volume 1, 2016

Bifurcation diagram for saddle/source bimodal linear dynamical systems

We continue the study of the structural stability and the bifurcations of planar bimodal linear dynamical systems (BLDS) (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we enlarge the study of the bifurcation diagram of saddle/spiral BLDS to saddle/source BLDS and in particular we study the position of the homoclinic bifurcation with regard to the new improper node bifurcatio

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Bifurcation diagram for saddle/source bimodal linear dynamical systems

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