Equivariant Versions of Odd Number Theorem Edward Hooton∗ Pavel Kravetc∗ Dmitrii Rachinskii∗ 7 1 0 2 Abstract n We consider the problem of stabilization of unstable periodic solu- a J tionstoautonomoussystemsbythenon-invasivedelayedfeedbackcontrol known as Pyragas control method. The Odd Number Theorem imposes 1 an important restriction upon the choice of the gain matrix by stating ] a necessary condition for stabilization. In this paper, the Odd Number S Theorem is extended to equivariant systems. We assume that both the D uncontrolled and controlled systems respect a group of symmetries. Two . types of results are discussed. First, we consider rotationally symmetric h systemsforwhichthecontrolstabilizesthewholeorbitofrelativeperiodic t a solutionsthatformaninvarianttwo-dimensionaltorusinthephasespace. m Second, we consider a modification of the Pyragas control method that [ has been recently proposed for systems with a finite symmetry group. This control acts non-invasively on one selected periodic solution from 1 the orbit and targets to stabilize this particular solution. Variants of the v Odd Number Limitation Theorem are proposed for both above types of 7 systems. The results are illustrated with examples that have been pre- 9 viously studied in the literature on Pyragas control including a system 2 0 oftwosymmetricallycoupledStewart-Landauoscillatorsandasystemof 0 two coupled lasers. . 1 Keywords: Stabilization of periodic orbits, Pyragas control, delayed feed- 0 back, S1-equivariance, finite symmetry group. 7 1 : 1 Introduction v i X Stabilization of unstable periodic solutions is an important problem in applied r nonlinearsciences. AnelegantmethodsuggestedbyPyragas[12]istointroduce a delayed feedback with the delay equal, or close, to the period T of the target unstable periodic solution x∗(t) to the uncontrolled system x˙(t) = f(t,x(t)). Thisfeedbackcontrolistypicallylinear,andthecontrolledsystemhastheform x˙(t)=f(t,x(t))+K(x(t−T)−x(t)), x∈Rn, (1) where K is an n×n gain matrix. Since the explicit form of the target cycle is notrequiredthismethodiseasytoimplementandwidelyapplicable[13,18,19]. ∗DepartmentofMathematicalSciences,TheUniversityofTexasatDallas,TX,USA 1 Pyragas control is often referred to as non-invasive, since x∗(t) is an exact solution of both the uncontrolled and controlled systems if the delay exactly equals the period of x∗. The question is how to choose the gain matrix K to ensure that x∗ is a stable solution of (1). Certain limitations to the method of Pyragas are known. It was proved in [8] that if f depends explicitly on t and the target periodic solution x∗ of the uncontrolled non-autonomous system is hyperbolic with an odd number of real Floquet multipliers greater than one, then for any choice of K, the function x∗ isanunstablesolutionof (1). In[6], thistheoremwasmodifiedtodealwiththe case of autonomous systems x˙(t)=f(x(t))+K(x(t−T)−x(t)). (2) In this case, the theorem provides necessary conditions on the control matrix K to allow stabilization of an unstable hyperbolic cycle x∗ of the autonomous system x˙(t)=f(x(t)). (3) These necessary conditions can be used as a guide when constructing the gain matrix K. In this paper, we consider systems (3), which respect some symmetry. Peri- odicsolutions(cycles)ofsuchsystemsnaturallycomeingrouptheoreticorbits, hence there are multiple cycles with the same period. This complicates the applicability of Pyragas control because the control acts non-invasively on all those cycles. In particular, for systems with a continuous group of symmetries, a connected continuum of cycles, which all have the same period, is generic. Thecyclesthatformthecontinuumarenothyperbolicandhencedonotsatisfy conditions of theorems from [6,8]. On the other hand, a modification of Pyragas control was proposed in [4] for systems with a finite group of symmetries in order to make the control non-invasive only on one selected target cycle, which has been chosen for sta- bilization. The symmetry of a cycle x∗ is described by a collection of pairs (A ,T ) where A ∈ GL(n) and T is a rational fraction of the period of x∗. g g g g The symmetry is expressed by the property that A x∗(t)=x∗(t+T ) (4) g g for all the pairs (A ,T ). To stabilize x∗, it was suggested in [15] to modify (2) g g by selecting one particular g and introducing control as follows: x˙(t)=f(x(t))+K(A x(t−T )−x(t)). (5) g g In [1,4,11] this type of control was applied to stabilize small amplitude cy- cles born via a Hopf bifurcation, while in [17] analysis of the stability of large amplitudecycleswasdonebyexploitingtheadditionalS1 symmetryofStewart- Landau oscillators. In this paper, we extend the odd-number limitation type results considered in[6]totreatthecasewhencontroloftheform(5)isappliedtoasystemwitha 2 finitegroupofsymmetries(Section2);and,thecasewhenthestandardPyragas controlsuchasin(2)isappliedtoatargetcycle,whichisnothyperbolic,because the system is S1-symmetric (Section 3). Analytical results are illustrated with examples. 2 Modified Pyragas control of systems with fi- nite symmetry group 2.1 Necessary condition for stabilization Supposethatsystem(11)hasaperiodicsolutionx∗ withperiodT. Assumethat thissystemrespectssomegroupofsymmetries,andforoneparticularg relation (4) holds. We denote by Φ(t) the fundamental matrix of the linearization y˙ =B(t)y, B(t):=Df (x∗(t)), (6) x ofsystem(11)nearx∗(t),whereDf denotestheJacobimatrixoff. Condition x (4) implies that A−1Φ(T )ψ(0)=ψ(0), ψ(t):=x˙∗(t), (7) g g i.e. the matrix A−1Φ(T ) has an eigenvalue 1. We assume that g g (H ) 1 is a simple eigenvalue for the matrix A−1Φ(T ). 1 g g Following [14], we introduce a modified Pyragas control as in (5), where we assume that KA =A K. (8) g g This commutativity property can be a natural restriction on feasible controls. For example, it is typical of laser systems. On the other hand, gain matri- ces, which are simple enough to allow for efficient analysis of stability of the controlled equation (5), also usually satisfy condition (8) (cf. [11,16]). Let D(cid:48) denote the transpose of a matrix D. Using (H ), denote by ψ† the 1 normalizedadjointeigenvectorwiththeeigenvalue1forthematrix[A−1Φ(T )](cid:48): g g [A−1Φ(T )](cid:48)ψ† =ψ†; ψ†·ψ(0)=1, g g 0 0 0 where dot denotes the standard scalar product in Rn. Furthermore, denote by ψ†(t) the solution of the initial value problem y˙ =−B(cid:48)(t)y; y(0)=ψ†. 0 Since the fundamental matrix of system y˙ =−B(cid:48)(t)y is (Φ−1(t))(cid:48), ψ†(t)=(Φ−1(t))(cid:48)ψ†. (9) 0 3 Note that relation (4) implies A ψ(t)=ψ(t+T ); A(cid:48)ψ†(t)=ψ†(t−T ). g g g g FinallydenotebyN thenumberofrealeigenvaluesµofthematrixA−1Φ(T ), g g which satisfy µ>1. Theorem 2.1 Assume that conditions (H ) and (8) hold. Let 1 (cid:32) (cid:33) (cid:90) Tg (−1)N 1+ ψ†(t)·Kψ(t)dt <0. (10) 0 Then, x∗(t) is an unstable periodic solution of the controlled system (5). Hence, the inequality opposite to (10) is a necessary condition for stabilization of the periodic solution x∗. This necessary condition restricts the choice of the gain matrix K. 2.2 Example AsanillustrativeexampleofTheorem2.1weconsiderthesystemoftwoidentical diffusely coupled Landau oscillators described in complex form by z˙ =(α+i+γ|z |2)z +a(z −z ), 1 1 1 2 1 (11) z˙ =(α+i+γ|z |2)z +a(z −z ) 2 2 2 1 2 with z ,z ∈ C. Here α and a > 0 are real parameters while γ is a complex 1 2 parameter with Reγ > 0. When α is treated as a bifurcation parameter, this system undergoes two sub-critical Hopf bifurcations, the first at α = 0 giving rise to a fully synchronized branch of solutions and the second at α=2a giving rise to an anti-phase branch. The anti-phase branch is defined for α < 2a and is given explicitly by the formula z∗(t)=−z∗(t)=r(α)eiω(α)t, (12) 1 2 (cid:112) where r =r(α):= (2a−α)/Reγ and ω =ω(α):=1+r2(α)Imγ. In [4] this branchwasstabilizedbyintroducingequivariantPyragascontroltosystem(11) in the following way: z˙ =(α+i+γ|z |2)z +a(z −z )+b(z (t−π/ω)−z )), 1 1 1 2 1 2 1 (13) z˙ =(α+i+γ|z |2)z +a(z −z )+b(z (t−π/ω)−z )) 2 2 2 1 2 1 2 with a complex gain parameter b. To study stability of the anti-phase branch close to the bifurcation point in system (13), linear stability analysis of the origin combined with explicit knowledge of the branch made it possible to find sufficient conditions under which for some interval of α sufficiently close to 2a the branch is stable. 4 Due to the simple nature of the Landau oscillator, explicit computation of the fundamental matrix of the linearization of system (11) near the solution (12)allowsustocomputeexpression(10). Thisgivesusthefollowingnecessary condition for stabilization of any of the anti-phase cycles (12): ReγReb+ImγImb 1+π <0. (14) ωReγ Atthepointα=2a,ω =1,thiscoincideswithformula(6.10)from[4]which, together with other conditions, defines the stability domain for small periodic orbits. In particular, for a fixed gain parameter b, condition (14) provides an upper bound for the interval of α where the anti-phase cycle is stable. 2.3 Proof of Theorem 2.1 Linearizing system (5) near x∗ gives y˙(t)=B(t)y(t)+K(A y(t−T )−y(t)). (15) g g Toprovethatx∗ isanunstableperiodicsolutionof (5)wewillshowthatsystem (15) has a solution y∗(t)=µt/Tgp(t), A p(t−T )=p(t) (16) µ g g with µ>1, where the relation A p(t−T )=p(t) ensures that p is periodic. It g g is easy to see that if the ordinary differential system y˙ =(cid:0)B(t)+(µ−1−1)K(cid:1)y (17) has a solution y of type (16), then y is also a solution of (15). Denote by µ µ Ψ (t) the fundamental matrix of (17). µ Lemma 2.2 If for some µ > 1 the matrix A−1Ψ (T ) has the eigenvalue µ, g µ g then system (17) has a solution of type (16) and hence the periodic solution x∗ of (5) is unstable. The proof of this lemma is presented in the next subsection. In order to use Lemma 2.2, we consider the characteristic polynomial F(µ):=det(cid:0)µId−A−1Ψ (T )(cid:1) g µ g of the matrix A−1Ψ (T ). Observe that equation (17) with µ = 1 coincides g µ g with (6), hence Ψ =Φ and therefore condition (H ) implies F(1)=0. We are 1 1 going to show that relation (10) implies F(1+ε)<0, 0<ε(cid:28)1. (18) Since F(µ) → +∞ as µ→+∞, relation (18) implies that F has a root µ > 1 and therefore the conclusion of Theorem 2.1 follows from (18) by Lemma 2.2. 5 Setting µ=1+ε and t=T in the identity g (cid:18) (cid:90) t (cid:19) Ψ (t)=Φ(t) Id+(µ−1−1) Φ−1(s)KΨ (s)ds µ µ 0 and using the fact that Ψ (T )=Φ(T )+O(ε), we obtain the expansion 1+ε g g (cid:90) Tg Ψ (T )=Φ(T )(Id−εQ)+O(ε2), Q:= Φ−1(t)KΦ(t)dt. 1+ε g g 0 Therefore, F(1+ε)=det(cid:0)Id−A−1Φ(T )+ε(cid:0)Id+A−1Φ(T )Q(cid:1)(cid:1)+O(ε2). (19) g g g g LetusdenotebyLthetransitionmatrixtoabasisinwhichthematrixA−1Φ(T ) g g assumes the Jordan form and agree that ψ(0) is the first vector of this basis (cf. (7)), i.e. Le =ψ(0), e :=(1,0,...,0)(cid:48) ∈Rn. (20) 1 1 Inthisbasis, thematrixId−L−1A−1Φ(T )LhastheJordanstructurewiththe g g diagonal entries 0,1−µ ,1−µ ,...,1−µ , where µ are the eigenvalues of 2 3 n k A−1Φ(T ) different from the simple eigenvalue 1. With this notation, formula g g (19) implies n (cid:89) F(1+ε)=εM (1−µ )+O(ε2), (21) 11 k k=2 where M :=Id+L−1A−1Φ(T )QL; M =e ·Me . (22) g g 11 1 1 Formula (20) implies that (L−1)(cid:48)e =ψ†, hence 1 0 (cid:90) Tg e ·L−1QLe = ψ†·Φ−1(t)KΦ(t)ψ(0)dt. 1 1 0 0 Combining this with (20) and ψ(t)=Φ(t)ψ(0), we obtain (cid:90) Tg e ·L−1QLe = ψ†(t)Kψ(t)dt. 1 1 0 Since the first row of the matrix L−1A−1Φ(T )L is (e )(cid:48) = (1,0,...,0), we see g g 1 from (31) that (cid:90) Tg M =1+e ·L−1QLe =1+ ψ†(t)Kψ(t)dt. 11 1 1 0 Hence (30) implies (cid:32) (cid:33) (cid:90) Tg sgnF(1+ε)=(−1)N 1+ ψ†(t)Kψ(t)dt , 0 where N is the number of eigenvalues µ satisfying µ >1. Thus, formula (10) k k indeed implies (18). 6 2.4 Proof of Lemma 2.2 Let us denote C(t):=B(t)+(µ−1−1)K. The main ingredient for the proof is the identity C(t+T )=A C(t)A−1, g g g whichisasimpleconsequenceofthefactsthatDf(A x)=A Df(x)A−1,x∗(t+ g g g T )=A x∗(t), and A K =KA . g g g g Tocompletetheproofdenotebyν theeigenvectorofA−1Ψ withtheeigen- g µ value µ and consider the solution y (t) of (17) with y (0) = ν. It is clear that 0 0 y (t+T ) satisfies the initial value problem 0 g (cid:40) y˙ =C(t+T )y =A C(t)A−1y , 1 g 1 g g 1 (23) y (0)=Ψ(T )ν. 1 g By the change of variables y = A−1y , we can see that the solution of (23) is 2 g 1 given by y (t+T )=y (t)=A Ψ(t)A−1Ψ(T )ν. 0 g 1 g g g However by assumption A−1Ψ(T )ν =µν, hence g g y (t+T )=µA Ψ(t)ν =µA y (t), 0 g g g 0 which proves the lemma. 3 Pyragas control of systems with S1 spatial sym- metry 3.1 Necessary condition for stabilization Suppose that equation (11) is S1-equivariant: f(eθJx)=eθJf(x) (24) for all θ ∈ R, x ∈ RN, where the skew-symmetric non-zero matrix J satisfies e2πJ = Id. We assume that (11) has a periodic solution x∗(t) of a period T, which is not a relative equilibrium. Hence, equation (11) has an orbit of T- periodicnon-stationarysolutionseθJx∗(t+τ)witharbitraryθ,τ. Therefore,the linearization (6) has two linearly independent zero modes (periodic solutions): ψ (t)=x˙∗(t), ψ (t)=Jx∗(t) (25) 1 2 with the Floquet multiplier 1. We additionally assume that (H ) The eigenvalue 1 of the monodromy matrix Φ(T) of system (6) has 2 multiplicity exactly 2. 7 Then, there are two adjoint eigenfunctions (periodic solutions of equation y˙ =−B(t)y) that can be normalized as follows: ψ†(t)·ψ (t)=ψ†(t)·ψ (t)≡1, ψ†(t)·ψ (t)=ψ†(t)·ψ (t)≡0. (26) 1 1 2 2 1 2 2 1 Theorem 3.1 Assume that condition (H ) holds. Let 2 (−1)N(cid:0)(1+c )(1+c )−c c (cid:1)<0, (27) 11 22 12 21 where N is the number of real eigenvalues µ of the monodromy matrix Φ(T), which satisfy µ>1, and (cid:90) T c = ψ†(t)·Kψ (t)dt. (28) ij j 0 Then, x∗(t) is an unstable periodic solution of the controlled system (1). 3.2 Proof of Theorem 3.1 Uptotheasymptoticexpansion(19)theproofofTheorem3.1isamodification of the proof of Theorem 2.1 where A is replaced by the identity matrix Id and g T is replaced by T. In this case, the counterpart of relation (19) is given by g (cid:32) (cid:32) (cid:33) (cid:33) (cid:90) T F(1+ε)=det Id−Φ(T)+ε Id+Φ(T) Φ−1(t)KΦ(t)dt +O(ε2) . 0 (29) We again denote by L the transition matrix to a basis in which the matrix Φ(T)assumestheJordanformandagreethatψ (0)andψ (0)arethefirstand 1 2 second vector of this basis (cf. (25)), i.e. Le =ψ (0), e :=(1,0,...,0)(cid:48) ∈Rn, 1 1 1 Le =ψ (0), e :=(0,1,...,0)(cid:48) ∈Rn. 2 2 2 In this basis, the matrix Id − L−1Φ(T)L has the Jordan structure with the diagonal entries 0,0,1−µ ,...,1−µ , where µ are the eigenvalues of Φ(T), 3 n k which are different from 1. With this notation, formula (29) implies n (cid:89) F(1+ε)=ε2(M M −M M ) (1−µ )+O(ε3), (30) 11 22 12 21 k k=3 where (cid:32) (cid:33) (cid:90) T M :=Id+L−1Φ(T) Φ−1(t)KΦ(t)dt L; M =e ·Me . (31) ij i j 0 The same argument as in the proof of Theorem 2.1 shows that sgnF(1+ε)=(−1)N(cid:0)(1+c )(1+c )−c c (cid:1), 11 22 12 21 where c is defined by (28). Combining this with the case of Lemma 2.2 where ij A =Id and T =T, and the fact that F(µ)→+∞ as µ→+∞ completes the g g proof. 8 3.3 Example In order to illustrate Theorem 3.1, we consider a model of two coupled lasers, see for example [23]. In dimensionless form, the rate equations describing this system can be written as E˙ = iδE +(1+iα)N E +ηe−iϕE , (32) 1 1 1 1 2 N˙ = ε(cid:2)J −N −(1+2N )|E |2(cid:3), (33) 1 1 1 1 E˙ = (1+iα)N E +ηe−iϕE , (34) 2 2 2 1 N˙ = ε(cid:2)J −N −(1+2N )|E |2(cid:3), (35) 2 2 2 2 where the complex-valued variables E , E are optical fields and the real- 1 2 valued variables N , N are carrier densities in two laser cavities, respectively. 1 2 This system is symmetric under the action of the group S1 of transformations (E ,N ,E ,N ) → (cid:0)eiθE ,N ,eiθE ,N (cid:1). Hence, the system admits relative 1 1 2 2 1 1 2 2 equilibria of the form (E ,N ,E ,N )=(cid:0)a eiωt,n ,a eiωt,n (cid:1) (36) 1 1 2 2 1 1 2 2 with ω,n ,n ∈ R and a ,a ∈ C. The problem of stabilization of unstable 1 2 1 2 relative equilibria for this system was considered in [5]. System (32)–(35) can also have relative periodic orbits, i.e. solutions of the form (eiωtE∗(t),N∗(t),eiωtE∗(t),N∗(t)), (37) 1 1 2 2 whereE∗(t),N∗(t),E∗(t),N∗(t)areT-periodic. Inthepresentsectionwechoose 1 1 2 2 a relative periodic solution as a target state for stabilization. 0.2 0.1 N1 0 -0.1 H -0.2 -2.5 -2 -1.5 -1 -0.5 0 ' Figure1: BifurcationdiagramobtainedwithnumericalpackageDDE-BIFTOOL[2,3] for system (32)–(35). Thin lines: relative equilibriua; thick lines: relative periodic solutions. Solid and dashed lines represent stable and unstable parts of the branches, respectively. H: subcritical Hopf bifurcation point; gray dot: unstable periodic orbit targetedforstabilizationbyPyragascontrol. Parametersareε=0.03,J =1,η=0.2, δ=0.3, α=2. 9 In order to stabilize the solution (37), we add the modified Pyragas control term E (t):=b eiβ(cid:0)e−iωTE (t−T)−E (t)(cid:1) (38) b 0 1 1 to the right hand side of equation (32). Here the parameters b >0 and β ∈R 0 measure the amplitude and phase of the control, respectively; and T, ω are the parameters of the target relative periodic solution (37). Introducing the rotat- ing coordinates (E˜ ,N˜ ,E˜ ,N˜ )=(e−iωtE ,N ,e−iωtE ,N ) transforms equa- 1 1 2 2 1 1 2 2 tions(32)–(35)toanautonomoussystemthathasanorbitofnon-stationaryT- periodicsolutions(eiθE˜ ,N˜ ,eiθE˜ ,N˜ )=(E∗(t+τ),N∗(t+τ),E∗(t+τ),N∗(t+ 1 1 2 2 1 1 2 2 τ)) with arbitrary θ,τ. This change of variables transforms the control term (38) to the standard Pyragas form E˜ (t)=b eiβ(E˜ (t−τ)−E˜ (t)). (39) b 0 1 1 Hence,wecanuseTheorem3.1toestablishthevaluesofb andβ forwhichthe 0 control cannot stabilize the solution (37). Following the analysis presented in [5], we use the phase ϕ of coupling be- tween the lasers as the bifurcation parameter. Varying ϕ one observes Hopf bifurcations on the branches of relative equilibria. These bifurcations give rise to branches of relative periodic solutions (which are just periodic solutions in the rotating coordinates). Figure 1 features the bifurcation diagram for system (32)–(35)withthesameparametersetasin[5]. Weareinterestedintheunsta- ble part of the branch of relative periodic solutions born via a subcritical Hopf bifurcation. 0.6 0.45 0 0.3 b 0.15 0. Π 7Π 9Π 11Π 13Π 3Π 5 5 5 5 Β Figure 2: Domains of stability of the target relative periodic solution. Param- eters correspond to the gray dot in Figure 1. Black region: sufficient condition (27) for instability is satisfied; white region: relative periodic solution is stable; gray region: relative periodic solution is unstable. Figure 2 shows three regions in the parameter space (β,b ). The black 0 regioncorrespondstothevaluesofb andβ forwhichcondition(27)issatisfied, 0 hence the target state is not stabilizable by control (38). The white and gray 10