FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS. Fritz Colonius. Marco Spadini

Transcription

1 FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS Fritz Colonius Institut für Mathematik, Universität Augsburg, Augsburg, Germany Marco Spadini Dipartimento di Matematica Applicata, Via S Marta 3, Firenze, Italy Abstract An algebraic semigroup describing the dynamic behavior is associated to compact, locally maximal chain transitive subsets The construction is based on perturbations and associated local control sets The dependence on the perturbation structure is analyzed 1 Introduction This paper introduces a classification of the dynamic behavior of autonomous ordinary differential equations We restrict attention to the dynamic behavior within a maximal chain transitive set, ie, a chain recurrent component In order to find a characterization which is robust with respect to certain perturbations, we allow all time-dependent perturbations taking values in a small set U R m This perturbed system may be viewed as a control system, and the chain transitive set blows up to a control set D U Using earlier constructions for control systems see Colonius/San Martin/Spadini [2]), we associate to the control set D U a semigroup describing the behavior of trajectories Then, taking the inverse it of the semigroups as U 0, we obtain a semigroup for the original differential equation A number of properties of this procedure are derived In particular, we study if the semigroup is independent of the perturbation structure and we show that the semigroup remains invariant under conjugacies of the differential equation, if the perturbation structure is respected We remark that the spirit of this paper is contrary to other contributions that emphasize the study of the global behavior between Morse sets hence, outside of the chain recurrent set); see, eg, Mischaikow [8] On the other hand, topological properties inside the chain recurrent set have recently also found interest see Farber et al [4]) The contents of the paper are as follows In Section 2 we collect some properties of algebraic semigroups and their inverse its Section 3 recalls results on the relation between maximal chain transitive sets and control sets and slightly generalizes them by considering local versions) Furthermore, the notion of a fundamental semigroup for local control sets of control Date: March 25,

2 2 F COLONIUS AND M SPADINI systems is recalled from [2] Then, for a given perturbation structure, the fundamental semigroup of a locally maximal chain transitive set is defined as the inverse it of such semigroups Section 4 gives conditions implying that this inverse it is independent of the perturbation structure It is shown that these conditions are met in a natural setting for higher order differential equations In Section 5 it is shown that the semigroup remains invariant under smooth conjugacies of the given differential equation Furthermore, a number of simple examples is presented Notation The set of compact and convex subsets of R m containing the origin in the interior is denoted by Co 0 = Co 0 R m ) 2 Preinary Facts on Semigroups and Inverse Limits This section collects some basic facts on algebraic) semigroups and their inverse its Main references are the classical book by Eilenberg/Steenrod [3] and the book by Howie [7] A semigroup Λ is given by an associative operation : Λ Λ Λ on a nonvoid set Λ A semigroup does not necessarily have a unity, ie, an element e with e g = g e = g for all g Λ If a unity exists, however, one easily sees that it is unique In order to define inverse its of semigroups, some preinaries are needed A quasi-order in a set A is a relation that is reflexive and transitive The set A along with the quasi-order is a directed set if for each pair α, β A there exists γ A such that α γ and β γ A subset A of A is cofinal in A if for each α A there exists β A such that α β In the present paper the following quasi-ordered set will be relevant Example 21 Let Co 0 = Co 0 R m ) denote the family of all compact convex subsets of R m that contain the origin in their interior For U, V Co 0, define U V if V U With this quasi-order Co 0 is a directed set; in fact, for any U, V Co 0, one has U U V and V U V Fix U Co 0 and, for ρ > 0, let U ρ be the set given by ρ U Then, the family of all the sets of the form U ρ for some ρ > 0 is cofinal in Co 0 In fact, for any V Co 0 there exists ρ 0 > 0 such that U ρ 0 V, ie, V U ρ 0 We define the notion of inverse it of a family of semigroups compare [3], Chapter VIII, for such constructions with general inverse systems) Let {Λ α } α A be a family of semigroups, where A is a directed set with ordering Assume that for all α α in A, there exist semigroup) homomorphisms λ α α : Λ α Λ α such that λ α α is the identity of Λ α and λ α α λ α α = λ α α for α α α In this case the family of semigroups {Λ α } α A along with the homomorphisms {λ α α } α,α A forms an inverse system {Λ α, λ α α } α,α A over the directed set A Denote by α A Λ α = { g α ) α A, g α Λ α for all α A } the product of the semigroups Λ α When no confusion is possible, we shall often use the compact notation g α ) for the more cumbersome g α ) α A

3 FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS 3 The inverse it of {Λ α, λ α α } α,α A is the subset of α A Λ α given by Λ α = {g α ) α A Λ α, λ α α g α ) = g α for all α α } α A α A Define the projection λ α : α A Λ α Λ α, λ α : g α ) α A g α In Λ α define an operation of composition as follows: Given g α ) α A and α A h α ) α A, let g α )h α ) = g α h α ) This is well defined, since for α α one has ) λ α α gα h α ) = λ α α gα ) λ α α hα = gα h α One easily verifies that the product and the inverse it again are semigroups and that the projections are homomorphisms Furthermore, if α α, then λ α = λ α α λ α, ie, the following diagram is commutative: Λ λ α α Λ α α A λ α Proposition 22 Consider inverse systems of semigroups {Λ α, λ α α } α,α A and {Γ β, γ β β } β,β B with directed sets A and B Assume that there are an order preserving map i : B A and homomorphisms φ β : Λ iβ) Γ β for β B, such that for all β β in B Λ α λ α α Λ iβ ) φ β Γ β λ iβ ) iβ) Λ iβ) φ β Γ β commutes Then there is a unique homomorphism φ such that for all β B the following diagram commutes: α A Λ α φ γ β β β B Γ β λ iβ) γ β φ β Λ iβ) Γ β Proof Define the homomorphism φ as follows: If g α A Λ α, set h β = φ β g iβ) ), β B

4 4 F COLONIUS AND M SPADINI Then φg) := h β ) β B β B Γ β since, for β β, γ β β h β ) = γβ β φβ g iβ )) ) = φ β λ iβ ) iβ) g iβ )) ) = φ β g iβ) ) = h β Now, commutativity of the diagram above follows immediately from the definition of φ This, in turn, implies the uniqueness of φ Proposition 23 Consider inverse systems of semigroups {Λ α, λ α α } α,α A and {Γ β, γ β β } β,β B with directed sets A and B Assume that there are order preserving maps i : B A and j : A B, and homomorphisms φ β : Λ iβ) Γ β and ψ α : Γ jα) Λ α such that for all β β in B and α α in A, the diagrams Γ jα ) ψ α Λ α Λ iβ ) φ β Γ β 21) γ jα ) jα) Γ jα) ψ α Λ α λ α α λ iβ ) iβ) Λ iβ) φ β Γ β γ β β commute Assume also that β jiβ)) and α ijα)) for every α A and β B, and that the diagrams ψ iβ) Γ jiβ)) Λ iβ) γ jiβ)) β φ β Γ β λ ijα)) α Λ ijα)) φ jα) Λ α ψ α Γ jα) 22) commute Then the unique homomorphisms φ : Λ α Γ β and ψ : Γ β Λ α, α A β B β B α A induced by the diagrams 21) as in Proposition 22 are inverses of each other Proof We shall refer to the notation of Proposition 22 It is enough to prove that for all h = h α ) Λ α and g = g β ) Γ β, one has α A β B λ α ψ φh) )) = h α and γ β φ ψg) )) = g β for all α A, β B Let us prove the first one, the second will follow from a similar argument Take h = h α ) Λ α Using Proposition 22 twice and 22), α A λ α ψ φh) )) = ψ α γ jα) φh) ) ) = ψ α φjα) λ ijα)) h) ) This concludes the proof = ψ α φjα) h ijα)) ) = λ ijα)) α h ijα)) = h α

5 FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS 5 The following lemma, roughly speaking, shows that the inverse it of an inverse system {Λ α, λ α α } α,α A is influenced only by Λ α for large of course in the sense of the quasi-order in A) values of α Lemma 24 Let A and B be directed sets and assume that B A is cofinal in A Consider a family of semigroups {Λ α } α A Then Λ α is isomorphic α A to Λ α α B Proof Corollary 316 Chap VIII in [3] provides the required semigroup isomorphism We now provide some results that can be useful for the computation of the inverse it of an inverse family of semigroups Proposition 25 Let {Λ α, λ α α } α,α A be an inverse system of semigroups i) If each Λ α has a unity e α, then e α ) α A is the unity of α A Λ α ii) Assume that B is cofinal in A and that Λ β has a unity for any β B Then α A Λ α has a unity Proof i) It is enough to observe that for any g α ) α A α A Λ α one has e α )g α ) = g α ) = g α )e α ) ii) Follows from part i) and Lemma 24 Proposition 26 Let {Λ α, λ α α } α,α A be an inverse system of semigroups i) Assume that for all α, α A, with α α, the map λ α α is an isomorphism Then for all α A α A Λ α Λ α ii) If B is cofinal in A and Λ β consists only of its unity for β B, then α A Λ α consists only of its unity Proof i) Follows from Theorem 34 in [3] ii) Follows from part i) and Lemma 24 Proposition 27 Suppose that the inverse it of an inverse system of semigroups {Λ α, λ α α } α,α A admits a unity and let B be cofinal in A Then for every β B the semigroup Λ β contains an idempotent element Proof By Lemma 24 it follows that β B Λ β contains the unity e = e β ) β B Since e β ) = e = e 2 = e β )e β ) = e 2 β ), the assertion follows

6 6 F COLONIUS AND M SPADINI We conclude this section with a characterization of inverse its using a universal property in the category of semigroups As constructed, the inverse it of {Λ α, λ α α } α,α A is given by a semigroup together with homomorphisms λ α to Λ α Propositions 22 and 26 i) imply the following property: If Γ is a semigroup and γ α : Γ Λ α are homomorphisms, such that for all α α γ α = λ α α γ α, then there is a unique homomorphism γ : Γ Λ α with α A λ α γ = γ α This follows, in fact, by considering the inverse system {Γ α, λ α α } α,α A where Γ α = Γ for all α A and λ α α is the identity for all α α This property characterizes the inverse it up to isomorphisms In fact, using commutative diagrams one can easily show that the following fact holds inverse its are a categorical construction) Proposition 28 Let {Λ α, λ α α } α,α A be an inverse system of semigroups Consider a semigroup together with homomorphisms δ α : Γ α such that for all α α δ α = λ α α δ α Assume that for every semigroup Γ and homomorphisms γ α : Γ Λ α, such that for all α α γ α = λ α α γ α there is a unique homomorphism γ : Γ with δ α γ = γ α Then there exists a unique isomorphism δ : α A Λ α such that for all α λ α δ = δ α Remark 29 Clearly, in Proposition 28, it suffices to require the conditions for a cofinal subset of A 3 Locally Maximal Chain Transitive Sets Consider a differential equation ẋ = f 0 x) 33) given by a C 1 -vector field f 0 : R d R d, and assume that global solutions ϕt, x), t R, exist for all considered initial conditions ϕ0, x) = x First recall that a chain transitive set M R d is a closed invariant set such that for all points x, y N and all ε, T > 0 there is an ε, T ) chain ζ given by n N, points x 0 = x, x 1,, x n = y in M and times T 0,, T n 1 T with dϕt i, x i ), x i+1 ) < ε for i = 0, 1,, n 1 34)

7 FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS 7 A chain transitive set M is called locally maximal, if it has a neighborhood N such that every chain transitive set M with M M N satisfies M = M The behavior within M will be analyzed via perturbations Let C 1 vector fields f 1,, f m on R d be given and let F x) = [f 1 x),, f m x)] For every U Co 0 R m ) we consider the perturbed system denoted by Σ U) m ẋ = f 0 x) + u i t)f i x) = f 0 x) + F x)u, 35) i=1 u i ) UU) = {u L R, R m ), ut) U for almost all t R} Again we assume that absolutely continuous) global solutions ϕt, x, u), t R, exist for all considered initial conditions ϕ0, x, u) = x and all u Interpreting u as a control function, 35) can be viewed as a control system The sets U Co 0 R m ) will be called admissible control ranges Definition 31 A subset D of R d with nonempty interior is a local control set if there exists a neighborhood N of cld such that for each x, y D and every ε > 0 there exist T > 0 and u U such that ϕt, x, u) N for all t [0, T ] and d ϕt, x, u), y ) < ε and for every D with D D N which satisfies this property, one has D = D The neighborhood N in the definition above will also be called an isolating neighborhood of D If the neighborhood N can be chosen as R d, we obtain the usual notion of a control set with nonvoid interior as considered, eg, in [1] Thus for local control sets the maximality property of control sets is replaced by a local maximality property, and we refer to the latter also as global control sets It is convenient to introduce the following notation for sets A R d O +,F A x, U) = { y R d, } there are T 0, u UU) with y = ϕt, x, u) and ϕt, x, u) A for 0 t T If A = R d or dependence on F is not relevant, we simply omit the index In this notation, a local control set for Σ U ) with isolating neighborhood N is a maximal subset D U of N with nonvoid interior such that D U cl O + N x, U) for all x DU Throughout, we assume that for every admissible control range U the control system Σ U ) is locally accessible, ie, for every T > 0 the sets O ± T x, U) := {y Rd, y = ϕ±t, x, u) with u UU) and t 0, T ]} have nonvoid interiors In this case we also say that the vector fields f 1,, f m specify a perturbation structure for 33) The following proposition shows that a locally maximal chain transitive set of the unperturbed equation is contained in local control sets for perturbed systems with admissible control range U

8 8 F COLONIUS AND M SPADINI Proposition 32 Let M be a compact locally maximal chain transitive set with isolating neighborhood N of the unperturbed equation 33) and consider for given vector fields f 1,, f m the control systems Σ U) depending on admissible control ranges U Co 0 Suppose that for all x M and all U there is T > 0 such that ϕt, x, 0) into + N x, U) 36) Then for every U there is a unique) local control set D U of Σ U) with M intd U and M = intd U = cld U U Co 0 U Co 0 Proof The statement of the proposition and its proof are minor modifications of [1], Corollary 472 Remark 33 Under the assumptions of Proposition 32, it is clear that for U, U Co 0 with U U M intd U intd U Remark 34 Special control ranges are obtained in the form U ρ = ρ U, where U is compact and convex with 0 intu This is a case considered eg in [1], Gayer [5] The latter reference also shows that the inner pair condition 36) is eg satisfied for oscillators Note that there the restriction to a neighborhood N is not explicitly taken into account However, all arguments are local, and hence immediately carry over to our situation We recall from [2] the following construction which associates algebraic semigroups to local control sets D reflecting the behavior of the trajectories in D Fix p 0 intd Define P D, p 0 ) as the set of all x, T ) W 1, [0, 1], R d ) 0, ) with the following properties: x0) = x1) = p 0, xt) intd for t [0, 1]; and there are 0 < γ γ + < and measurable functions u : [0, 1] U, γ : [0, 1] [γ, γ + ] such that ẋt) = γt)f xt), ut) ), t [0, 1], and T = Endow P D, p 0 ) with the metric structure given by 1 0 γt) dt dx 1, T 1 ), x 2, T 2 )) = max x 1 x 2, T 1 T 2 ) for x 1, T 1 ), x 2, T 2 ) P D, p 0 ) The set P D, p 0 ) consists of the T periodic trajectories in intd of 35) starting at p 0 and reparametrized to [0, 1] Two elements x 0, T 0 ), x 1, T 1 ) P D, p 0 ) are homotopic, written x 0, T 0 ) x 1, T 1 ), if there exists a continuous map a homotopy) H : [0, 1] P D, p 0 ) such that H0) = x 0, T 0 ) and H1) = x 1, T 1 ) One can check that this is an equivalence relation Denote by ΛD, p 0 ) the quotient P D, p 0 )/ The operation on P D, p 0 ) given by x, T ) y, S) = x y, S + T ) with { x2t) t [0, 1/2] x y)t) = y2t 1) t [1/2, 1], 37)

9 FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS 9 is compatible with homotopy equivalence Hence, this operation induces a semigroup structure on ΛD, p 0 ) called the fundamental semigroup of the pointed local control set D, p 0 ) Let p 0 be an element of the given locally maximal chain transitive set N of 33) and let all assumptions above be satisfied Then consider the fundamental semigroups Λ U D U, p 0 ) of Σ U ) Note that by Remark 33 one has D U D U for U U Hence P U D U, p 0 ) can be regarded as a subset of P U D U, p 0 ) The map λ U U : ΛU D U, p 0 ) Λ U D U, p 0 ) associating [α] Λ U D U, p 0 ) to the class of α in P U D U, p 0 ) is, clearly, a well defined homomorphism Furthermore, the set Co 0 of admissible control ranges U is a directed system with respect to set inclusion Thus we can define inverse its and the following semigroup is well defined Definition 35 Let M be a compact locally maximal chain transitive set for 33) Then the fundamental semigroup of M with respect to the perturbation structure given by 35) is ΛM, f 1,, f m, p 0 ) := U Λ U D U, p 0 ) This notation will be shortened to ΛM, p 0 ) whenever an explicit reference to f 1,, f m is not needed The fundamental semigroup provides an algebraic description of the behavior of trajectories in perturbed locally maximal chain transitive sets In the next section we will discuss the dependence of this notion on the perturbation structure 4 Compatible Perturbation Structures In this section we will indicate a condition which guarantees that the fundamental semigroup is independent of the perturbing vector fields f 1,, f m It is convenient to write 35) in the form ẋ = f 0 x) + F x)u, u UU) 48) Along with 48) consider the family of perturbed systems ẋ = f 0 x) + Gx)v, v UV ) 49) with Gx) = [g 1 x),, g l x)] and C 1 vector fields g i and admissible control ranges V Co 0 R l ) Denote the corresponding trajectories by ϕ F t, x, u) and ϕ G t, x, v), respectively, and assume their global existence Throughout this section a compact locally maximal chain transitive set M with compact isolating neighborhood N of the system ẋ = f 0 x) is given The following property will be crucial in order to show that the fundamental semigroups for the perturbation structures given by F and G coincide It requires that the trajectories for F are reproducible by those for G and conversely

10 10 F COLONIUS AND M SPADINI Definition 41 The perturbation structures given by F and G are called compatible, whenever the following two conditions are satisfied: For all admissible control ranges U Co 0 R m ) there exists an admissible control range V = iu) Co 0 R l ) such that for all x N and v UV ) with ϕ G t, x, v) N for t 0, there is u UU) with ϕ F t, x, u) = ϕ G t, x, v) for all t 0; conversely, for all V Co 0 R l ) there exists U = jv ) Co 0 R m ) such that for all x N and u UU) with ϕ F t, x, u) N for t 0, there is v UV ) with ϕ F t, x, u) = ϕ G t, x, v) for all t 0 Remark 42 Obviously, one may assume that ijv )) V and jiu)) U Thus in the order on admissible control ranges cp Example 21) one has ijv )) V and jiu)) U The following result shows that compatible perturbation structures lead to the same fundamental semigroup Denote the control sets containing M for 48) and 49) with control ranges U and V by D U,F and D V,G, respectively Theorem 43 Let M be a compact locally maximal chain transitive set for 33) and consider compatible perturbation structures F and G given by 48) and 49), respectively Assume that for every x M and every U Co 0 R m ) there is T > 0 such that ϕt, x) into +,F N x, U) and, similarly, that for every x M and every V Co 0 R l ) there is T > 0 with ϕt, x) into +,G N x, V ) Then for p 0 M the fundamental semigroups ΛM, F, p 0 ) = U Co 0 R m ) ΛU D U,F, F, p 0 ) and are isomorphic ΛM, G, p 0 ) = Proof Let the corresponding loop sets be Λ V D V,G, G, p 0 ) V Co 0 R l ) P U D U,F, F, p 0 ) and P V D V,G, G, p 0 ), respectively Since the perturbation structures are compatible, there exist natural injective) maps P jv ) D jv ),F, F, p 0 ) P V D V,G, G, p 0 ), P iu) D iu),g, G; p 0 ) P U D U,F, F, p 0 ) These maps are easily seen to pass to the quotient and to induce maps ψ V : Λ jv ) D jv ),F, F, p 0 ) Λ V D V,G, G, p 0 ), φ U : Λ iu) D iu),g, G, p 0 ) Λ U D U,F, F, p 0 )

11 FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS 11 By Remark 42, we can assume ijv )) V and jiu)) U One can also check that the diagrams 21) and 22) in Proposition 23 commute Thus, passing to the inverse its, one obtains an isomorphism as claimed U Co 0 R m ) ΛU D U,F, F, p 0 ) Λ V D V,G, G, p 0 ), V Co 0 R l ) Next we turn to the question, how one can guarantee that two perturbation structures are compatible We can answer this only in the special case of higher order differential equations in R N of the following form: x n 1) 1 + h 1 x,, x n 1 1) ) = 0, x n N ) N + h N x,, x n N 1) ) = 0, 410) where n 1,,n N N are given, and x j) i denotes the j-th derivative of x i ; all h i are C 1 functions Associate to 410) the control system x n 1) 1 + h 1 x,, x n 1 1) ) = f 1 x,, x n 1 1) )u 1 t), x n N ) N + h N x,, x n N 1) ) = f N x,, x n N 1) )u N t), 411) where the functions f i are C 1 and the controls u = u 1,, u N ) are measurable and take values in an admissible control range U Co 0 R N ) Both systems 410) and 411) can, in the standard way, be considered as first order systems in R n 1++n N In particular, for 411) one obtains as a special case of 35) with x i,j = x j+1) i, i = 1,, N, j = 1,, n i, ẋ 1,1 = x 1,2, ẋ 1,n1 + h 1 x 1,1,, x N,n1 ) = f 1 x 1,1,, x N,n1 )u 1 t), ẋ N,1 = x N,2, ẋ N,nN + h N x 1,1,, x N,nN ) = f N x 1,1,, x N,nN )u N t), 412) Fix a compact, locally maximal chain transitive sets M R n 1++n N of system 410) As we shall see, when f 1,, f N are bounded away from 0, there are local control sets as in Proposition 32 for 412); and, as long as this condition holds, the inverse it of the semigroups associated to these local control sets does not depend on the choice of the functions f 1,, f N Let x = x 1,1,, x 1,n1,, x N,nN ) be a point in R n 1++n N Denote by ϕt, x, u) the solution of 412) at time t starting from x at time t = 0 and driven by the control function u

12 12 F COLONIUS AND M SPADINI Remark 44 Consider the family Co 0 R N ) of all compact convex neighborhoods of the origin with the partial order defined in Example 21 Given any ρ 0 > 0, the family of ρ-cubes { ξ1,, ξ N ) R N : ξ i [ ρ, ρ] for i = 1,, N } for 0 < ρ < ρ 0, is cofinal Therefore, by Lemma 24, the inverse it in Theorem 46 does not change if we consider only the control sets relative to these control ranges For simplicity we indicate dependence on such a control range by a superfix ρ Again a compact locally maximal chain transitive set M of system 412) is considered Lemma 45 If in system 412) the functions f 1,,f N are nonzero on M, then there is a compact neighborhood K of M such that for every ρ > 0 and every x R n 1++n N there exists T > 0 such that ϕt, x, 0) into ρ,+ K x) Proof This follows by inspection of the proof of Theorem 19 in Gayer [5] Observe that by continuity the functions f 1,, f N are bounded away from 0 in a compact neighborhood of M By Proposition 32 one has that for every U there exists a local control set D U containing M in its interior with M = D U U Co 0 Fix a point p 0 M and consider for 412) the family of fundamental semigroups {Λ U D U, f 1,, f N, p 0 )} U Co0 Then the corresponding inverse it exists We shall show that it does not depend on the choice of the functions f 1,, f N Theorem 46 Let M R n 1++n N be a compact locally maximal chain transitive set for 410) Assume that the functions f 1,, f N in 411) are bounded away from zero in M Then the inverse it Λ U D U, f 1,, f N, p 0 ) U Co 0 is independent up to an isomorphism) of the choice of f 1,, f N Proof For the sake of simplicity, we shall perform the proof only in the simple case of a single second-order equation The proof of the general case can be performed along the same lines By Remark 44 one can restrict attention to systems having only small intervals centered at the origin as control ranges More precisely, for ρ > 0, consider the following two second order scalar) control systems ẍ + hx, ẋ) =fx, ẋ)ut), ut) [ ρ, ρ], 413a) ẍ + hx, ẋ) =gx, ẋ)vt), vt) [ δ, δ] 413b) with f and g nonzero on a compact locally maximal chain transitive set M for the following uncontrolled system:

13 FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS 13 { ẋ = y, ẏ + hx, y) = 0 414) Since f and g are nonzero on M, continuity implies the existence of a compact neighborhood K of M on which both f and g are bounded away from zero By Lemma 45 and Proposition 32 there exist families of local control sets D ρ,f and D δ,g for the control systems 413a) and 413b) respectively, such that M = intd ρ,f = intd δ,g ρ>0 δ>0 One may assume that ρ and δ are so small that D ρ,f K and D δ,g K Let p 0 M and consider the semigroups Λ ρ,f D ρ,f, f, p 0 ) and Γ δ,g D δ,g, g, p 0 ) associated to these control sets We shall verify that the perturbation structures for f and g are compatible Hence Theorem 43 shows that the corresponding inverse its are isomorphic, as claimed Denote by ϕ f t, x, y, u) and ϕ g t, x, y, v) the solutions of the first order systems corresponding to 413a) and 413b) starting at x, y) for t = 0 driven by u and v, respectively We claim that there exists α > 0 with the following property: Let x, y) R 2 and let v be a control with vt) V = [ δ, δ] and ϕ g t, x, y, u) K for all t 0; then there exists a control function u with ut) U = [ αδ, αδ] such that ϕ f t, x, y, u) = ϕ g t, x, y, v) If this claim is verified, the first compatibility property is satisfied with iu) = i[ ρ, ρ]) = [ ρ/α, ρ/α] The map j is constructed in a similar way To prove the claim, choose α := ut) = f max g x, ȳ) min f x, ȳ), where the minimum and maximum are taken for x, ȳ) K Define ut) for t 0 by g ϕ g t, x, y, u )) ϕ g )) vt) t, x, y, u Then ut) [ αρ, αρ] and, moreover, the function t φ f t, x, y, u ) satisfies equation 413b) with initial condition x, y) Therefore, by uniqueness, φ f t, x, y, u ) = φ g t, x, y, u ) 5 Invariance under conjugacy In this section we consider a chain recurrent component M for dynamical systems induced by equations of the form 33), and prove that the associated fundamental semigroup is invariant under C 1 conjugacies provided that the perturbation structure is preserved Consider two differential equations on R d ẋ = fx) and ẋ = gx) 515)

14 14 F COLONIUS AND M SPADINI with global flows ϕ f t and ϕ g t, t R Assume that they are C1 conjugate, ie, there exists a C 1 diffeomorphism H such that the following diagram commutes for all t R: R d ϕ t R d H H R d ψ t R d Consider a compact, locally maximal chain transitive set M of ẋ = fx) Then N := HM) is a compact locally maximal chain transitive set for ẋ = g 0 x) For perturbation structures F x) = [f 1 x),, f m x)] and Gx) = [g 1 x),, g l x)], one obtains the control systems ẋ = fx) + F x)u, u UU), 516) ẋ = gx) + Gx)v, v UV ), with U Co 0 R m ), V Co 0 R l ), and corresponding fundamental semigroups ΛM, F, p 0 ) and ΛHM), G, Hp 0 )) We will show that these semigroups are isomorphic if the perturbation structures are compatible under the conjugacy H in an appropriate sense Remark 51 Consider a C 1 conjugacy H of ϕ f t and ϕ g t Then, differentiating the relation H ϕ f t x)) = ϕ g ) t Hx) with respect to t at t = 0, one gets H x)fx) = H x) ϕ f 0 x)) = ) ) 0 ϕg Hx) = g Hx) ; here H x) denotes the Fréchet derivative of H at x Thus for every x R d fx) = H x) 1 g Hx) ), In order to compare the perturbation structures we transport the vector fields f i via H and compare them to the g i Lemma 52 Let H be a C 1 conjugacy of the differential equations 515) and consider the perturbed equations 516) with perturbation structures F and G Define perturbation structures by H 1 G := [H x) 1 g 1 Hx) ),, H x) 1 g m Hx) ) ], HF := [H x)f 1 H 1 x) ),, H x)f m H 1 x) ) ] 517) Then the perturbation structures HF and G are compatible if and only if F and H 1 G are compatible Proof It is enough to show that the diffeomorphism H maps solutions ϕ F, x, u) of ẋ = fx) + F x)u to solutions ϕ HF, x, u) of ẋ = gx) + HF x)u,

15 FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS 15 and that, conversely, H 1 maps solutions of ẋ = fx) + Gx)u to solutions of ẋ = gx) + H 1 Gx)u, Let us prove the first assertion, the second can be proven in a similar way The composition of a C 1 map with an absolutely continuous function is absolutely continuous Moreover, for aa t R, one has d dt H ϕ F t, x, u) ) = H ϕ F t, x, u) ) d dt ϕf t, x, u) = H ϕ F t, x, u) ) f ϕ F t, x, u) ) + H ϕ F t, x, u) ) F ϕ F t, x, u) ) ut) = g H 1 H ϕ F t, x, u) )) + HF H ϕ F t, x, u) )) This implies the assertion by uniqueness The following theorem shows that the fundamental semigroups remain invariant under C 1 conjugacies H provided that H respects the perturbation structures Theorem 53 Assume that H is a C 1 conjugacy of the flows for the differential equations 515), and consider a compact, locally maximal chain transitive set M with isolating neighborhood N of ẋ = fx) and perturbation structures F and G given by 516) Assume that for every x M and every U Co 0 R m ) there is T > 0 such that ϕ F T, x, 0) into +,F N x, U) and, similarly, that for every x M and every V Co 0 R l ) there is T > 0 with ϕ G T, Hx), 0) into +,G HN) Hx), V ) Then for p 0 M the fundamental semigroups Λ ) M, F, p 0 and Λ HM), G, Hp0 ) ) are isomorphic if the perturbation structures G and F H given by 517) are compatible Proof By Theorem 43, the semigroups Λ HM), HF, Hp 0 ) ) and Λ HM), G, Hp 0 ) ) are isomorphic, since they correspond to compatible perturbation structures Hence it remains to show that the semigroup remains invariant under the C 1 conjugacy H As in Lemma 52, trajectories of ẋ = fx) + F x)u are mapped via H onto trajectories of ẋ = gx) + HF x)u, preserving time, and conversely Thus for every U Co 0 R m ) the corresponding local control sets D U,F and D U,HF of these two systems are mapped onto each other and the loop sets for both systems are in a natural

16 16 F COLONIUS AND M SPADINI bijective correspondence Hence, H induces an isomorphism between the semigroups Λ U D U,F, F, p 0 ) and Λ U D U,HF, HF, Hp 0 )), so that their inverse its are isomorphic As in the preceding section the assumptions of this theorem are satisfied for higher order equations If the conjugacy respects the form of these equations, then compatibility of the perturbation structures is automatically satisfied More precisely, in addition to 410) with perturbation structure 411), consider a higher order equation with perturbation structure of the form x n 1) 1 + k 1 x,, x n1 1) ) = g 1 x,, x n1 1) ) v 1 t), x n N ) N + k N x,, x n N 1) ) = g N x,, x n N 1) ) v N t) 518) Consider C 1 conjugacies H : R n 1++n N R n 1++n N of the special form Hx 1,1,, x N,nN ) ) = x 1,1,, x 1,n1 1, H 1 x1,1,, x N,nN,, x N,1,, x N,nN 1, H N x1,1,, x N,nN ) ) 519) where x ij, i and j are taken as in 412) Note that this kind of conjugacies can be seen as phase-space diffeomorphisms for systems of the form 410) The following result shows that for higher order differential equations the fundamental semigroup is essentially independent of the perturbations Theorem 54 Let M R n 1++n N be a compact locally maximal chain transitive set for the first order system associated to 410), and take p 0 M Suppose that H, of the form 519), is a C 1 conjugacy of the flow associated to this system to the one induced in R n 1++n N by 518) when v i t) = 0 for all t and all i Suppose that the f i s in 411) do not vanish on M and that the g i s in 518) do not vanish on HM) Then the fundamental semigroups are isomorphic ΛM, f 1, f N, p 0 ) and ΛHM), g 1,, g N, Hp 0 )) Proof Due to the special form 519) of the conjugacy H, the perturbation structure HF also corresponds to a higher order differential equation of the form 518) Clearly, by Remark 51, the functions in front of the controls corresponding to the perturbation structure HF are bounded away from zero The assertion follows from Theorem 46 and Theorem 53 Let us now see some simple examples illustrating the relation between the dynamics and the semigroup

17 FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS 17 Example 55 Consider the linear differential equation ẋ = Ax and additive perturbations leading to the linear control system ẋ = Ax + Bu, ut) U, with A R d d, B R d m Assume that A, B) is controllable, ie, rank[b, AB, A d 1 B] = d It is known see [2, Proposition 47]) that for U compact, convex with 0 int U, there exists a unique local control set D U, and the semigroup Λ U D U, 0) consists of just its unity Assuming also that A is hyperbolic ie, its eigenvalues are not purely imaginary), one has that the local chain recurrent component M for ẋ = Ax is compact as it reduces to the origin Therefore, it makes sense to consider the inverse it ΛM, 0) = Λ U D U, 0) By Proposition 26 U Co 0 ii), ΛM, 0) is just the trivial semigroup Note that Example 55 shows that the converse of Theorem 53 does not hold In fact, as shown above, the fundamental semigroup for a linear system, when defined, is trivial regardless its conjugacy class The following two examples show that even for situations where the chain recurrent components exhibit the same topological structure, the fundamental it semigroup may not be the same Example 56 Let U R m be a compact and convex set containing 0 in its interior Consider control-affine systems of the form m ẋ = f 0 x) + u i t)f i x), u U ρ, 520) i=1 where U ρ denotes the set of measurable functions on R with values in ρu, ρ 0 Suppose that the uncontrolled system with u 0) has a homoclinic orbit given by ϕt, p 1, 0), t R, with ϕt, p 1, 0) = p 0, t ± where p 0 p 1 is an equilibrium Suppose that H := {p 0 } {ϕt, p 1, 0), t R} is a chain recurrent component of the uncontrolled system and that the controllability condition span {ad k f 0 f i x), i = 1,, m, k = 0, 1, } = R d 521) holds for all points x H Then for every ρ > 0 there is a control set D ρ containing H in its interior and D ρ = H; ρ>0 see Corollary 476 in [1] the controlled Takens-Bogdanov oscillator is a system where these conditions can be verified; cp Häckl/Schneider [6] or Section 94 in [1]) It is known see [2]) that the fundamental semigroup

18 18 F COLONIUS AND M SPADINI Λ U ρ D U ρ, p 0 ) contains a unity Therefore, by Proposition 25 ii) the inverse it ΛH, p 0 ) = Λ U D U, 0) contains the unity U Co 0 Example 57 Consider a system 520) where the uncontrolled system has a periodic trajectory Ĥ = {ϕt, p 0, 0), t [0, T ]} If Ĥ is a local chain recurrent component of the uncontrolled system and 521) holds on Ĥ, then again Corollary 476 in [1] implies the existence of control sets ˆD U ρ containing Ĥ in the interiors with ˆD ρ>0 U ρ = Ĥ It is known see [2]) that the fundamental semigroup Λ U ρ D U ρ, p 0 ) does not contain a unity We want to show that this is true for the inverse it as well For ρ > 0 small enough, given any trajectory of 520) starting at p 0 and lying in ˆD U ρ one can associate to any xt) its orthogonal projection ˆxt) onto Ĥ this can be proved directly or deduced from the well-known Tubular Neighborhood Theorem) Thus, it makes sense to consider the function that associates the arc length on Ĥ measured in the natural trajectory direction) between the point p 0 and ˆxt) to t Reducing ρ if necessary, we can assume that this function is strictly increasing Now, topological considerations similar to those used in [2, Example 48] show that, for ρ > 0 small enough, Λ U ρ D U ρ, p 0 ) cannot contain idempotent elements Consequently, by Proposition 27, in this situation ΛĤ, p 0) = Λ U D U, 0) does not contain a unity U Co 0 Clearly, the periodic solution in Example 57 and the homoclinic orbit together with the equilibrium in Example 56 are homeomorphic maximal chain transitive sets However, the corresponding semigroups are not isomorphic, since one contains a unity, while the other does not These systems are not conjugate illustrating Theorem 54 Example 58 Let Ĥ and ΛĤ, p 0) be as in Example 57 Let γ ρ be the map that associates to any n N the class of maps in Λ U ρ D U ρ, p 0 ) corresponding to the periodic trajectory Ĥ gone through n times Topological considerations show that this map is injective The discussion preceding Proposition 28 shows that there exists a unique homomorphism γ : N ˆΛ such that the following diagram commutes γ λ ρ N ΛĤ, p 0) Λ U ρ D U ρ, p 0 ) λ γ ρ ρ λ ρ Λ U ρ D U ρ ρ, p 0 ) An inspection of the above diagram shows that γ is actually an injection of N into ΛĤ, p 0) Similar considerations in the case of a homoclinic orbit H described in Example 56 show that here an injection of N {0} into the it semigroup ΛH, p 0 ) exists

19 FUNDAMENTAL SEMIGROUPS FOR DYNAMICAL SYSTEMS 19 References [1] F Colonius, W Kliemann, The Dynamics of Control, Birkhäuser, 2000 [2] F Colonius, L San Martin, M Spadini, Fundamental semigroups for local control sets To appear in Annali Mat Pura App [3] S Eilenberg, N Steenrod Fundations of Algebraic Topology Princeton Univ Press, 1952 [4] M Farber, T Kappeler, J Latschev, E Zehnder, Lyapunov 1 forms for flows, ETH Zürich 2003, [5] T Gayer, Control sets and their boundaries under parameter variation To appear in J Diff Equations 2004) [6] G Häckl, K Schneider, Controllability near Takens-Bogdanov points, J Dyn Control Sys, ), pp [7] J Howie, An Introduction to Semigroup Theory, Academic Press, 1976 [8] K Mischaikow, Topological techniques for efficient numerical computation in dynamics, Acta Numerica ), address, F Colonius: colonius@mathuni-augsburgde address, M Spadini: spadini@dmaunifiit