THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS

Transcription

1 THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS GUILLAUME LAJOIE Contents 1. Introduction 2 2. The Hartman-Grobman Theorem Preliminaries The discrete-time Case The continuous-time case 4 3. Topological equivalence for linear systems Preliminaries Demonstration 7 4. Conclusion and Discussion 12 References 14 Date: April 18,

2 2 GUILLAUME LAJOIE 1. Introduction To study dynamical systems in continuous time, one needs to be creative to find ways of simplifying problems. Indeed, most of the time, interesting systems come with their share of complexity that prevents us in finding explicit solutions. Although there is a plethora of different structures to which dynamical systems can relate, most of the problematic ones will share an attribute : non-linearity. Although this is a vast category, it gains some importance due to the fact that linear problems and techniques to analyze them are well known. Hence, if one could find a way to linearise a system, a big step would be accomplished. This is precisely what we will study here. We will first be looking at the Hartman- Grobman Theorem. In short, this result gives us the ability, in certain cases, to locally reduce a system of differential equations to its linear part near equilibrium points. After doing so, we can find explicit solutions to this linear problem and show that there is a local topological equivalence between the flow of the original problem and the one resulting from the linear problem. This ends up being a quite powerful tool in the sense that it gives us an insight on the behavior of a continuous-time flow where it usually matters : near equilibrium points. It is important to note that this is not a simple stability result. The topological equivalence of the flows gives us an accurate portrait of the flow resulting from the original problem near equilibrium points. Secondly, we will study the topological equivalence of linear systems. Indeed, we will show that if two systems have the form ẋ = Ax, where x R n and A is an n n matrix, then their flows are topologically equivalent if the matrices have the same number of eigenvalues with positive and negative real parts respectiveley and none on the imaginary axis. This result is central to the study of continuous-time dynamical systems since it enables us to catalogue an infinite number of linear systems (as many as there are possibilities for the non-imaginary spectrum of an n n matrix) into a small collection of equivalent systems. In turn, this tool is a building block of a phase diagram of a flow. With both the Hartman-Grobman theorem and the equivalence of the above described linear systems, one can draw portraits of the flow of any hyperbolic system near equilibrium point modulo some homeomorphism and some smooth changes of coordinates. Analogous results are known and proved using similar techniques for discrete-time dynamical systems. 2. The Hartman-Grobman Theorem 2.1. Preliminaries. In this section, we will construct the outline of a proof for the Hartman-Grobman theorem. Here, both the discrete-time and continuous-time cases

3 THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS 3 will be treated as we will use one to prove the other. In both cases, we will study the dynamics around an equilibrium point (i.e.: fixed point of the associated flows). Without loss of generality, lets assume that the studied equilibrium point is the origin. A simple translation is enough to move any equilibrium point to the origin without affecting the dynamics of the flow. We will construct the outline of the proof of the Hartman-Grobman Theorem (H-G). To do so, we will need the concept of a properly defined cut-off function The cut-off function. In this subsection, the reader will find a description of the use of a cut-off function to which we will refer to in the following demonstrations. Only the statement of this construction is given here, but a rigourous demonstration can be found in [2]. Lets note that in the following, represents the C 1 -norm of the appropriate function space. Let f : R n R n be sufficiently smooth and such that f(0) = 0 and Df(0) = 0. Let ε > 0. Both f and Df are continuous, therefore we can choose r > 0 such that the open ball centred at the origin of radius r (B r ) has the following property sup x B r Df(x) < ε 3. Applying the mean value theorem to f, we get sup f(x) < εr x B r 3. Then there is a cut-off function η : R n R with properties (P.1) η(b r/3 ) = {1} (P.2) η(r n B r ) = {0} (P.3) η = 1 and Dη 2 r. Let f = ηf, then D(η f) Dη f + η D f < 2 r εr 3 + ε 3 = ε We call this inequality an estimate for D(η f).

4 4 GUILLAUME LAJOIE This concludes the preliminaries of this section. Lets now construct an outline of the proof of the Hartman-Grobman Theorem The discrete-time Case. In this sub-section, we refer to the discrete-time flow defined by the iteration of a diffeomorphism f on R n. Theorem 1 (H-G for discrete-time). If the origin is a hyperbolic fixed point of the diffeomorphism f : R n R n, then there exists open neighbourhoods of the origin, U, V R n and an homeomorphism H : U V such that f H(x) = H(Ax) x U where A = Df(0). We will first need this lemma. Lemma 1. Let A be an n n matrix with non-zero real part eigenvalues and ρ : R n R n a smooth function. If 0 < α < 1 and ρ is sufficiently small (where is the C 1 norm), then there exists a unique continuous function h : R n R n such that h α, h(0) = 0 and h(ax) Ah(x) = ρ(x + h(x)) x R n. The proof of this Lemma can be found in [2]. We can now sketch the proof of Theorem 1. Proof(1). The idea here is to define a functionnal equation of which our desired homeomorphism H is a solution. We do this using the concept of a cut-off function. We know A = Df(0) is invertible. Choose α such that 0 < α < 1. Define k(x) = f(x) Ax. It is clear that k is continuous and k(0) = 0. Hence, there exists an open neighbourhood Ũ of the origin in Rn such that k Ũ < α 3. We can choose an appropriate cut-off function on to define the map k on R n such that k = k on Ũ and k < 3 k Ũ < α. If we apply Lemma1 with ρ = k, we get the existence of the map h described in the statement of the same Lemma. Now define H(x) = x + h(x). We then get f(h(x)) = H(Ax) x R n. It can be shown that there exists a neighbourhood U of the origin such that U Ũ and H U is an homeomorphism The continuous-time case. Theorem 2 (H-G for continuous-time). Let f : R n R n be sufficiently smooth function that has a root at the origin. Then the origin is a fixed point of the flow ϕ t associated with the system ẋ = f(x). Let A = Df(0) be a n n matrix with all eigenvalues having non-zero real part. Then there exists open neighbourhoods of the

5 THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS 5 origin U, V R n, an open interval containing zero I 0 R and an homeomorphism H : U V such that H(ϕ t (x)) = e At H(x) x U, t I 0. The construction of the homeomorphism H is somewhat similar to the discret-time case. We will first need this Lemma. Lemma 2. If A is an n n matrix as stated above and H : R n R n is an homeomorphism, then the operator defined via χ(g)(x) = Ag(x) g(h(x)) is a bounded linear transformation with bounded inverse in the Banach space C(R n ) with respect to the supremum norm. The proof of this Lemma can be found in [2]. We can now sketch the proof of Theorem2. Proof(2). Define T 1 to be the time-one map of the flow associated to ẋ = Ax, e At. T 1 = e A Choose an appropriate cut-off function η and let f = ηf. Then the system ẋ = f has a flow ϕ t and the following holds (i) f is Lipschitz (ii) There is a neighbourhood of the origin Ũ on which the time-one maps ϕ 1 and ϕ 1 agree. (iii) The function ρ(x) = ϕ 1 T 1 (x) is such that ρ is small enough to meet the prerequisites of Lemma1. Then (iii) and Lemma 1 imply that there exists a continuous function g such that g(0) = 0 and g < 1 for which g(t 1 x) T 1 g(x) = ρ(x + g(x)) x R n We can now apply Lemma 2 to get the existence of a unique function h C(R n ) such that T 1 h(x) H( ϕ 1 (x)) = ρ(x) Define H(x) = x + h(x). It is clear that H( ϕ 1 (x)) = T 1 H(x). Now define H = 1 0 e sa H( ϕ s (x))ds. Since e At is a linear map and H commutes with the

6 6 GUILLAUME LAJOIE time-one maps T 1 and ϕ 1, we can write the following. Consider the reparametrisation of time τ = s + t 1, then e ta H( ϕ 1 (x)) = = = 1 0 t t 1 0 t 1 e ( t s)a H( ϕ s+t (x))ds e τa H( ϕ τ (x))dτ e τa H( ϕ τ (x))dτ + t 0 e τa H( ϕ τ (x))dτ. Reparametrise time for the first integral of the last line : σ = τ + 1 e ta H( ϕ 1 (x)) = 1 t = H(x). e ( σ+1)a H( ϕ 1+σ (x))dσ + t 0 e τa H( ϕ τ (x))dτ And so we can write H( ϕ t (x)) = e ta H(x). If we set t = 1, this implies that H( ϕ 1 (x)) = T 1 H(x). Note that we already have H( ϕ 1 (x)) = T 1 H(x). Using a unicity argument coming from Lemma 2, it can be shown that H = H. We must now show that there exists an open neighbourhood of the origin U R n such that H U is an homeomorphism. Let G(x) = x+g(x) where g(x) is the function in (iii). Theorem 1 gives us that ϕ 1 (G(x)) = G(T 1 x). Hence we get H(G(T 1 x)) = H( ϕ 1 (G(x))) = T 1 H(G(X)) Lemma 2 gives us that G H is the identity on R n. Since we know that there exists a neighbourhood W on which G is an homeomorphism (from Theorem 1). Then H G (W ) is the inverse of G W. Hence U = G(W ) is our desired neighbourhood and H U : U H(U) is our desired homeomorphism. This concludes the proof of Theorem 2. We have now shown that the flow of a system of the form ẋ = f(x) is topologically equivalent to the flow of ẋ = Df(ξ)(x ξ) in a neighbourhood of ξ, where ξ is a hyperbolic fixed point of these systems. This concludes this section. 3. Topological equivalence for linear systems 3.1. Preliminaries. In this section, we investigate the topological equivalence of two linear systems that leave the origin as an hyperbolic equilibrium point and

7 THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS 7 present the same number of eigenvalues having positive and negative real part respectively. Ultimately and in conjunction with the Hartman-Grobman Theorem, this result will enable us to declare a topological equivalence between two arbitrary systems that present linearized systems as stated above. Lets consider two linear systems ẋ = Ax and ẏ = By where x and y are in R n. Suppose that A and B do not have eigenvalues with zero real part and both have m + eigenvalues with positive real part and m eigenvalues with negative real part (m + + m = n). With an appropriate change of basis, we can write ( ) ( ) QA 0 QB 0 A = and B = 0 P A 0 P B where Q A, Q B are m + m + matrices with the positive real part eigenvalues of A and B respectively and P A, P B are m m matrices with the negative real part eigenvalues of A and B respectively. Hence, it is easy to see that without loss of generality, we can restrict ourselves to show the equivalence of the systems ẋ = Q A x and ẏ = Q B y where x and y are in R m +. Equivalently, we could show the equivalence for the systems with P A,B in a similar manner. Therefore, lets suppose that m = Demonstration. Theorem 3. Let A be a linear operator from R n to R n with eigenvalues of strictly positive real parts, then the system ẋ = Ax is topologically equivalent to the system ẋ = x. To show this result, we will construct a Lyapunov function that will enable us to design the proper homeomorphism between the flows of the two systems. To do this, we will first need this lemma. Lemma 3. There exits a positive definite quadratic form r on R n such that its derivative along the vector field Ax is always positive for any x 0. Proof. To facilitate the proof, suppose A is a complex linear operator on C n with eigenvalues λ i with positive real parts. We will show the existence of a positive definite quadratic form r : R C n R with a derivative that stays positive along the vector field R Az for z 0. Here, the symbol R represents the realification of a complex vector space. For example, R C n = R 2n. Essentially, this permits us to keep the linear and geometric structure of a complex space and express it over the real numbers. For a complex operator A on C n, its realification gives an operator R A on R 2n which corresponds point to point to its complex counterpart.[1] Lets choose r to be the following

8 8 GUILLAUME LAJOIE r(z) = n z i 2 = i=1 n z i z i It is easy to see that r is positive definite. We must now evaluate the derivative of r along the vector field R Az. i=1 (1) R Azr(z, z) =< Az, z > + < z, Ā z >= 2Re(< Az, z >) This expression defines a quadratic form. We must show that it is also positive definite. To do so, it would help to choose a good basis for C n in which the expression of R Azr can simplified. In the case where A is diagonalisable, we can take the eigenbasis associated to A in which the new form reads R Azr(z, z) = 2 n Re(λ i ) z i 2 Since we assumed all the eigenvalues of A had positive real parts, this expression is clearly positive definite. We must now find a way to generalize this result in the case where A is not diagonalisable. To do this, we will make use of this Lemma. Lemma 4. If A is a linear operator on C n and ɛ > 0, then there exists a basis of C n such that the expression of A in that basis is of the form i=1 λ λ λ n λ n in which every term is smaller than ɛ. This lemma is a direct consequence of the Jordan normal form theorem and I will omit its proof in this document. Lets now suppose that we express r in such a basis. We need to show that R Azr is positive definite. In order to achieve this, lets consider the space of all quadratic forms on R 2n denoted by Q 2n. We first show that the set of all positive definite quadratic forms on R 2n is open in Q 2n.

9 THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS 9 Suppose the following form is positive definite ρ(x) = 1 i,j 2n ρ ij x i x j More specifically, x on the unit sphere S 2n 1, ρ(x) > 0. Also, since S 2n 1 is compact and ρ is continuous, it reaches its minimum and maximum on S 2n 1. Hence there are m, M > 0 such that m ρ(x) M, x S 2n 1. Now, choose 0 < ɛ < m and µ Q (2n) 2 2n such that µ ij ɛ, 1 i, j 2n. Then x S 2n 1, µ(x) µ ij (2n) 2 ɛ < m 1 i,j 2n Consider the new quadratic form ρ + µ. It is easy to see that this new form, that is close to ρ, is also positive on S 2n 1 since we get x S 2n 1, ρ(x) + µ(x) ρ(x) µ(x) > ρ(x) m 0 In turn, this implies positive definiteness everywhere since for any x R 2n, there is a x S 2n 1 such that x = x x and hence, ρ(x) + µ(x) = x 2 (ρ( x) + µ( x)) > 0. From this reasoning and the above argument, we get that for any positive definite quadratic form ζ, there are m, M > 0 such that x 2 m ζ(x) x 2 M, x R 2n. Now, lets go back to our map r and its derivative R Azr which are both quadratic forms on R C n. From equation (1), we write n R Azr(z, z) = 2Re(< Az, z >) = 2Re( λ i z i 2 ) + 2Re( r ij z i z j ) i=1 i<j where the r ij s come from the upper triangular terms ( ) in the expression of A with respect to the basis described in Lemma 4. Notice now that the action of realification of C n sends z = (z 1,..., z n ) = (x 1 + iy 1,..., x n + iy x ) onto R z = (x 1, y 1,..., x n, y n ) R C n = R 2n. Considering that the terms r ij are as small as we like along with the above argument, we can conclude that the quadratic from R Azr is positive definite. This then concludes the proof of Lemma 3. Note that the above result was formulated for a linear operator A on C n but is also valid for any operator on R n which is mearly the restriction of a complex operator on a linear subspace. We are now equipped to prove Theorem 3. Proof(3).

10 10 GUILLAUME LAJOIE We now want to construct a topological equivalence between the flows of the equations ẋ = Ax and ẋ = x, e At and e t respectively. To do so, we must construct an homeomophism H : R n R n such that H e At = e t H. First, lets define the following set Λ = {x R n r(x) = 1}. (2) Now, consider the following candidate for our homoemorphism { H(e At x 0 ) = e t x 0, t R, x 0 Λ H(0) = 0. To show that H represents the desired function, we must verify the following assertions (A.1) equation (2) is well defined x R n (A.2) equation (2) is bijective and bicontinuous (A.3) H e At = e t H It is important to understand that the equation (2) is built using the idea that for any x R n, we can find a couple (x 0, t 0 ) Λ R such that e At 0 x 0 = x. We can now see that H leaves Λ fixed and sends any other point of R n onto a certain trajectory of a point in Λ. To show this rigorously, we will need the following Lemma. Lemma 5. Let γ(t) be a non-zero solution to ẋ = Ax and let ξ(t) = ln(r(γ(t)), then ξ is a diffeomorphism and there exists α, β > 0 such that α dξ dt β. Proof. By the unicity of solutions of a linear ODE, γ(t) 0 for some t implies γ(t) 0 t. We also have dξ = Axr. Since r and dt r Ax r are both positive definite quadratic forms, there exists α 1,2, β 1,2 > 0 such that and hence α 1 r(x) x β 1 and α 2 Axr(x) x β 2 r(x) α 2 Axr(x) x β 1 x r(x) x β 2 α 1

11 THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS 11 which yields α 2 r(x) Ax r(x) β 2 r(x) α 2 dξ β 1 α 1 β 1 dt β 2. α 1 It is not a hard task to verify that ξ is a diffeomorphism from R onto itself. This comes from the fact that the modulus of any non zero solution γ of our linear system will range over (0, ) for t (, ). This completes the proof of Lemma 5. We can now verify our three assertions. (A.1) For x R n {0}, consider the solution γ where γ(0) = x. Lemma 5 gives us that there exists a t such that r(γ(t)) = 1. Also, since ξ is monotone (it has a positive derivative everywhere), this implies that this t is unique. We have shown well definedness. (A.2) We have then constructed a bijective map Γ : R Λ R n {0} (t, x 0 ) e At x 0 This map and its inverse are clearly continuous. In the same way, we can define a similar map for our system ẋ = x Ψ : R Λ R n {0} (t, x 0 ) e t x 0. We can write the first part of expression (2) as H = Ψ Γ 1 : R n {0} R n {0}. The continuity of H everywhere but at zero is hence demonstrated. To see that it is also continuous at zero, we use the limit of H at zero along with Lemma 5. (A.3) Let x R n, then there is a unique t 0 R and a unique x 0 Λ such that x = e At 0 x 0. Then H(e At x = H(e At (e At 0 x 0 )) = H(e A(t+t 0) x 0 ) = e t+t 0 x 0 = e t H(x 0 ) t R

12 12 GUILLAUME LAJOIE and clearly H(e At 0) = H(0) = 0 = e t 0 = e t H(0) t R. We have now shown that the map H is the homeomorphism we were looking for. This concludes the proof of Theorem 3. Since this result is applicable to both the systems ẋ = Ax and ẏ = By, if we restrict ourselves to the case where m = 0, we can conclude that both systems are topologically equivalent to the system ẋ = x and are therefore topologically equivalent to each other. It is also easy to see that a similar reasoning can be applied to the case where m + = 0. Only in this case, we need to show negative definiteness for the map r since for increasing time, solutions will tend to zero. Hence, since we noted earlier that we can separate the two case (m = 0 and m + = 0) via a change of basis and a simple restriction to a linear subspace of R n, we can therefore conclude that two linear systems with the same number of eigenvalues with positive real parts, negative real parts and none on the imaginary axis, will be topologically equivalent. 4. Conclusion and Discussion In the light of the results presented in this paper, we can conclude the following. The Hartman-Grobman Theorem for a continuous-time dynamical gives us the equivalence of the flow of any ODE to the flow of its linearised system around hyperbolic fixed points. In turn, we have shown that two linear systems are equivalent if they present spectrums with no eigenvalues on the imaginary axis and the same number of eigenvalues having positive and negative real parts respectively. Combining these results yields a quite remarkable tool. Indeed, we get that around hyperbolic fixed points, if the linearisation of two different systems on R n have the same spectrum structure as described above, then there are neighbourhoods of the fixed points in which both systems are topologically equivalent. This results is quite useful for cataloguing different types of dynamical systems. Indeed, when doing a bifurcation analysis of some system for example, we are interested in the qualitative change of behaviour in the flow around equilibrium points. We can then restrict ourselves to the linear part of the system around hyperbolic fixed points to give us an idea of the dynamics that can be found there. Furthermore, we do not care for the actual linear part but only on the structure of its spectrum and the parameter values for which this structure changes. Nevertheless, these results have limits. Indeed, in the event where an equilibrium point is not hyperbolic, i.e.: there is at least one zero real part eigenvalue in the

13 THE HARTMAN-GROBMAN THEOREM AND THE EQUIVALENCE OF LINEAR SYSTEMS 13 spectrum of the linear part of the system, then these results fail. But that does not mean we cannot use them. In fact, the tools we have to analyse such a situation makes explicit use of them. One of these tools is called Center manifold reduction. This technique makes use of the idea that there are invariant manifolds tangent to the generalized eigenspaces associated to the linear part of a system. On the manifolds corresponding to the eigenvalues with non-zero real part, we can apply the above results to characterize the behaviour of the flow. Hence, to study the complete dynamics of a system around a non-hyperbolic fixed point, one can concentrate on understanding the dynamics occurring on the centre manifold associated to zero real part eigenvalues. In many cases, this reduces considerably the dimension of the problem and is the only way we can approach it. To conclude this paper, another remark is worth mentioning. Recall that when we outlined the proof of the Hartman-Grobman theorem, we sketched the proof of the version of this Theorem concerning discrete-time dynamical systems. There is also an equivalent version for the second result presented. Indeed, suppose we have a discrete-time dynamical system defined via the iteration of a linear operator on R n. Then the structure of the spectrum of the operator characterizes the dynamics around the origin. In the discrete time case, we have to look at eigenvalues inside and outside the unit circle of the complex plane. Eigenvalues on the unit circle are the analogous of the zero real part eigenvalues in the continuous case. Therefore, the origin is an hyperbolic fixed point if there are no eigenvalues on the unit circle and the dynamics close to it are completely characterized by the number of eigenvalues inside and outside (respectively) of the unit circle. The reader should understand that these results are part of the building blocks of the study of dynamical systems. Specialists nowadays use them intuitively. In such cases, it is sometimes interesting to take some distance from such tools and to look at their origins as we have done in this paper. It is in this manner that mathematics become truly natural.

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