Lecture 11 2012/12/10
Outline
1. Motivation 2. Hartman-Grobman Theorem 3. Center Manifold Theorems 4. Summary
1. Motivation
• The Hartman-Grobman theorem is another very important result in the local qualitative theory of ODE. The theorem shows that x′ = fx() with f (0)= 0
and its linearized system x′ = Df(0) x have the same qualitative structures
near a hyperbolic equilibrium. • The center manifold theorem is a natural extension of the stable manifold theorem, but it has an important difference. The proof is much complicated. This is an advanced topic in both qualitative analysis and stability theory.
2. Hartman-Grobman Theorem
1) Equivalence
Consider x′ = fx(), (11.1)
where x = 0 is a hyperbolic equilibrium. The linearized system x′ = Ax, (11.2)
where A= Df(0) .
Definition 11.1 Let A and B be subsets of R n . A homemorphism A onto B is a continuous one-to-one map of A onto B , HA: → B, such that
HBA−1 : → is continuous. The two sets A and B are called homemorphic or topologically equivalent if there is a homemorphism from A onto B .
Remark 11.1 Notice the difference between isomorphism and homemorphism. The latter has the requirement of H having a continuous property.
1 Definition 11.2 Two dynamic systems such as (11.1) and (11.2) are said to be topologically equivalent in a neighborhood of the origin if there is a homemorphism H , mapping an open set U containing the origin onto an open set V containing the origin, which maps trajectories of (11.1) in U onto trajectories of (11.2) in V and preserves the orientation. If the homemorphism H preserves parameterization by time t , then (11.1) and (11.2) are said topologically conjugate in the neighborhood of the origin.
Remark 11.2 Notice the difference between topologically equivalence and topologically conjugate.
2) An Illustrative Example for the Topologically Conjugate
Example 11.1 Consider two linear systems x′ = Ax and x′ = Bx with
−−13 20 A = and B = . −−31 04−
Let H() x= Rx, where
1 11− −1 1 11 R = and R = . 2 11 2 −11
Then B= RAR−1 and letting y= H() x = Rx or x= Ry−1 gives
yRARyBy=−1 = .
At Thus, if xt()= e x0 is the solution of x′ = Ax through x 0 , then
At Bt yt()= Hxt ( ()) = Rxt () = Rex00 = eRx
is the solution of x′ = Bx; i.e. H maps trajectories of x′ = Ax onto trajectories of
x′ = Bx and it preserves the parameterization by t since
HeAt= e Bt H.
Therefore, H is a homemorphism from A onto B . x′ = Ax and x′ = Bx are
topologically conjugate.
Remark 11.3 The mapping H() x= Rx is in fact a rotation through 45 0 and it is clearly a homemorphism. See Fig. 11.1 as follows.
2
Fig. 11.1
3) Statement of Hartman-Grobman Theorem
Theorem 11.1 If x = 0 is a hyperbolic equilibrium of (11.1) and (11.2), then there exists a homemorphism H of an open set U containing the origin onto an open set
V containing the origin such that for each xU0 ∈ , there exists an open interval
IR0 ⊂ containing the origin such that for all tI∈ 0
At Hϕ t () x00= e Hx ().
Remark 11.4 The proof of Theorem 11.1 is not presented. One may consults the book of “Differential Equations and Dynamical Systems” 3rd ed. by Lawrence Perko at pp. 121-124.
4) An Example Showing Hartman-Grobman Theorem
Example 11.2 Consider yy′ = − ; z′ = zy + 2 . (11.3)
The solution with yy(0) = 0 and zz(0) = 0 is solved by
y 2 yt()= y e−t ; zt()=+− z et0 ( e tt e−2 ). 0 0 3 The linearized system is given by yy′ = − , zz′ = . (11.4)
− t Its solution with yy(0) = 0 and zz(0) = 0 is easily solved by yt()= y0 e ,
3 y t 2 zt()= z0 e . The homemorphism H is defined as H(,) yx = y . Then, we z + 3 verify the result of the Hartman-Grobman Theorem as follows. Let the solution of the original system as ye−t 0 ϕ (,)yz= 2 t 00 y 0 − zet+−() e tt e2 0 3
e −t 0 At At = and e of the linearized system is given by e t . Since 0 e
−t −t y 0 ey0 At e 0 e Hy(,) z = 2 = 2 ; 00 t y 0 t y 0 0 e z 0 + ez()+ 3 0 3 and −t ye0 y 2 2 Hϕ (,) yz= H = −t t 00 y y yye= t0 tt−2 + 0 ze0 +−() e e z y 2 3 zze=+−t0 () e tt e−2 3 0 3 ye−−tte y 00 = 2−t 22= , y00()ye y 0 zet+( e tt −+ e−2 )et () z+ 00333 we have
At Hϕ t (,) yz00= eHyz (,)00 for all t ≥ 0 . The nonlinear system (11.3) and the linearized system (11.4) of this example are shown in Fig. 11.2 as follows,
Fig. 11.2
4 y 2 where “ Ws (0)= {( yz , ) ∈=− R2 | z } maps onto Es ={( yz , ) ∈= R2 | z 0} and 3
Wu (0)= {( yz , ) ∈= R2 | y 0} maps onto Eu ={( yz , ) ∈= R2 | y 0} by H .
1 y 2 Trajectories, such as z = − of (11.3) maps onto trajectories, such as y 3
1 z = of (11.4), by H and H preserves the parameterization of t .” (The details y
of computation “” are left for homework)
At Remark 11.5 How to find a homemorphism H such that Hϕ t () x00= e Hx () is difficult. In fact, the Hartman-Grobman Theorem only assures the existence of H . It doesn’t tell us any information on how to find H . It is a qualitative property!
Remark 11.6 If f is of C r , r ≥1, then it can be proved that there exists a C 1 -
At homemorphism H such that Hϕ t () x00= e Hx () by Hartman in 1960. It should
be noted that assuming higher derivatives of f doesn’t imply the existence of
higher derivatives of H . In general even if f is analytic, there does not exist a
2 At mapping H of class C satisfying Hϕ t () x00= e Hx (). If the eigenvalues of
Df(0) satisfy some extra conditions, it may increase the smoothness of H . The
discussion on this matter may consults the book of “Differential Equations and Dynamical Systems” by L. Perko, 3rd, Springer, pp. 127, 2001,.
3. Center Manifold Theorems
The stable manifold theorem gives a complete description of the dynamics of (11.1) in a neighborhood of a hyperbolic equilibrium. The center manifold theorem does provide such a description if we determine the behavior of solutions on the
center manifold W c .
1) The Linearized System with Zero Real Part
Consider (11.1) and (11.2). If A in (11.2) has k eigenvalues with zero real
5 parts, j eigenvalues with positive real parts and nk−− j eigenvalues with
negative real parts, then REEEncu=⊕⊕ s with dim Ekc = , dim Eju = and
dim Es =−− nk j. This situation can also be extended to (11.1) near x = 0 .
First we consider the case where E u is trivial, i.e. A has no eigenvalues with positive real parts. In this case, REEnc= ⊕ s with dim Ekc = and dim Es = nk − .
2) Center Manifold Theorem
Theorem 11.2 Suppose that A in (11.2) has k eigenvalues with zero real parts and nk− eigenvalues with negative real parts. Then, there exist 1. an k -dimensional center manifold W c (0) of class C 1 for (11.1) with
dimWEcc (0)= dim , tangent to the center subspace E c at x = 0 , which is
invariant under the flow ϕ t of (11.1);
2. an nk− -dimensional stable manifold W s (0) of class C 1 for (11.1) with
dimWEss (0)= dim , tangent to the stable subspace E s at x = 0 , which is
s invariant under the flow ϕ t of (11.1) and ϕ t that starts on W (0) is exponentially decay as t → +∞ .
3) An illustrative Example with Many Local Center Manifolds
Consider the following typical system for showing many center manifolds. x′ = fx(),
x2 where fx()= 1 . The origin is only equilibrium. The linearized system is given − x 2 by x′ = Df(0) x,
00 s where A= Df(0) = . Obviously, the stable subspace E is x 2 -axis and the 01−
c T center subspace E is x1 -axis. The solution with x(0)= c = ( cc12 , ) is easily
6 solved by
c1 −t xt1()= ; x22() t= ce . 1− ct1
11 − cx11 Eliminating t yields x22= ce e . Then, for any c1 < 0 , c 2 , the function
11 − cx11 ce21 e,0 x< x2=ψ (,, xcc 112 )= (11.5) = 0,x1 0
gives the plane curves which are invariant under the flow ϕ t , satisfying
∂ψ ψ (0,cc12 , )= 0 ; (0,cc12 , )= 0 . ∂ x1
c1 Notice that cx11<⇔00 < by xt1()= for all t ≥ 0 . Therefore, when 1− ct1
c x1 < 0 is sufficiently small, (11.5) is a local center manifold, which is tangent to E
at the origin. There are infinite many center manifolds because of any c1 < 0 and
c 2 . However, these manifolds keep the same orientation. See Fig. 11.3 as follows.
Fig. 11.3
4) Center Manifold and Reduced System
For simplicity, we assume that f in (11.1) is of C 2 . Then, (11.1) can be written as x′ = Ax + g() x , (11.6)
7 ∂g where g(0)= 0 and (0)= 0 . ∂x Then, there exists an invertible matrix C s.t.
− AO1 C AC 1 = . OA2
where an kk× matrix A1 has eigenvalues with zero real parts and an
()()nk−×− nk matrix A2 has eigenvalues with negative real parts. The change of variable
y k nk− = Cx, yR∈ ; zR∈ z transform (11.6) into the form
y′ = Ay11 + g(,) y z ; (11.7a)
z′ = Ay22 + g(,) yz, (11.7b)
where g 1 and g 2 inherit properties of g . They satisfy
∂g ∂g g (0, 0)= 0 ; j (0, 0)= 0 ; j (0, 0)= 0 for j =1, 2 . j ∂ y ∂z
Definition 11.3 If z= hy() is an invariant manifold for (11.7) and h is of C 1 , then it is said a center manifold if ∂h h(0)= 0 and (0)= 0. ∂ y
Then, by the definition 11.3, The local center manifold, tangent to E c at x = 0 , is given by Wc(0)= {( yz , ) ∈× Rk R nk− | z = hy ( ), || y || ≤δ } .
Theorem 11.3 For (11.7), there exist δ > 0 and hy() of C 1 , defined for all
||y || < δ such that z= hy() is a center manifold for (11.7).
The flow on the center manifold with z(0)= hy ( (0)) satisfies the following equation
8 y′ = Ay11 + g( yh , ( y )) , (11.8) which we refer to as the reduced system. However, if hy() is unknown, (11.8) is also unknown.
On the other hand, the change of variables yy = w z− hy() Transforms (11.6) into
y′ =++ Ay11 g( y , w hy ( )) ; (11.9a)
∂ h w′ =++ A[ w hy ( )] g [ yw , +− hy ( )] ( y )[ Ay + g ( yw , + hy ( ))]. (11.9b) 2 2 ∂ y 11 In the new coordinates, the center manifold is w = 0 . The flow on the center manifold is characterized by wt()≡ 0 ⇒ wt′()≡ 0 for all t ≥ 0 .
Substituting wt()≡ 0 and wt′()≡ 0 into (11.9b) results in
∂ h 0=+−+Ahy ( ) g ( yhy , ( )) ( y )[ Ay g ( yhy , ( ))], (11.10) 2 2 ∂ y 11 which is a center manifold equation for hy() must satisfy, with boundary conditions ∂h h(0)= 0 ; (0)= 0 . (11.11) ∂x
Denote ∂ h Nhy( ( ))= ( y )[ Ay + g ( yhy , ( ))] −− Ahy ( ) g ( yhy , ( )) , ∂ y 11 2 2 and (11.10) becomes Nhy( ( ))= 0 .
Remark 11.7 (11.10) for hy() can’t be solved in most cases (to do so would imply that (11.7) can be solved), but its solution can be approximated arbitrarily closely as a Taylor series in y .
9 ∂ϕ Theorem 11.4 If ϕ()y is of C 1 with ϕ(0)= 0 and (0)= 0 can be found such ∂ y that N(ϕ ( y ))= Oy (|| ||p ) for some p >1, then for ||y ||<< 1,
hy( )−=ϕ ( y ) O (|| y ||p ) and the reduced system (11.7) can be represented as
p+1 y′ =++ Ay11 g( y ,ϕ ( y )) O (|| y || ) .
Remark 11.8 The order of magnitude notation O()⋅ , it is enough to think of
ρ (y )= Oy (|| ||p ) as ||ρ (y ) ||≤ ky || || p for ||y ||<< 1.
5) Stability of Center Manifold
When E u is nontrivial, it is definitely unstable by Lyapunov stability. When E u is trivial, the stability of the full system (11.7) or (11.6) is determined by the reduced system (11.8).
Theorem 11.5 If y = 0 of the reduced system (11.8) is asymptotically stable
(respectively unstable), then x = 0 of the full system (11.7), or (11.6) is also asymptotically stable (respectively unstable).
Corollary 11.6 Under the conditions of Theorem 11.5, if y = 0 of the reduced
system (11.8) is stable and there is a Lyapunov function Vy() of C 1 such that
∂V [Ay+≤ g ( y , hy ( ))] 0 , ∂ y 11
near y = 0, then, x = 0 of the full system (11.7), or (11.6) is also stable.
6) Examples
Example 11.3 Consider
xx′12= , (11.12) ′ =−++2 x2 x 2 ax 1 bx 12 x where a ≠ 0 .
10 The system has a unique equilibrium x = 0 . The linearization at x = 0 results in 01 0 A= Df(0) = and gx()= , 2 + 01− ax1 bx 12 x where A has eigenvalues at 0 and −1 , and gx() satisfies g(0)= 0 and
∂g (0)= 0 . Let M be a matrix whose columns are the eigenvectors of A ; that is, ∂x
11 M = , 01− and take CM= −1 . Then,
−1 00 C AC = . 01− The change of variables
y x1 xx 12+ =C = − z xx22 puts the system (11.12) into the form y′ = a()( y +− z22 b yz + z );
z′ =−− z a()( y + z22 + b yz + z ). The center manifold equation (11.10) with the boundary condition (11.11) becomes Nhy( ( ))= h′ ( y )[ ay ( +− hy ( ))22 byhy ( ( ) + h ( y ))]
+hy() ++ ay ( hy ())22 − byhy ( () + h ())0 y =;
hh(0)=′ (0) = 0 .
23 We set hy()=++ hy23 hy and substitute it in the center manifold equation to
find unknown coefficients hh23,, by matching coefficients of like powers in y . We don’t know in advance how many terms of series we need. Let us start with the simplest approximation hy()≈ 0. The reduced system is
y′ = ay23 + O(| y | ) .
The term ay2 is the dominant term when ||y << 1. Since a ≠ 0 , the reduced
11 system is unstable. By Theorem 11.5, the full system is unstable.
Remark 11.9 Notice that an Oy(| |2 ) error in hy() results in an Oy(| |3 ) error in the right-hand side of the reduced system. This is a consequence of the fact that
g1(,) yz, which appears on the right-hand side of the reduced system as g1( yhy , ( )) , has a partial derivative with respect to z that vanishes at the origin. Clearly, this observation is also valid for higher order approximations.
Example 11.4 Consider the system y′ = yz; z′ =−+ z ay2 . The center manifold equation (11.10) with the boundary condition (11.11) is Nhy(())= hy′ ()[ yhy ()] + hy () −= ay2 0; hh(0)=′ (0) = 0 .
We start by trying ϕ()y = 0. The reduced system is y′ = Oy(| |3 ) , for which, we can’t reach any conclusion about the stability of the origin.
24 Then, we try hy( )= hy2 + O (| y | ) and substitute it into the center manifold
2 equation and calculate h2 , by matching coefficients of y , to obtain ha2 = . The reduced system is y′ = ay34 + O(| y | ) . Therefore, the origin is asymptotically stable if a < 0 and unstable if a > 0 . Consequently, by Theorem 11.5, the full system is asymptotically stable if a < 0 and unstable if a > 0 . If a = 0 , the center manifold equation (11.10) with the boundary condition (11.11) reduces to Nhy(())= hy′ ()[ yhy ()] += hy () 0, hh(0)=′ (0) = 0 ,
which has the exact solution hy()= 0. The reduced system y′ = 0 is stable with
Vy()= y2 as a Lyapunov function. Therefore, by Corollary 11.6, the origin of the full system is stable if a = 0 .
7) General Center Manifold Theorem
Theorem 11.7 (Center Manifold Theorem) Suppose that A in (11.2) has k eigenvalues with negative real part, j eigenvalues with positive real part and
nk−− j eigenvalues with zero real part. Then, there exist
12 1. an mnk=−− j-dimensional center manifold W c (0) of class C 1 for (11.1)
with dimWEcc (0)= dim , tangent to the center subspace E c at x = 0 , which is
invariant under the flow ϕ t of (11.1);
2. an k -dimensional stable manifold W s (0) of class C 1 for (11.1) with
dimWEss (0)= dim , tangent to the stable subspace E s at x = 0 , which is
s invariant under the flow ϕ t of (11.1) and ϕ t that starts on W (0) is exponentially decay as t → +∞ ; 3. an j -dimensional unstable manifold W u (0) of class C 1 for (11.1) with
dimWEuu (0)= dim , tangent to the unstable subspace E u at x = 0 , which is
u invariant under the flow ϕ t of (11.1) and ϕ t that starts on W (0) is exponentially decay as t → −∞ .
Remark 11.10 Theorem 11.2 is a general form of the center manifold theorem. It is local. The proof of the center manifold theorem is a bit harder and tedious. We will not give the proof. It may consult the textbook of “Elements of Differentiable Dynamics and Bifurcation Theory” Academic Press, New York, 1989 at p.32, by D. Ruelle.
Remark 11.11 The treatment of the case where E u is trivial can be generalized to the case where E u is nontrivial. It is omitted.
4. Summary
• Although the stable manifold theorem and the linearization characterize that x′ = fx() and x′ = Df(0) x have the same stability property near a hyperbolic
equilibrium, the stable manifold theorem gives much more information on geometric structures.
• The stable manifold theorem uses a geometric way to characterize the local property near a hyperbolic equilibrium. The linearization uses an analytical way to characterize the local property near a hyperbolic equilibrium.
• The center manifold approach treats the case where equilibrium is not hyperbolic, but it is much complicated.
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Homework
11.1 Consider the system given by
3 xx′11= − , xx′22= − .
1) Find E s and E c ;
2) Show that for any any c1 and c 2 , the function
1 2 − x1 −
gives one dimensional invariant manifold, which satisfy
∂ψ (0,cc12 , ) ψ (0,cc12 , )= 0 ; = 0 . ∂ x1
3) For ||x1 << 1, this invariant manifold is local center manifold. Show that there are infinite many local center manifolds at the origin.
11.2 Study the stability of the following system y′ = yz22 − y; z′ =−+ zy2 by the center manifold approach.
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Remark 11.7 Theorem 11.2 is a general form of the center manifold theorem. It is local. The proof of the center manifold theorem is a bit harder and tedious. We will not give the proof. It may consult the textbook of “Elements of Differentiable Dynamics and Bifurcation Theory” Academic Press, New York, 1989 at p.32, by D. Ruelle.
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