chap.2. introduction to dynamical systemscas.ensmp.fr/~levine/enseignement/2dynsys.pdf ·...
TRANSCRIPT
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Chap.2. Introduction to Dynamical Systems
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1. Flow, Orbits and StabilityNon controlled, Stationary System:
x = f (x)
onX, C∞ manifold of dimensionn, andf vector fieldC∞.The velocityf (x) at everyx doesn’t depend on timet at which wepass.A system is calledtime-varyiing , whenf depends on time :
x = f (t, x).
When the integral curves off are defined on the wholeR, we say thatthe vector fieldf is complete.
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1.1. Flow, Phase Portrait
We note :– Xt(x) : the integral curve at timet starting from the initial statex
at time0 ;– Xt : the mappingx 7→ Xt(x)
Properties :
1. for all t whereXt is defined,Xt is a local diffeomorphism ;
2. the mappingt 7→ Xt is C∞ ;
3.Xt ◦ Xs = Xt+s for all t, s ∈ R andX0 = IdX (one-parametergroup).
The mappingt 7→ Xt is calledflow associated to the system.
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Time-varying case : we can setx = (x, t) andt = 1.f (x) = (f (t, x), 1) is stationary onX × R.=⇒ Flow defined onX × R (of dim n + 1).
We call orbit of the differential equationx = f (x) an equivalenceclass for the relation∼ :“x1 ∼ x2 iff ∃t t.q.Xt(x1) = x2 or Xt(x2) = x1”.
In other words,x1 ∼ x2 iff x1 andx2 belong to the same maximalintegral curve.
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Phase Portrait : partition of X by the orbits with their sense ofmotion.Example :
x = −x
Xt(x) = e−tx. Flow : t 7→ e−t·Orbit of x :
O(x) = {y ∈ R|∃t ∈ R, y = etx} =
R−∗ if x < 00 if x = 0R+∗ if x > 0
The phase portrait thus corresponds to the 3 classesR−∗ , {0} andR+∗ .
•O
x<0 x>0
Phase Portrait ofx = x : inverse arrows.
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Definitions independent of the choice of coordinates :if z = ϕ(x) the flows satisfyZt ◦ ϕ = ϕ ◦Xt.
Recall : in a neighborhood of aregular or transientpoint x0, (i.e.such thatf (x0) 6= 0), there exists alocal diffeomorphismϕ thatstraightens outf .
The transformed integral curves are given by :
z1(t) = z01, . . . , zn−1(t) = z0
n−1, zn(t) = t + z0n
withz0i = ϕi(x0), i = 1, . . . , n
thusx(t) = ϕ−1(ϕ1(x0), . . . , ϕn−1(x0), t + ϕn(x0)).
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1.2. Equilibrium Point
An equilibrium point (or singular point, or permanent regime)of fis a pointx such thatf (x) = 0 (or such thatXt(x) = x : fixed pointof the flow).A vector field doesn’t necessarily have an equilibrium point.Ex : f (x) = 1 ∀xVariational, or Tangent linear Equation :
∂
∂x
dXt(x)
dt=
d
dt
∂Xt(x)
∂x=
∂f
∂x(x)
∂Xt(x)
∂x.
SetA = ∂f∂x(x) et z =
∂Xt(x)∂x . We have :
z = Az.
Eigenvalues ofA : characteristic exponentsof x.Without loss of generality, we can assume thatx = 0.
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The equilibriumx is saidnon degeneratedif A doesn’t have 0 aseigenvalue (A invertible).It is saidhyperbolic if A doesn’t have eigenvalues on the imaginaryaxis.left-top: saddlex1 = x1x2 = −2x2right-top : stable nodex1 = −x1x2 = −2x2left-bottom: stable focusx1 = −x2x2 = 2x1 − 2x2right-bottom: centerx1 = x2x2 = −x1
-40 -30 -20 -10 0 10 20 30-2
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-0.5
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-1 -0.5 0 0.5 1-1
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•
-0.01 -0.005 0 0.005 0.01
-0.01
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-1.5 -1 -0.5 0 0.5 1 1.5
-1
-0.5
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1.3. Periodic Orbit, Poincare’s Map
We callcycleor periodic orbit an isolated integral curve ofx = f (x)not reduced to a point and closed (diffeomorphic to a circle, i.e.suchthat there existsT > 0 satisfyingXT (x) = x).
Some remarkable classes of systems don’t admit periodic orbits :The vector fieldf on the manifoldX is calledgradient if and onlyif there exists a functionV of classC2 from X to R such that
f (x) = −∂V
∂x(x) , ∀x ∈ X .
Proposition A gradient vector field doesn’t have periodic orbits.
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Poincare’s map
Let γ be a periodic orbit ofx = f (x) of periodT .
• ••
W
•
x—xP(x)•
Let W be a submanifold of dimen-sion n − 1 transverse toγ at x ∈ γ(i.e.such that the tangent spaceTxWto W at the pointx and the lineR.f (x) tangent tox to the orbitγ aresupplementary).
ThePoincare mapor First Return Map associated toW andx, isthe mapping, notedP , that maps everyz ∈ W close tox to P (z) firstintersection of the orbit of dez with W .
The Poincare map neither depends on the transverse submani-fold W nor on the point x.
P is a local diffeomorphism ofW to itself.
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The convergent or divergent behavior in a neighborhood ofx is des-cribed by the recursion
zk+1 = P (zk)
which admitsx as fixed point.
Flow : Zk(z) for z ∈ W : Zk+1(z) = P (Zk(z)).
Tangent linear map :∂Zk+1
∂z(x) =
∂P
∂z(x)
∂Zk
∂z(x).
Setζk =∂Zk∂z (x) andA = ∂P
∂z (x). We have
ζk+1 = Aζk.
Then− 1 eigenvalues ofA are called thecharacteristic multipliersof P at x.The orbit γ is hyperbolic iff P has no characteristic multiplier onthe unit circle ofC.
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Example : simple pendulum
θ = −g
lsin θ
Notex1 = θ andx2 = θ :{x1 = x2x2 = −g
l sin x1 .
The vector fieldf (x1, x2) = x2∂
∂x1− g
l sin x1∂
∂x2
is defined on the cylinderX = S1 × R.
Two equilibrium points in X :(x1, x2) = (0, 0) and(x1, x2) = (π, 0).
The mechanical energy :E(x1, x2) =1
2x2
2 +g
l(1 − cos x1) is a first
integral :LfE = 0.
Orbit equation :E(x1, x2) = E0 = E(x1(0), x2(0)).
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choosing the initial conditionsx1(0) = θ0 andx2(0) = θ(0) = 0 thecorresponding orbit
x2 = ±√
2g
l(cos x1 − cos θ0)
is closed and its periodT (θ0) is (integratedt = 1x2
dx1) :
T (θ0) =
√2l
g
∫ 0
θ0
dζ√cos ζ − cos θ0
elliptic integral.Choose forW the half line{ x1 ≥ 0 , x2 = 0 }.Each orbit intersectsW at x1 = θ0 and all the orbits are closed andgo back to their initial point :P (x1) = θ0.The distance between two orbits remains constant after one roundthusA = ∂P
∂x1= 1 is non hyperbolic.
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Now add the air friction−εθ,with ε > 0 small : {
x1 = x2x2 = −g
l sin x1 − εx2 .
We get
fε(x1, x2) = x2∂
∂x1−(g
lsin x1 + εx2
) ∂
∂x2
andLfεE = −εx2
2 < 0 : the energy decreases along the orbits and
Pε(x1) < θ0, thusAε = ∂Pε∂x1
< 1 : the orbits are nowhyperbolic.
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2. Stability of Equilibrium Points and Or-bits2.1. AttractorA setA is invariant (resp.positively invariant ) if contains its imageby the flow for allt (resp. for allt ≥ 0) :
Xt(A) ⊂ A , ∀t ∈ R (resp.∀t ≥ 0).
invariant manifold : invariant set which is a submanifold ofX.Examples– The orbit of a closed first integral is an invariant manifold.– If X = Rn andA : compact with non empty interior ofX with
differentiable and orientable boundary∂A,A : positively invariant with respect tof iff : < f, ν >|∂A< 0 withν outward normal to∂A (f inward on∂A).
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Limit Sets : ⋂t∈R
Xt(A),⋂t≥0
Xt(A),⋂t≤0
Xt(A)
whereA : closure ofA.
Attractor : B =⋂t≥0
Xt(A)
with A : invariant,A compact.
Property : B = Xt(B) ∀t ≥ 0.
Example : for x1 = x2, x2 = −x1, the compact submanifoldB ={x2
1 + x22 = 1} is invariant forf and−f :
B =⋂t∈R
Xt(B) =⋂t≥0
Xt(B) =⋂t≥0
X−t(B).
Remark : an attracteur can be made of an infinite union of submani-folds ofX (strange attractor or fractal).
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2.2. Lyapunov’s Stability
U1
U2
x—•
Xt(x)
x•U1
U2x—•
Xt(x)
x•
The equilibrium pointx is Lyapunov–stable, or L–stable, if forevery neighborhoodU1 of x there exists a neighborhoodU2 of x,U2 ⊂ U1, such thatXt(x) ∈ U1, ∀t ≥ 0, ∀x ∈ U2.
x is Lyapunov–asymptotically stable, or L–asymptotically stable,if it is L–stable and iflimt→∞Xt(x) = x, ∀x ∈ U2.
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Theorem Let x be a non degenerated equilibrium off .
1. If all its characteristic exponents have strictly negative real part,thenx is L–asymptotically stable.
2. If at least one characteristic exponent has a positive real part, thenx isn’t L–stable.
Example :
The integral curves ofx = ax3 are given byx(t) = (x−20 − 2at)−
12.
If a = −1, 0 is an attractor : all the integral curves are well-definedwhent → +∞ and converge to 0.
If a = 1, 0 is not L-stable : the integral curves don’t exist aftert =x−2
02 , but start from 0 att = −∞.
Opposite behaviors for the same eigenvalue 0 !
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Example :Same problem for linear systems in dim≥ 2 :
x =
(0 00 0
)x
is L–stable sincex(t) = x0 for all t, but not L–asymptotically stable.Conversely,
x =
(0 10 0
)x
isn’t L–stable since
x1(t) = x2(0)t + x1(0), x2(t) = x2(0).
In both cases, 0 is a double eigenvalue.
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Stability of a fixed point, discrete case
Theorem Let x be a fixed point of the diffeomorphismf .
1. If all its characteristic multipliers have their modulus strictly smal-ler than 1, thenx is L–asymptotically stable.
2. If at least one of the characteristic multipliers has its modulusstrictly greater than 1, thenx isn’t L–stable.
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2.3. On the Stability of Time-Varying SystemsIn the time-varying case, stability and eigenvalues are no more rela-ted :
x =
(−1 + 3
2 cos2 t 1− 32 sin t cos t
−1− 32 sin t cos t −1 + 3
2 sin2 t
)x .
Unique uniform equilibrium point :x = 0.
For all t, the eigenvalues are−14 ± i
√7
4 : independent oft and withnegative real part−1
4 < 0.
But, for the initial condition(−a, 0) at t = 0 (a ∈ R) :
x(t) =
(−ae
t2 cos t
aet2 sin t
).
Thus limt→+∞
‖x(t)‖ = +∞ ∀a 6= 0 : the origin isn’t L–stable.
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In fact, if we set (Floquet’s theory) :
x =
(− cos t sin tsin t cos t
)y
y is the solution of the stationary system
y =
(12 00 −1
)y
which isn’t L-stable.
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2.4. Lyapunov’s and Chetaev’s Fonctions
f
•= min V
V=Cte
x—
Let X0 be a bounded invariant manifold. ALyapunov’s functionassociated toX0 is a mappingV from an open boundedU of XcontainingX0 to R+, of classC1, satisfying :
(i) V reaches its minimum inU ;
(ii) V is non increasing along the integral curves off : LfV ≤ 0 inU .
If U is not bounded, we may assume in addition that
(iii) lim‖x‖→∞, x∈X
V (x) = +∞. (proper function)
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LaSalle’s Invariance PrincipleTheorem Let C be a compact ofX = Rn, positively invariant forf , C ⊂ U open ofX.Let V be aC1 function satisfying
LfV ≤ 0 in U.
LetW0 = {x ∈ U |LfV = 0}
andX0 the largest invariant set byf contained inW0.Thus, for every initial condition inC, X0 is an attractor, i.e.⋂
t≥0
Xt(C) ⊂ X0.
If the level setV −1((−∞, c]) = {x ∈ X|V (x) ≤ c} is bounded forsomec ∈ R, one can choose
C = V −1((−∞, c]).
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Particular Cases
– If W0 = X0 and ifLfV < 0 in U\X0, thenX0 is an attractor.– If furthermoreV is a quadratic form onX and if LfV ≤ −αV in
U , then the convergence ofXt(x) to X0 is exponential forx ∈ C.
Example :
x = −x3, V (x) =1
2x2, LfV = −x4 = −4(V (x))2
but we don’t haveLfV ≤ −αV : convergence to 0 non exponential :
x(t) =1√
1x2
0+ 2t
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Pendulum with dissipation (followed):
{x1 = x2x2 = −g
l sin x1 − εx2.
V (x1, x2) =1
2x2
2 +g
l(1− cos x1).
We haveLfV = −εx22 ≤ 0 in S1 × R and
W0 = {(x1, x2)|LfV = 0} = {x2 = 0} = R.
ChooseC = ]− θ, +θ[⋂
V −1(]−∞, c]), for θ ∈]0, π[, compact po-sitively invariant.
W0 ∩ C : f|W0∩C = −(gl sin x1)∂
∂x2admits 0 as unique equilibrium
point, thus the convergence to 0∀ε > 0.
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Theorem Consider a hyperbolic equilibrium (resp. fixed) pointx ofthe vector fieldf . The following conditions are equivalent :
1. The equilibrium (resp. fixed) pointx has all its characteristic ex-ponents (resp. multipliers) with strictly negative real part (resp.inside the unit disc).
2. There exists a strong Lyapunov’s function (LfV ≤ −αV ), (resp.V (fk(x)) ≤ (1− α)kV (x)) in a neighborhood ofx.
Moreover, if one of the above conditions is satisfied,x is exponen-tially stable.
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Chetaev’s Function
W= Cste
Γf
•
∂Γ
x—
∂Γ
Let x be an equilibrium point off . The mappingW from U , neigh-borhood ofx, to R+, of classC1, is aChetaev’s function if
(i) U contains a coneΓ with non empty interior, with vertexx andpiecewise regular boundary∂Γ, such thatf is inwardΓ on∂Γ ;
(ii) limx→x,x∈Γ
W (x) = 0, W > 0 andLfW > 0 in Γ.
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Theorem An equilibrium pointx for which a Chetaev’s functionexists is unstable.In particular, if x is hyperbolic and has at least one characteristicexponent with positive real part, in a suitably chosen conic neighbo-rhood, the function
W (x) = ‖π+(x− x)‖2
whereπ+ is the projection on the eigenspace corresponding to theeigenvalues with positive real part, is a Chetaev’s fonction.
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2.5. Hartman-Grobman’s Theorem, Centre ManifoldThe vector field (resp. diffeomorphism)f having 0 as equilibrium(resp. fixed) point istopologically equivalent to its tangent linearmapping Az if there exists a homeomorphismh from a neighbo-rhoodU of 0 to itself, that maps every orbit off in an orbit of itstangent linear mapping and that preserves the sense of motion.In other words, such that
Xt(h(z)) = h(eAτ (t,z)z) ∀z ∈ U
(resp.fk(h(z)) = h(Aκ(k,z)z), ∀z ∈ U ) with τ strictly increasingreal function for allz (resp.κ strictly increasing integer function forall z).
The scalar fields−x and−kx are topologically equivalent.But x and−x are not.
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Eigenspace Decomposition of the Tangent Linear MappingIf A : hyperbolic, withk < n eignevalues with strictly positive realpart, counted with their multiplicity, andn− k with strictly negativereal part, also counted with their multiplicity.Eigenspaces ofA :
E+, of dimensionk, associated to the eigenvalues with strictly posi-tive real part ;
E−, of dimensionn − k, associated to the eigenvalues with strictlynegative real part.
E+ andE− are supplementary and invariant byA :
E+ ⊕ E− = Rn, AE+ ⊂ E+, AE− ⊂ E−.
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Non Linear Extension, Hyperbolic CaseWe call local stable manifold of f at the equilibrium point 0 thesubmanifold
W−loc(0) = {x ∈ U | lim
t→+∞Xt(x) = 0 andXt(x) ∈ U ∀t ≥ 0} .
We call local unstable manifold off at the equilibrium point 0 thesubmanifold
W+loc(0) = {x ∈ U | lim
t→+∞X−t(x) = 0 andX−t(x) ∈ U ∀t ≥ 0} .
Theorem (Hartman-Grobman) If 0 is a hyperbolic equilibriumpoint off , thenf is topologically equivalent to its tangent linear map-ping.Moreover, there exists local stable and unstable manifolds off at 0with dim W+
loc(0) = dim E+ anddim W−loc(0) = dim E−, tangent at
the origin toE+ andE− respectively and having the same regularityasf .
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Non Linear Extension, Non Hyperbolic CaseTheorem (Shoshitaishvili) If f is of classeCr and admits 0 as equi-librium (resp. fixed) point, it admits local stable, unstable and centremanifolds notedW−
loc(0), W+loc(0) andW 0
loc(0), of classCr, Cr andCr−1 respectively, tangent at the origin toE−, E+ andE0.W−
loc(0) andW+loc(0) are uniquely defined, whereasW 0
loc(0) isn’t ne-cessarily unique.Moreover,f is topologically equivalent, in a neighborhood of the ori-gin, to the vector field
−x1∂
∂x1+ x2
∂
∂x2+ f0(x3)
∂
∂x3
wherex1 : local coordinates ofW−, x2 : local coordinates ofW+
andf0(x3) : restriction off to W 0loc(0).
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Example of invariant manifold computation{x1 = −x1 + f1(x1, x2)x2 = f2(x1, x2)
f1 andf2 satisfy :fi(0, 0) = 0 and∂fi
∂xj(0, 0) = 0 for i, j = 1, 2.
Tangent linear system :
z =
(−1 00 0
)z
def= Az .
Eigenvalues : -1 and 0.
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Centre Manifold : x1 = h(x2) = h2x22 + h3x
32 + O(x4
2) in a neighbo-rhood of 0.
x1 = −h(x2) + f1(h(x2), x2) =dh
dx2f2(h(x2), x2).
with
fi((h(x2), x2) = 12∂2fi∂x2
2(0)x2
2 +
(∂2fi
∂x1∂x2(0)h2 + 1
6∂3fi∂x3
2(0)
)x3
2 + O(x42) i = 1, 2.
Identifying the 2nd and 3rd degree monomials :
h(x2) = 12∂2f1∂x2
2(0)x2
2 +
(12
(∂2f1
∂x1∂x2(0)− ∂2f2
∂x22(0)
)∂2f1∂x2
2(0) + 1
6∂3f1∂x3
2(0)
)x3
2 + O(x42)
and thex2-dynamics (central dynamics) :
x2 = 12∂2f2∂x2
2(0)x2
2 +
(12
∂2f2∂x1∂x2
(0)∂2f1
∂x22(0) + 1
6∂3f2∂x3
2(0)
)x3
2 + O(x42).
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The lowest degree term of the central dynamics is thus quadratic.
Seta = 12∂2f2∂x2
2(0) and assume thata 6= 0. The integral curve of the
central dynamics is, forx2(0) sufficiently small :
x2(t) = (x2(0)−1 − at)−1
not L-stable in a neighborhood of 0 (blow up in finite-time atT =1
ax2(0)for sign(x2) = sign(a)).
If, on the contrarya = 0, the central dynamics is given at the order 3by
x2 =
(12
∂2f2∂x1∂x2
(0)∂2f1
∂x22(0) + 1
6∂3f2∂x3
2(0)
)x3
2 + O(x42)
L-stable if
(∂2f2
∂x1∂x2(0)∂
2f1∂x2
2(0) + 1
3∂3f2∂x3
2(0)
)< 0 and unstable other-
wise.The stability or the instability of the central dynamics implies thelocal stability or instability of the overall system.