Chap.2. Introduction to Dynamical Systems

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.

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.

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.

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.

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)).

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.

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

-1.5

-1

-0.5

0

0.5

1

1.5

•

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

•

-0.01 -0.005 0 0.005 0.01

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

•

-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.5

0

0.5

1

•

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.

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.

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.

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)).

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.

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.

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).

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).

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.

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 !

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.

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.

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.

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.

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)

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]).

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

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.

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.

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.

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.

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.

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−.

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 .

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).

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.

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).

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.