Jos e Ferreira AlvesJos e F. Alves (CMUP) Ergodic properties Impulsive semi ows 16 / 32 The next...

Propriedades ergódicas de semifluxos impulsivos

José Ferreira Alves

Impulsive dynamical systems

Dynamical systems with impulse effects seem to be an adequatemathematical model to describe real world phenomena that exhibitsudden changes in their states.

An impulsive dynamical system is prescribed by three ingredients:

a continuous semiflow ϕ on a space X which governs the state of thesystem between impulses;

a set D ⊂ X where the flow undergoes some abrupt perturbations;a function I : D → X which specifies how a jump event happens eachtime a trajectory of the flow hits D.

José F. Alves (CMUP) Ergodic properties Impulsive semiflows 2 / 32

The major developments so far on the theory of impulsive dynamicalsystems have been to extend the classical theorem on existence anduniqueness of solutions and to establish sufficient conditions to ensure acharacterization and some asymptotic stability of the limit sets [Băınov,Bonotto, Ciesielski, Federson, Kaul, Lakshmikantham, Simeonov...].

Meanwhile, a significant progress in the study of dynamical systems hasbeen achieved due to a remarkable sample of the so-called ergodictheorems which concern the connection between the time and the spatialaverages of observable measurable maps along orbits, and whosefundamental request is the existence of an invariant probability measure.

The results we present here give a first approach to the theory of impulsivedynamical systems from a probabilistic point of view, providing:

conditions for the existence of invariant probability measures;

a suitable notion of topological entropy;

conditions for the validity of a variational principle.

.


Semiflows

Let X be a compact metric space. We say that ψ : R+0 × X → X is asemiflow if for all x ∈ X and all s, t ∈ R+0 we have

1 ψ0(x) = x ,

2 ψt+s(x) = ψt(ψs(x)),

where ψt(x) stands for ψ(t, x). The curve defined for t ≥ 0 by ψt(x) iscalled the ψ-trajectory of the point x ∈ X .

Given a continuous semiflow ϕ : R+0 × X → X , a compact set D ⊂ X anda continuous function I : D → X , define τ1 : X → [0,+∞] as

τ1(x) =

{inf {t > 0 : ϕt(x) ∈ D} , if ϕt(x) ∈ D for some t > 0;+∞, otherwise.

τ1 is called the first impulsive time.


Assume τ1(x) > 0 for all x ∈ X . We define the impulsive trajectory γx(t)and the subsequent impulsive times as follows:

For 0 ≤ t < τ1(x), define

γx(t) = ϕt(x).

If τ1(x)

Impulsive semiflows

In general, T (x) < +∞ or T (x) = +∞ are possible for x ∈ X . Under thecondition I (D) ∩ D = ∅, for instance, we have T (x) =∞ for all x ∈ X .

We say that (X , ϕ,D, I ) is an impulsive dynamical system if

τ1(x) > 0 and T (x) = +∞, for all x ∈ X .

The semiflow ψ of an impulsive dynamical system (X , ϕ,D, I ) is defined as

ψ : R+0 × X −→ X(t, x) 7−→ γx(t),

where γx(t) is the impulsive trajectory of x determined by (X , ϕ,D, I ).It is straightforward to check that ψ is indeed a semiflow.


Invariant probability measures

A map between two topological spaces is called measurable if thepre-image of any Borel set is a Borel set. An invertible map is calledbimeasurable if both the map and its inverse are measurable.

Notice that the measurability of a semiflow ψ : R+ × X → X gives inparticular that ψt is measurable for each t ≥ 0.

A probability measure µ on the Borel sets of a topological space X is saidto be invariant by a semiflow ψ (or ψ-invariant) if ψ is measurable and

µ(ψ−1t (A)) = µ(A),

for every Borel set A ⊂ X and every t ≥ 0.

Let M(X ) be the set of all probability measures on the Borel sets of Xand Mψ(X ) the set of those measures in M(X ) which are ψ-invariant.


Given a measurable map f : X → Y between two topological spaces Xand Y we introduce the push-forward map

f∗ :M(X ) −→ M(Y )µ 7−→ f∗µ

with f∗µ defined for any µ ∈M(X ) and any Borel set B ⊂ Y as

f∗µ(B) = µ(f−1(B)).

We clearly have for µ ∈M(X )

µ ∈Mψ(X ) ⇐⇒ (ψt)∗µ = µ, for all t ≥ 0.


Non-wandering sets

A point x ∈ X is said to be non-wandering for a semiflow ψ if, for everyneighborhood U of x and any T > 0, there exists t ≥ T such that

ψ−1t (U) ∩ U 6= ∅.

The non-wandering set of ψ is defined as

Ωψ = {x ∈ X : x is non-wandering for ψ }.

It follows that Ωψ is closed. Moreover, Ωψ contains the set of limit pointsof the semiflow, which is clearly nonempty when X is compact. Therefore,Ωψ is always nonempty and compact.

The support of a measure µ ∈M(X ) is defined as the set of points x ∈ Xsuch that µ(U) > 0 for any neighborhood U of x . It’s a general fact thatany invariant probability measure for a semiflow ψ necessarily has itssupport contained in the non-wandering set Ωψ.


Example: system with no invariant probability measure

Consider the phase space

X ={

(r cos θ, r sin θ) ∈ R2 : 1 ≤ r ≤ 2, θ ∈ [0, 2π]},

D I(D)..

and define ϕ as the flow associatedto the vector field in X given by{

r ′ = f (r)

θ′ = 1,f (r) = 1− r , 1 ≤ r ≤ 2.

Trajectories of ϕ are curves spiralingto the inner border circle of X . Take

D = {(1, 0)} and I (1, 0) = (2, 0) .


Let ψ be the semiflow of the impulsive dynamical system (X , ϕ,D, I ).

D I(D)..

It is not difficult to see that

Ωψ = {(cos θ, sin θ) : 0 ≤ θ ≤ 2π}

and that this set is not forwardinvariant under ψ. We claimthat ψ has no invariant probabilitymeasure. Actually, if µ were sucha measure, then we would have

1 = µ(Ωψ) = µ(ψ−12π (Ωψ)).

However,

ψ−12π (Ωψ) = ∅.

Remark

The non-wandering set of an impulsive semiflow may not be forwardinvariant, as the previous example illustrates.


We introduce a function τΩ : Ωψ → [0,+∞], defined as

τΩ(x) =

{τ1(x), if x ∈ Ωψ \ D;0, if x ∈ Ωψ ∩ D.

Theorem 1 (A.-Carvalho, 14)

Let ψ be the semiflow of an impulsive dynamical system (X , ϕ,D, I ) s.t.

1 I (Ωψ ∩ D) ⊂ Ωψ \ D;2 τΩ is continuous.

Then ψ has some invariant probability measure.

If Ωψ ∩ D = ∅, then ψ has some invariant probability measure.The continuity of τΩ essentially means that there are no points in Ωψon the “right hand side” of D.

In the previous example we have τΩ : Ωψ → [0, 2π] given by

τΩ(cos θ, sin θ) = 2π − θ,

clearly discontinuous at the point (1, 0).


Example: system with an invariant probability measure

Consider the phase space X as the annulus

X ={

(r cos θ, r sin θ) ∈ R2 : 1 ≤ r ≤ 2, θ ∈ [0, 2π]}

DI(D)

and define ϕ as theflow of the vector field in X given by{

r ′ = 0

θ′ = 1.

Trajectories of ϕ are circlesspinning around zero. Take

D = {(r , 0) ∈ X : 1 ≤ r ≤ 2}

and define I : D → X by

I (r , 0) =

(−1

2− 1

2r , 0

).


Example: system with an invariant probability measure

Let ψ be the semiflow of the impulsive dynamical system (X , ϕ,D, I ).We have

Ωψ = {(cos θ, sin θ) : π ≤ θ ≤ 2π} .

DI(D)

Ωψ is not forward invariantunder ψ: the trajectory of (1, 0) ∈ Ωψis not contained in Ωψ. Still, we have

I (Ωψ∩D) = I ({(1, 0)}) = {(−1, 0)} ⊂ Ωψ\D.

Moreover,

τΩ(cos θ, sin θ) = 2π − θ,

which is clearly continuous in Ωψ.

By Theorem 1, ψ has an invariant probability measure.


Theorem 2 (A.-Carvalho, 14)

Let ψ be the semiflow of an impulsive dynamical system (X , ϕ,D, I ) s.t.

1 I (Ωψ ∩ D) ⊂ Ωψ \ D;2 τΩ is continuous.

Then there are a compact metric space X̃ , a continuous semiflow ψ̃ in X̃and a continuous bimeasurable map g : Ωψ \ D → X̃ such that

1 ψ̃t ◦ g |Ωψ\D = g ◦ ψt |Ωψ\D for all t ≥ 0;2 (ι ◦ g−1)∗ :Mψ̃(X̃ )→Mψ(X ) is a bijection, where ι : Ωψ \ D → X

is the inclusion map.

Theorem 1 is a corollary of Theorem 2. In fact, as ψ̃ is continuousand X̃ is a compact metric space, then ψ̃ has some invariantprobability measure, by Kryloff-Bogoliouboff Theorem.

Under our assumptions we have µ(D) = 0 for any ψ-invariantmeasure µ, and so the second conclusion follows from the first one.


Quotient dynamics

Given an impulsive dynamical system (X , ϕ,D, I ), we consider thequotient space X/∼, where ∼ is the equivalence relation given by

x ∼ y ⇔ x = y , y = I (x), x = I (y) or I (x) = I (y).

We shall use x̃ to represent the equivalence class of x ∈ X .Consider X/∼ with the quotient space and the natural projection

π : X → X/∼.

In general, as Ωψ is compact, π(Ωψ) is a compact pseudometric space.

Lemma 1

π(Ωψ) is a compact metric space.

It’s enough to prove that π(Ωψ) is a T0 space: given two distinct points inπ(Ωψ), there is an open set containing one and not the other.


The next proposition gives the first conclusion of Theorem 2 withX̃ = π(Ωψ) and g = π.

Proposition

Assume that I (Ωψ ∩D) ⊂ Ωψ \D and τΩ is continuous. Then π|Ωψ\D is acontinuous bimeasurable bijection onto π(Ωψ) and there exists acontinuous semiflow ψ̃ : R+0 × π(Ωψ)→ π(Ωψ) such that for all t ≥ 0

ψ̃t ◦ π|Ωψ\D = π ◦ ψt |Ωψ\D .

Assuming that I (Ωψ ∩D) ⊂ Ωψ \D, from the definition of the equivalencerelation one easily deduces that

π(Ωψ \ D) = π(Ωψ).

Additionally, for any x , y ∈ Ωψ \ D we have x ∼ y if and only if x = y .This shows that π|Ωψ\D is a continuous bijection (not necessarily ahomeomorphism) from Ωψ \ D onto π(Ωψ).


Setting for each x ∈ Ωψ \ D and t ≥ 0

ψ̃(t, x̃) = π(ψ(t, x)),

then ψ̃ : R+ × π(Ωψ)→ π(Ωψ) satisfies for all t ≥ 0

ψ̃t ◦ π|Ωψ\D = π ◦ ψt |Ωψ\D .

We are left to prove that ψ̃ is continuous.

Consider for each x̃ ∈ π(Ωψ) the map ψ̃x̃ : R+0 → π(Ωψ) defined by

ψ̃x̃(t) = ψ̃(t, x̃).

It is enough to prove that the maps ψ̃x̃ and ψ̃t are continuous.


Case 1. ψ̃x̃ is continuous for each x ∈ Ωψ \ D.Assume first that t0 ≥ 0 is not an impulsive time for x .We have for t in a small neighborhood of t0 in R+0 ,

ψ̃x̃(t) = π(ϕ(t, x))

As ϕ is continuous, this obviously gives the continuity of ψ̃x̃ at t0.If t0 is an impulsive time for x , then we have

limt→t−0

ψ̃x̃(t) = limt→t−0

π(ψ(t, x)) = limt→t−0

π(ϕ(t, x)) = π(ϕ(t0, x)).

As ϕ(t0, x) ∈ D, it follows from the definition of ψ(t0, x) and theequivalence relation that

π(ϕ(t0, x)) = π(I (ϕ(t0, x))) = π(ψ(t0, x)) = ψ̃x̃(t0).

Continuity on the right hand side of t0 follows easily from the fact that theimpulsive trajectories are continuous on the right hand side.


Case 2. ψ̃t is continuous for each t ≥ 0.As we are considering the quotient topology on π(Ωψ \ D), then

ψ̃t continuous ⇐⇒ ψ̃t ◦ π|Ωψ\D continuous.

By definitionψ̃t ◦ π|Ωψ\D = π ◦ ψt |Ωψ\D .

The continuity of ψ̃t ◦ π|Ωψ\D is an immediate consequence of Lemma 2.

Lemma 2

Assume that I (Ωψ ∩ D) ⊂ Ωψ \ D and τΩ is continuous.Then π ◦ ψt |Ωψ\D is continuous for all t ≥ 0.

D I(D)

... .

x

y


Variational principle

The continuous bimeasurable map g given by Theorem 2 allows us toexchange information between the semiflows ψ̃ in X̃ and ψ in Ωψ.

For instance, the topological entropy of ψ̃ is given by

htop(ψ̃) = htop(ψ̃1) = sup{

hν(ψ̃1) : ν ∈ Mψ̃(X̃ )},

where hν(ψ̃1) stands for the measure-theoretic entropy of the map ψ̃1 withrespect to the probability ν. It follows from our second theorem that

sup {hµ(ψ1) : µ ∈ Mψ(X )} = htop(ψ̃).

Problems1 Variational principle: sup {hµ(ψ1) : µ ∈ Mψ(X )} = htop(ψ)?2 What is the definition of htop(ψ)?


Topological entropy: classical definition

Let ϕ be a continuous semiflow on a compact metric space X .Given x ∈ X , T > 0 and � > 0 we define the dynamic ball

B(x , ϕ,T , �) = {y ∈ X : dist(ϕt(x), ϕt(y)) < �, for every t ∈ [0,T ]} .

The continuity of ϕ implies that B(x , ϕ,T , �) is an open set of X since itis the open ball centered at x of radius � for the metric

distϕT (x , y) = max0≤t≤T{dist(ϕt(x), ϕt(y))}.

A set E ⊆ X is said to be (ϕ,T , �)-separated if, for each x ∈ E there is noother points of E inside the ball B(x , ϕ,T , �) besides x .


As a consequence of the compactness of X and the continuity of ϕ, anyset E ⊆ X which is (ϕ,T , �)-separated is finite.

If we denote by |E | the cardinality of E , then we define

s(ϕ,T , �) = max{|E | : E is (ϕ,T , �)-separated},

and the growth rate of this number as

h(ϕ, �) = lim supT→+∞

1

Tlog s(ϕ,T , �).

The topological entropy of ϕ is then given by

htop(ϕ) = lim�→0+

h(ϕ, �).


Topological entropy: a new definition

Let X be a compact metric space and ψ : R+0 × X → X a (not necessarilycontinuous) semiflow. Consider a function τ

X 3 x 7−→ (τn(x))n∈A(x),

where

either A(x) = {1, . . . , `} for some ` ∈ N or A(x) = N;(τn(x))n∈A(x) is a strictly increasing (possibly finite) sequence in R+.

We say that τ is admissible with respect to Z ⊂ X if there exists η > 0such that for all x ∈ Z

τ1(x) ≥ η;and for all x ∈ X

τn(ψs(x)) = τn(x)− s, for all n ∈ N and all s ≥ 0;τn+1(x)− τn(x) ≥ η, for all n ∈ N with n + 1 ∈ A(x).


For each admissible function τ , x ∈ X , T > 0 and 0 < δ < η/2, we define

JτT ,δ(x) = (0,T ] \

nT (x)⋃j=1

]τj(x)− δ, τj(x) + δ [

,where

nT (x) = max{n ≥ 1 : τn(x) ≤ T}.

The τ -dynamical ball of radius � > 0 centered at x is the set

Bτ (x , ψ,T , �, δ) ={

y ∈ X : dist(ψt(x), ψt(y)) < �, ∀t ∈ JτT ,δ(x)}.

A set E ⊆ X is called (ψ, τ,T , �, δ)-separated if, for each x ∈ E , we have

y /∈ Bτ (x , ψ,T , �, δ), ∀y ∈ E \ {x}.


As before, define

sτ (ψ,T , �, δ) = sup{|E | : E is a finite (ψ, τ,T , �, δ)-separated set},

and the growth rate

hτ (ψ, �, δ) = lim supT→+∞

1

Tlog sτ (ψ,T , �, δ),

As the function � 7→ hτ (ψ, �, δ) is decreasing, the following limit exists

hτ (ψ, δ) = lim�→0+

hτ (ψ, �, δ).

Finally, as the function δ 7→ hτ (ψ, δ) is also decreasing, we define theτ -topological entropy of ψ

hτtop(ψ) = limδ→0+

hτ (ψ, δ).

Theorem 3 (A.-Carvalho-Vásquez, 14)

Let ϕ : R+0 × X → X be a continuous semiflow on a compact metricspace X and τ an admissible function on X . Then hτtop(ϕ) = htop(ϕ).


Impulsive semiflows

Let now ψ be a semiflow of an impulsive dynamical system (X , ϕ,D, I ).Given ξ > 0, define

Dξ =⋃x∈D{ϕt(x) : 0 ≤ t < ξ}.

We say that D satisfies a half-tube condition if there is ξ0 > 0 such that:

1 ϕt(x) ∈ Dξ0 ⇒ ∃0 ≤ t ′ < t with ϕt′(x) ∈ D;2 for all x1, x2 ∈ D with x1 6= x2

{ϕt(x1) : 0 < t < ξ0} ∩ {ϕt(x2) : 0 < t < ξ0} = ∅.

This is to ensure that removing Dξ from X we have a ψ-invariant regionon which ψ has the same topological entropy.


For small enough ξ > 0 we define

Xξ = X \ Dξ.

Lemma

If D satisfies a half-tube condition, then ψt(Xξ) ⊆ Xξ for all t ≥ 0 and

hτtop(ψ) = hτtop(ψ|Xξ).

We say that I (D) is transverse if there are s0 > 0 and ξ0 > 0 such that

ϕt(x) ∈ I (D) ⇒ ϕt+s(x) /∈ I (D), ∀ 0 < s < s0;for all x1, x2 ∈ I (D) with x1 6= x2

{ϕt(x1) : 0 < t < ξ0} ∩ {ϕt(x2) : 0 < t < ξ0} = ∅.

This property holds, for instance, when ϕ is a C 1 semiflow and I (D) istransversal to the flow direction.


Quotient dynamics

Given an impulsive dynamical system (X , ϕ,D, I ), consider the quotientspace X/∼ with the quotient topology, where ∼ is as before

x ∼ y ⇔ x = y , y = I (x), x = I (y) or I (x) = I (y).

Let π : X → X/∼ be the natural projection. If d denotes the metric onX , the metric d̃ in π(X ) that induces the quotient topology is given by

d̃ (x̃ , ỹ) = inf {d (p1, q1) + d (p2, q2) + · · ·+ d (pn, qn)},

where p1, q1, . . . , pn, qn is any chain of points in X such that p1 ∼ x ,q1 ∼ p2, q2 ∼ p3, ... qn ∼ y . In particular, we have

d̃ (x̃ , ỹ) ≤ d (x , y), ∀x , y ∈ X .


The length n of chains needed to evaluate d̃ (x̃ , ỹ) may be arbitrarily large,preventing us from comparing d̃ (x̃ , ỹ) with d (p, q) for p ∼ x and q ∼ y .This difficulty can be overcome if the map I does not expand distances, forinstance: I is called 1-Lipschitz if

dist (I (x), I (y)) ≤ dist (x , y), for all x , y ∈ D.

Lemma

If I is 1-Lipschitz, then for all x̃ , ỹ ∈ π(X ) there exist p, q ∈ X such that

p ∼ x , q ∼ y and d(p, q) ≤ 2 d̃ (x̃ , ỹ).

Consider now τξ : Xξ ∪ D → [0,+∞] defined by

τξ(x) =

{τ1(x), if x ∈ Xξ;0, if x ∈ D


Theorem 4 (A.-Carvalho-Vásquez, 14)

Let ψ be the semiflow of an impulsive dynamical system (X , ϕ,D, I ) suchthat D satisfies a half-tube condition, I is 1-Lipschitz, I (D) ∩ D = ∅, I (D)is transverse and τξ is continuous. Then there exists a continuous semiflowψ̃ in π(Xξ) and a continuous invertible bimeasurable map h : Xξ → π(Xξ)such that ψ̃t ◦ h = h ◦ ψt for all t ≥ 0 and

hτtop(ψ) = htop(ψ̃).

Corollary

Let ψ be the semiflow of an impulsive dynamical system (X , ϕ,D, I )satisfying the assumptions of Theorem 4. Then

hτtop(ψ) = sup {hµ(ψ1) : µ ∈Mψ(X )}.


References

J. F. Alves, M. CarvalhoInvariant probability measures and non-wandering sets for impulsivesemiflowsJ. Stat. Phys. 157 (2014), no. 6, 1097–1113

J. F. Alves, M. Carvalho, C. VásquezA variational principle for impulsive semiflowsarxiv.org/pdf/1410.2372


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