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Non-compact global attractors for slowlynon-dissipative equations
Juliana F. S. Pimentel
Instituto Superior TécnicoUniversidade de Lisboa
June 2014
Joint work with C. Rocha
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 1 / 35
Outline
1 IntroductionDefinitions - dissipative and non-dissipative systemsMotivation - dissipative system and its global attractor
2 Problem statementSlowly non-dissipative systemMain objective
3 Preliminary resultsEquilibria at infinity
4 Main resultsNon-compact global attractorAssociated permutation
5 Strategy of the proof
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 2 / 35
Outline
1 IntroductionDefinitions - dissipative and non-dissipative systemsMotivation - dissipative system and its global attractor
2 Problem statementSlowly non-dissipative systemMain objective
3 Preliminary resultsEquilibria at infinity
4 Main resultsNon-compact global attractorAssociated permutation
5 Strategy of the proof
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 3 / 35
Reaction-diffusion equation
1 Consider the scalar reaction-diffusion equation{ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
(1)
where f : [0, π]× R2 → R is a C2 function.2 Equation (1) generates a local semiflow that is
I dissipative: global existence and ultimately boundedness for allsolutions;
I fast non-dissipative: at least one solution blows-up in finite-time; orI slowly non-dissipative: global existence for all solutions and there
exists a solution whose norm grows to infinity with time.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 4 / 35
Reaction-diffusion equation
1 Consider the scalar reaction-diffusion equation{ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
(1)
where f : [0, π]× R2 → R is a C2 function.2 Equation (1) generates a local semiflow that is
I dissipative: global existence and ultimately boundedness for allsolutions;
I fast non-dissipative: at least one solution blows-up in finite-time; orI slowly non-dissipative: global existence for all solutions and there
exists a solution whose norm grows to infinity with time.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 4 / 35
Reaction-diffusion equation
1 Consider the scalar reaction-diffusion equation{ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
(1)
where f : [0, π]× R2 → R is a C2 function.2 Equation (1) generates a local semiflow that is
I dissipative: global existence and ultimately boundedness for allsolutions;
I fast non-dissipative: at least one solution blows-up in finite-time; orI slowly non-dissipative: global existence for all solutions and there
exists a solution whose norm grows to infinity with time.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 4 / 35
Reaction-diffusion equation
1 Consider the scalar reaction-diffusion equation{ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
(1)
where f : [0, π]× R2 → R is a C2 function.2 Equation (1) generates a local semiflow that is
I dissipative: global existence and ultimately boundedness for allsolutions;
I fast non-dissipative: at least one solution blows-up in finite-time; orI slowly non-dissipative: global existence for all solutions and there
exists a solution whose norm grows to infinity with time.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 4 / 35
Reaction-diffusion equation
1 Consider the scalar reaction-diffusion equation{ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
(1)
where f : [0, π]× R2 → R is a C2 function.2 Equation (1) generates a local semiflow that is
I dissipative: global existence and ultimately boundedness for allsolutions;
I fast non-dissipative: at least one solution blows-up in finite-time; orI slowly non-dissipative: global existence for all solutions and there
exists a solution whose norm grows to infinity with time.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 4 / 35
Dissipative system1 An example of sufficient conditions for (1) to be dissipative:
f (x ,u,0).u < 0, for |u| large enough,
and|f (x ,u,p)| ≤ c(1 + |p|γ),
with c > 0 and 0 ≤ γ < 2, uniformly for x and u in compact sets.2 There exists a global attractor A, i.e., a maximal compact invariant
set attracting each bounded set in the appropriate state space.3 Permutation σ associated to (1): assume hyperbolicity and let
E = {v1, ..., vn} be the set of equilibria of (1) with
v1(0) < v2(0) < ... < vn(0),
then σ is defined by
vσ(1)(π) < vσ(2)(π) < ... < vσ(n)(π).
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 5 / 35
Dissipative system1 An example of sufficient conditions for (1) to be dissipative:
f (x ,u,0).u < 0, for |u| large enough,
and|f (x ,u,p)| ≤ c(1 + |p|γ),
with c > 0 and 0 ≤ γ < 2, uniformly for x and u in compact sets.2 There exists a global attractor A, i.e., a maximal compact invariant
set attracting each bounded set in the appropriate state space.3 Permutation σ associated to (1): assume hyperbolicity and let
E = {v1, ..., vn} be the set of equilibria of (1) with
v1(0) < v2(0) < ... < vn(0),
then σ is defined by
vσ(1)(π) < vσ(2)(π) < ... < vσ(n)(π).
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 5 / 35
Dissipative system1 An example of sufficient conditions for (1) to be dissipative:
f (x ,u,0).u < 0, for |u| large enough,
and|f (x ,u,p)| ≤ c(1 + |p|γ),
with c > 0 and 0 ≤ γ < 2, uniformly for x and u in compact sets.2 There exists a global attractor A, i.e., a maximal compact invariant
set attracting each bounded set in the appropriate state space.3 Permutation σ associated to (1): assume hyperbolicity and let
E = {v1, ..., vn} be the set of equilibria of (1) with
v1(0) < v2(0) < ... < vn(0),
then σ is defined by
vσ(1)(π) < vσ(2)(π) < ... < vσ(n)(π).
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 5 / 35
Global attractor
Proposition (Henry(1981))
The global attractor A associated to the dissipative equation (1) iscomposed by the set of equilibria E and their heteroclinic connections.
Proposition (Fusco and Rocha(1991), Fiedler and Rocha(1996))The permutation σ ∈ S(n) determines which equilibria are connectedand which are not.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 6 / 35
Outline
1 IntroductionDefinitions - dissipative and non-dissipative systemsMotivation - dissipative system and its global attractor
2 Problem statementSlowly non-dissipative systemMain objective
3 Preliminary resultsEquilibria at infinity
4 Main resultsNon-compact global attractorAssociated permutation
5 Strategy of the proof
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 7 / 35
Slowly non-dissipative equation
Consider {ut = uxx + bu + g(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0.
(2)
Assumptions and notation:f (x ,u,ux) = bu + g(x ,u,ux)
b > 0 and g : [0, π]× R2 → R is a C2 functiong is bounded and g(x ,u,p) is globally Lipschitz in (u,p)X = L2([0, π]) with norm ‖ · ‖A = −∂xx − bIXα are the associated fractional power spaces
Xα := D((A + (b + 1)I)α).
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 8 / 35
Slowly non-dissipative equation
Consider {ut = uxx + bu + g(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0.
(2)
Assumptions and notation:f (x ,u,ux) = bu + g(x ,u,ux)
b > 0 and g : [0, π]× R2 → R is a C2 functiong is bounded and g(x ,u,p) is globally Lipschitz in (u,p)X = L2([0, π]) with norm ‖ · ‖A = −∂xx − bIXα are the associated fractional power spaces
Xα := D((A + (b + 1)I)α).
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 8 / 35
Slowly non-dissipative equation
Consider {ut = uxx + bu + g(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0.
(2)
Assumptions and notation:f (x ,u,ux) = bu + g(x ,u,ux)
b > 0 and g : [0, π]× R2 → R is a C2 functiong is bounded and g(x ,u,p) is globally Lipschitz in (u,p)X = L2([0, π]) with norm ‖ · ‖A = −∂xx − bIXα are the associated fractional power spaces
Xα := D((A + (b + 1)I)α).
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 8 / 35
Slowly non-dissipative systemIf b > 0 then equation (2) generates a slowly non-dissipative system.
1 Global existence for every initial condition.2 {ϕj}j∈N0 orthonormal basis in L2 and λj = j2 − b.3 Any solution u(t , x) is represented by u(t , x) =
∑∞j=0 uj(t)ϕj(x).
4 Apply the E0-projection to equation (2) and obtain
ddt
u0(t) = −λ0u0(t) + g0(t). (3)
5 The solution of (3) is given by
u0(t) = uh0(0)e
−λ0t +
∫ t
∞e−λ0(t−s)g0(s)ds.
6 Since λ0 = −b < 0, if take an initial condition u0 such thatuh
0(0) 6= 0 then u0(t)→∞ as t →∞.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 9 / 35
Slowly non-dissipative systemIf b > 0 then equation (2) generates a slowly non-dissipative system.
1 Global existence for every initial condition.2 {ϕj}j∈N0 orthonormal basis in L2 and λj = j2 − b.3 Any solution u(t , x) is represented by u(t , x) =
∑∞j=0 uj(t)ϕj(x).
4 Apply the E0-projection to equation (2) and obtain
ddt
u0(t) = −λ0u0(t) + g0(t). (3)
5 The solution of (3) is given by
u0(t) = uh0(0)e
−λ0t +
∫ t
∞e−λ0(t−s)g0(s)ds.
6 Since λ0 = −b < 0, if take an initial condition u0 such thatuh
0(0) 6= 0 then u0(t)→∞ as t →∞.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 9 / 35
Slowly non-dissipative systemIf b > 0 then equation (2) generates a slowly non-dissipative system.
1 Global existence for every initial condition.2 {ϕj}j∈N0 orthonormal basis in L2 and λj = j2 − b.3 Any solution u(t , x) is represented by u(t , x) =
∑∞j=0 uj(t)ϕj(x).
4 Apply the E0-projection to equation (2) and obtain
ddt
u0(t) = −λ0u0(t) + g0(t). (3)
5 The solution of (3) is given by
u0(t) = uh0(0)e
−λ0t +
∫ t
∞e−λ0(t−s)g0(s)ds.
6 Since λ0 = −b < 0, if take an initial condition u0 such thatuh
0(0) 6= 0 then u0(t)→∞ as t →∞.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 9 / 35
Slowly non-dissipative systemIf b > 0 then equation (2) generates a slowly non-dissipative system.
1 Global existence for every initial condition.2 {ϕj}j∈N0 orthonormal basis in L2 and λj = j2 − b.3 Any solution u(t , x) is represented by u(t , x) =
∑∞j=0 uj(t)ϕj(x).
4 Apply the E0-projection to equation (2) and obtain
ddt
u0(t) = −λ0u0(t) + g0(t). (3)
5 The solution of (3) is given by
u0(t) = uh0(0)e
−λ0t +
∫ t
∞e−λ0(t−s)g0(s)ds.
6 Since λ0 = −b < 0, if take an initial condition u0 such thatuh
0(0) 6= 0 then u0(t)→∞ as t →∞.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 9 / 35
Slowly non-dissipative systemIf b > 0 then equation (2) generates a slowly non-dissipative system.
1 Global existence for every initial condition.2 {ϕj}j∈N0 orthonormal basis in L2 and λj = j2 − b.3 Any solution u(t , x) is represented by u(t , x) =
∑∞j=0 uj(t)ϕj(x).
4 Apply the E0-projection to equation (2) and obtain
ddt
u0(t) = −λ0u0(t) + g0(t). (3)
5 The solution of (3) is given by
u0(t) = uh0(0)e
−λ0t +
∫ t
∞e−λ0(t−s)g0(s)ds.
6 Since λ0 = −b < 0, if take an initial condition u0 such thatuh
0(0) 6= 0 then u0(t)→∞ as t →∞.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 9 / 35
Slowly non-dissipative systemIf b > 0 then equation (2) generates a slowly non-dissipative system.
1 Global existence for every initial condition.2 {ϕj}j∈N0 orthonormal basis in L2 and λj = j2 − b.3 Any solution u(t , x) is represented by u(t , x) =
∑∞j=0 uj(t)ϕj(x).
4 Apply the E0-projection to equation (2) and obtain
ddt
u0(t) = −λ0u0(t) + g0(t). (3)
5 The solution of (3) is given by
u0(t) = uh0(0)e
−λ0t +
∫ t
∞e−λ0(t−s)g0(s)ds.
6 Since λ0 = −b < 0, if take an initial condition u0 such thatuh
0(0) 6= 0 then u0(t)→∞ as t →∞.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 9 / 35
Slowly non-dissipative systemIf b > 0 then equation (2) generates a slowly non-dissipative system.
1 Global existence for every initial condition.2 {ϕj}j∈N0 orthonormal basis in L2 and λj = j2 − b.3 Any solution u(t , x) is represented by u(t , x) =
∑∞j=0 uj(t)ϕj(x).
4 Apply the E0-projection to equation (2) and obtain
ddt
u0(t) = −λ0u0(t) + g0(t). (3)
5 The solution of (3) is given by
u0(t) = uh0(0)e
−λ0t +
∫ t
∞e−λ0(t−s)g0(s)ds.
6 Since λ0 = −b < 0, if take an initial condition u0 such thatuh
0(0) 6= 0 then u0(t)→∞ as t →∞.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 9 / 35
Non-compact global attractor
1 Equation (2) possesses at least one solution blowing-up in infinitetime, i.e., a grow-up solution.
2 Non-compact global attractor : non-empty minimal set in the statespace Xα attracting all bounded sets of Xα.
3 Objective:I Obtain a decomposition for the non-compact global attractor.
F Ben-Gal (2010) - for g = g(u).I Describe the heteroclinic connections in terms of an associated
permutation.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 10 / 35
Non-compact global attractor
1 Equation (2) possesses at least one solution blowing-up in infinitetime, i.e., a grow-up solution.
2 Non-compact global attractor : non-empty minimal set in the statespace Xα attracting all bounded sets of Xα.
3 Objective:I Obtain a decomposition for the non-compact global attractor.
F Ben-Gal (2010) - for g = g(u).I Describe the heteroclinic connections in terms of an associated
permutation.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 10 / 35
Non-compact global attractor
1 Equation (2) possesses at least one solution blowing-up in infinitetime, i.e., a grow-up solution.
2 Non-compact global attractor : non-empty minimal set in the statespace Xα attracting all bounded sets of Xα.
3 Objective:I Obtain a decomposition for the non-compact global attractor.
F Ben-Gal (2010) - for g = g(u).I Describe the heteroclinic connections in terms of an associated
permutation.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 10 / 35
Outline
1 IntroductionDefinitions - dissipative and non-dissipative systemsMotivation - dissipative system and its global attractor
2 Problem statementSlowly non-dissipative systemMain objective
3 Preliminary resultsEquilibria at infinity
4 Main resultsNon-compact global attractorAssociated permutation
5 Strategy of the proof
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 11 / 35
Asymptotic behavior
There exists a Lyapunov functional for equation (2).Let u(t , ·) be a solution of equation (2). Then u(t , ·) eitherconverges to some (bounded) equilibrium as t goes to infinity oru(t , ·) is a grow-up solution.Obtain the limits of the unbounded solutions.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 12 / 35
Asymptotic behavior
There exists a Lyapunov functional for equation (2).Let u(t , ·) be a solution of equation (2). Then u(t , ·) eitherconverges to some (bounded) equilibrium as t goes to infinity oru(t , ·) is a grow-up solution.Obtain the limits of the unbounded solutions.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 12 / 35
Asymptotic behavior
There exists a Lyapunov functional for equation (2).Let u(t , ·) be a solution of equation (2). Then u(t , ·) eitherconverges to some (bounded) equilibrium as t goes to infinity oru(t , ·) is a grow-up solution.Obtain the limits of the unbounded solutions.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 12 / 35
Grow-up solutions
Let {ϕj}j∈N be an orthonormal basis in L2([0, π]) comprised of theeigenfunctions of A = −∂xx − bI with Neumann boundaryconditions.If u(t , ·) is a grow-up solution then
u(t , ·)‖u(t , ·)‖
−→ ϕ±j (·) in L2,
with j ≤√
b and ϕ±j := ±ϕj .
The original orbit grows-up to infinity in the direction of ϕ±j .
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 13 / 35
Grow-up solutions
Let {ϕj}j∈N be an orthonormal basis in L2([0, π]) comprised of theeigenfunctions of A = −∂xx − bI with Neumann boundaryconditions.If u(t , ·) is a grow-up solution then
u(t , ·)‖u(t , ·)‖
−→ ϕ±j (·) in L2,
with j ≤√
b and ϕ±j := ±ϕj .
The original orbit grows-up to infinity in the direction of ϕ±j .
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 13 / 35
Grow-up solutions
Let {ϕj}j∈N be an orthonormal basis in L2([0, π]) comprised of theeigenfunctions of A = −∂xx − bI with Neumann boundaryconditions.If u(t , ·) is a grow-up solution then
u(t , ·)‖u(t , ·)‖
−→ ϕ±j (·) in L2,
with j ≤√
b and ϕ±j := ±ϕj .
The original orbit grows-up to infinity in the direction of ϕ±j .
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 13 / 35
Poincaré Projection
Figure: Projection of M = (u,1) ∈ Xα × {1}.
H = {(χ, z) ∈ Xα × R|〈χ, χ〉2α + z2 = 1, z ≥ 0}.Equilibrium points on He = {(χ,0) ∈ H}: ±Φj , for j ∈ N.
I Hell (2009).
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 14 / 35
Equilibria at infinity
Φ±j are equilibrium points on He.
We thus define objects Φ±,∞j at infinity as
P(Φ±,∞j ) = Φ±j , for j = 0,1, ..., [√
b],
and we refer to these as equilibria at infinity.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 15 / 35
Equilibria at infinity
Φ±j are equilibrium points on He.
We thus define objects Φ±,∞j at infinity as
P(Φ±,∞j ) = Φ±j , for j = 0,1, ..., [√
b],
and we refer to these as equilibria at infinity.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 15 / 35
Outline
1 IntroductionDefinitions - dissipative and non-dissipative systemsMotivation - dissipative system and its global attractor
2 Problem statementSlowly non-dissipative systemMain objective
3 Preliminary resultsEquilibria at infinity
4 Main resultsNon-compact global attractorAssociated permutation
5 Strategy of the proof
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 16 / 35
Definitions - zero number and adjacency
1 Let u be a continuous function defined on [0, π]. We denote byz(u) the zero number of u, that is, the number of strict signchanges of u.
2 Let u, v ∈ Ef with z(v − u) = j and u(0) < v(0). We say that u andv are adjacent if there does not exist w ∈ Ef satisfying
u(0) < w(0) < v(0)
and z(v − w) = z(w − u) = j .
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 17 / 35
Definitions - zero number and adjacency
1 Let u be a continuous function defined on [0, π]. We denote byz(u) the zero number of u, that is, the number of strict signchanges of u.
2 Let u, v ∈ Ef with z(v − u) = j and u(0) < v(0). We say that u andv are adjacent if there does not exist w ∈ Ef satisfying
u(0) < w(0) < v(0)
and z(v − w) = z(w − u) = j .
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 17 / 35
The non-compact global attractor
Theorem
LetEc
f denote the set of (bounded) equilibria andE∞f denote the set of equilibria at infinity.
Then the non-compact global attractor Af of (2) is composed by theset of equilibria Ef = Ec
f ∪ E∞f and the heteroclinic connectionsbetween the equilibria,
Af = Ef ∪ {heteroclinic connections}.
Moreover, for any u, v ∈ Ef , there exists an orbit connecting them if,and only if, u and v are adjacent.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 18 / 35
Permutation related to the non-dissipative equation
Theorem
There exists a permutation related to the non-dissipative equation (2)determining which equilibria in Ef are connected and which are not.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 19 / 35
Outline
1 IntroductionDefinitions - dissipative and non-dissipative systemsMotivation - dissipative system and its global attractor
2 Problem statementSlowly non-dissipative systemMain objective
3 Preliminary resultsEquilibria at infinity
4 Main resultsNon-compact global attractorAssociated permutation
5 Strategy of the proof
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 20 / 35
Permutation related to f
1 Generic assumption: all the equilibria are hyperbolic.2 Let Ec
f = {v1, v2, ..., vn} be the set of equilibria of equation (2),where
v1(0) < v2(0) < ... < vn(0).
We then define the permutation σf ∈ S(n) by
vσf (1)(π) < vσf (2)(π) < ... < vσf (n)(π).
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 21 / 35
Meander associated to σf
1 Consider the initial value problem
ux = v (4)vx = −f (x ,u,ux) = −bu − g(x ,u,ux)
u(0) = u0, v(0) = 0.
2 The set of solutions u = u(x ,u0), v = v(x ,u0) of (4) defines thetwo-dimensional manifold in [0, π]× R2
M = {(x ,u, v) ∈ [0, π]× R2|u = u(x ,u0), v = v(x ,u0) solve (4)}.
3 Let γf be the section curve of M at x = π
γf = {(x ,u(x ,u0), v(x ,u0))|u0 ∈ R, x = π}.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 22 / 35
Example of a meander permutation
1 2 3 4 5 6 7 8 9
9 8 3 6 5 4 7 2 1
Figure: Meander related to σf = {9,8,3,6,5,4,7,2,1} ∈ S(9) considering thenonlinearity f (u) = 10u + 16 sin u.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 23 / 35
Suspension
Definition
Let σ ∈ S(n) and k be a positive integer. We define the suspension σk
of the permutation σ as the permutation σk ∈ S(n + 2) which satisfies:(i) σk (j) = σ(j − 1) + 1, for j ∈ {2, ...,n + 1}; and(ii) if k is odd
σk (1) = 1 and σk (n + 2) = n + 2 ,
and if k is even
σk (1) = n + 2 and σk (n + 2) = 1 .
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 24 / 35
Example of suspension
Figure: k -th suspension of the meander γf for k = [√
10] + 1 = 4 andf (u) = 10u + 16 sin u.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 25 / 35
Suspension of σf
1 The Morse indices of v1 and vn, i.e., the dimension of theircorresponding unstable manifolds, satisfy
i(v1) = i(vn) = [√
b] + 1 =: k .
2 Let σ1f ∈ S(n + 2k) be the k -th suspension of σf .
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 26 / 35
Sturm permutation method
1 σ1f is a Sturm permutation, i.e., there exists h realizing σ1
f .2 There exists a function h such that the dynamical system induced
by {ut = uxx + h(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0
(5)
I is dissipative andI the permutation σh defined by
wσh(1)(π) < wσh(2)(π) < ... < wσh(n+2k)(π),
where Eh = {w1, ...,wn+2k} is the set of equilibria of (5) ordered bytheir values at x = 0, satisfies
σh = σ1f .
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 27 / 35
Global attractor Ah
Proposition (Wolfrum (2002))Given any two equilibria u, v ∈ Eh, there exists an orbit connectingthem if, and only if, u and v are adjacent.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 28 / 35
Morse indices and zero numbers given by σh
1 The Morse indices i(wm) are given in terms of σh by
i(wm) =m−1∑j=1
(−1)j+1 sign(σ−1h (j + 1)− σ−1
h (j)).
2 The zero numbers z(wl − wm) for 1 ≤ m < l ≤ n + 2k are given interms of σh by
z(wl − wm) =i(wm) +12[(−1)l sign(σ−1
h (l)− σ−1h (m))− 1]
+l−1∑
j=m+1
(−1)j sign(σ−1h (j)− σ−1
h (m)).
I Rocha (1985)I Fusco and Rocha (1991)I Rocha (1991)
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 29 / 35
Correspondence between the equilibria
1 Consider the set of equilibria for the dissipative equation
Eh = {w1, ...,wn+2k}.
2 Consider the set of equilibria for the non-dissipative equation
Ef = {Φ−,∞0 , ..., Φ−,∞k−1 , v1, ..., vn, Φ+,∞k−1 , ..., Φ
+,∞0 }.
3 We make the correspondence
Φ−,∞j ↔ wj+1 and Φ+,∞j ↔ wn+2k−j ,
for 0 ≤ j ≤ k − 1.4 Since h = f on B where B ⊂ Xα contains {v1, ..., vn},
wk+l = vl for l = 1, ...,n.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 30 / 35
Morse indices for the dissipative equation
Lemma
The Morse indices of the equilibria satisfy(i) i(wk+l) = i(vl), for any 1 ≤ l ≤ n(ii) i(wj) = i(Φ−,∞j−1 ) = j − 1, for 1 ≤ j ≤ k
(iii) i(wn+2k−j) = i(Φ+,∞j ) = j , for 0 ≤ j ≤ k − 1
where i(Φ±,∞j ) := i(ϕ±j ) = j .
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 31 / 35
Zero numbers for the dissipative equation
LemmaFor any 1 ≤ l ≤ n, the following hold
z(wk+l − wj) = z(Φ−,∞j−1 − vl) = j − 1, for 1 ≤ j ≤ k
z(wn+2k−j − wk+l) = z(Φ+,∞j − vl) = j , for 0 ≤ j ≤ k − 1.
LemmaFor any 1 ≤ r < l ≤ n, the following holds
z(wk+l − wk+r ) = z(vl − vr ).
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 32 / 35
Heteroclinic connections in Af in terms of σh
1 σh determines the Morse indices and zero numbers of theequilibria in Eh.
2 The Morse indices and zero numbers are preserved by thecorrespondence.
3 The Sturm permutation σh explicitly determines the zero numbersand Morse indices of the equilibria in Ef .
4 The correspondence preserves the connections between theequilibria.
5 The permutation σh = σ1f determines, through the adjacency
notion, which equilibria in Ef = Ecf ∪ E∞f are connected and which
are not.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 33 / 35
Heteroclinic connections in Af in terms of σh
1 σh determines the Morse indices and zero numbers of theequilibria in Eh.
2 The Morse indices and zero numbers are preserved by thecorrespondence.
3 The Sturm permutation σh explicitly determines the zero numbersand Morse indices of the equilibria in Ef .
4 The correspondence preserves the connections between theequilibria.
5 The permutation σh = σ1f determines, through the adjacency
notion, which equilibria in Ef = Ecf ∪ E∞f are connected and which
are not.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 33 / 35
Heteroclinic connections in Af in terms of σh
1 σh determines the Morse indices and zero numbers of theequilibria in Eh.
2 The Morse indices and zero numbers are preserved by thecorrespondence.
3 The Sturm permutation σh explicitly determines the zero numbersand Morse indices of the equilibria in Ef .
4 The correspondence preserves the connections between theequilibria.
5 The permutation σh = σ1f determines, through the adjacency
notion, which equilibria in Ef = Ecf ∪ E∞f are connected and which
are not.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 33 / 35
Heteroclinic connections in Af in terms of σh
1 σh determines the Morse indices and zero numbers of theequilibria in Eh.
2 The Morse indices and zero numbers are preserved by thecorrespondence.
3 The Sturm permutation σh explicitly determines the zero numbersand Morse indices of the equilibria in Ef .
4 The correspondence preserves the connections between theequilibria.
5 The permutation σh = σ1f determines, through the adjacency
notion, which equilibria in Ef = Ecf ∪ E∞f are connected and which
are not.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 33 / 35
Heteroclinic connections in Af in terms of σh
1 σh determines the Morse indices and zero numbers of theequilibria in Eh.
2 The Morse indices and zero numbers are preserved by thecorrespondence.
3 The Sturm permutation σh explicitly determines the zero numbersand Morse indices of the equilibria in Ef .
4 The correspondence preserves the connections between theequilibria.
5 The permutation σh = σ1f determines, through the adjacency
notion, which equilibria in Ef = Ecf ∪ E∞f are connected and which
are not.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 33 / 35
Cut-off functionConsider the non-dissipative equation{
ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
with f (x ,u,ux) = bu + g(x ,u,ux).We construct a cut-off function h such that
I h = f on B ⊂ Xα containing Ecf = {v1, ..., vn},
I h = cu, for c < 0, outside a larger set B, andI σ1
f is used to define h in the remaining portion of the domain.
Analyze the equation{ut = uxx + h(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
using the theory for dissipative systems.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 34 / 35
Cut-off functionConsider the non-dissipative equation{
ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
with f (x ,u,ux) = bu + g(x ,u,ux).We construct a cut-off function h such that
I h = f on B ⊂ Xα containing Ecf = {v1, ..., vn},
I h = cu, for c < 0, outside a larger set B, andI σ1
f is used to define h in the remaining portion of the domain.
Analyze the equation{ut = uxx + h(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
using the theory for dissipative systems.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 34 / 35
Cut-off functionConsider the non-dissipative equation{
ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
with f (x ,u,ux) = bu + g(x ,u,ux).We construct a cut-off function h such that
I h = f on B ⊂ Xα containing Ecf = {v1, ..., vn},
I h = cu, for c < 0, outside a larger set B, andI σ1
f is used to define h in the remaining portion of the domain.
Analyze the equation{ut = uxx + h(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
using the theory for dissipative systems.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 34 / 35
Cut-off functionConsider the non-dissipative equation{
ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
with f (x ,u,ux) = bu + g(x ,u,ux).We construct a cut-off function h such that
I h = f on B ⊂ Xα containing Ecf = {v1, ..., vn},
I h = cu, for c < 0, outside a larger set B, andI σ1
f is used to define h in the remaining portion of the domain.
Analyze the equation{ut = uxx + h(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
using the theory for dissipative systems.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 34 / 35
Cut-off functionConsider the non-dissipative equation{
ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
with f (x ,u,ux) = bu + g(x ,u,ux).We construct a cut-off function h such that
I h = f on B ⊂ Xα containing Ecf = {v1, ..., vn},
I h = cu, for c < 0, outside a larger set B, andI σ1
f is used to define h in the remaining portion of the domain.
Analyze the equation{ut = uxx + h(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
using the theory for dissipative systems.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 34 / 35
Cut-off functionConsider the non-dissipative equation{
ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
with f (x ,u,ux) = bu + g(x ,u,ux).We construct a cut-off function h such that
I h = f on B ⊂ Xα containing Ecf = {v1, ..., vn},
I h = cu, for c < 0, outside a larger set B, andI σ1
f is used to define h in the remaining portion of the domain.
Analyze the equation{ut = uxx + h(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,
using the theory for dissipative systems.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 34 / 35
Thank you for your attention.
Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 35 / 35
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