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Outline
Bridging steady states with renormalizationgroup analysis
Yueheng LanDepartment of Physics
Tsinghua University
April, 2013
Yueheng Lan Bridging steady states with RG analysis
Outline
Main contents
1 IntroductionPhysics and heteroclinic connectionsRenormalization groupThe RG and differential equations
2 An extension of the RG analysis
3 Several examplesThe Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
4 Summary
Yueheng Lan Bridging steady states with RG analysis
Outline
Main contents
1 IntroductionPhysics and heteroclinic connectionsRenormalization groupThe RG and differential equations
2 An extension of the RG analysis
3 Several examplesThe Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
4 Summary
Yueheng Lan Bridging steady states with RG analysis
Outline
Main contents
1 IntroductionPhysics and heteroclinic connectionsRenormalization groupThe RG and differential equations
2 An extension of the RG analysis
3 Several examplesThe Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
4 Summary
Yueheng Lan Bridging steady states with RG analysis
Outline
Main contents
1 IntroductionPhysics and heteroclinic connectionsRenormalization groupThe RG and differential equations
2 An extension of the RG analysis
3 Several examplesThe Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
4 Summary
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Main contents
1 IntroductionPhysics and heteroclinic connectionsRenormalization groupThe RG and differential equations
2 An extension of the RG analysis
3 Several examplesThe Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
4 Summary
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Nonlinear dynamics and phase space
State and dynamics
x1 = f1(x1 , x2 , · · · , xn)x2 = f2(x1 , x2 , · · · , xn)· · · = · · ·xn = fn(x1 , x2 , · · · , xn)
The phase space - a geometricrepresentationVector field and trajectoriesInvariant set and organizationof trajectories
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Connections
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Pendulum orbits
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
State transition in chemical reactions
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Transition orbit of Chang E I
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Computation of heteroclinic connections
Exact analytic solutions under certain conditions: nonlinearintegrable systems, partially integrable systems.
Asymptotic methods for analytic approximations: localstability analysis plus interpolation.
Numerical methods: two-point boundary problem; shootingmethod; relaxation method.
Challenge:(1) Need to know orbit existence and both end points;(2) Need to know the local behavior near two ends;(3) Hard to represent the dynamics on the connection;(4) Hard to derive analytic expressions.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Computation of heteroclinic connections
Exact analytic solutions under certain conditions: nonlinearintegrable systems, partially integrable systems.
Asymptotic methods for analytic approximations: localstability analysis plus interpolation.
Numerical methods: two-point boundary problem; shootingmethod; relaxation method.
Challenge:(1) Need to know orbit existence and both end points;(2) Need to know the local behavior near two ends;(3) Hard to represent the dynamics on the connection;(4) Hard to derive analytic expressions.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Computation of heteroclinic connections
Exact analytic solutions under certain conditions: nonlinearintegrable systems, partially integrable systems.
Asymptotic methods for analytic approximations: localstability analysis plus interpolation.
Numerical methods: two-point boundary problem; shootingmethod; relaxation method.
Challenge:(1) Need to know orbit existence and both end points;(2) Need to know the local behavior near two ends;(3) Hard to represent the dynamics on the connection;(4) Hard to derive analytic expressions.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Computation of heteroclinic connections
Exact analytic solutions under certain conditions: nonlinearintegrable systems, partially integrable systems.
Asymptotic methods for analytic approximations: localstability analysis plus interpolation.
Numerical methods: two-point boundary problem; shootingmethod; relaxation method.
Challenge:(1) Need to know orbit existence and both end points;(2) Need to know the local behavior near two ends;(3) Hard to represent the dynamics on the connection;(4) Hard to derive analytic expressions.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Main contents
1 IntroductionPhysics and heteroclinic connectionsRenormalization groupThe RG and differential equations
2 An extension of the RG analysis
3 Several examplesThe Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
4 Summary
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Renormalization group in physics
RG investigates changes ofphysical laws at differentscales.RG and scale invariance: arenormalizable system at onescale consists of self-similarcopies of itself at a smallerscale, with convergent couplingparameters when scaled up.In statistical physics: blockspin; In quantum physics:renormalization equation∂g/∂ lnµ = β(g); In nonlineardynamics: the universal routeto chaos.
Block spin renormalizationgroup for a spin systemdescribed byH(T, J):(T, J) → (T ′, J ′) →(T ′′, J ′′). Resummation?
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Renormalization group in physics
RG investigates changes ofphysical laws at differentscales.RG and scale invariance: arenormalizable system at onescale consists of self-similarcopies of itself at a smallerscale, with convergent couplingparameters when scaled up.In statistical physics: blockspin; In quantum physics:renormalization equation∂g/∂ lnµ = β(g); In nonlineardynamics: the universal routeto chaos.
Block spin renormalizationgroup for a spin systemdescribed byH(T, J):(T, J) → (T ′, J ′) →(T ′′, J ′′). Resummation?
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Renormalization group in physics
RG investigates changes ofphysical laws at differentscales.RG and scale invariance: arenormalizable system at onescale consists of self-similarcopies of itself at a smallerscale, with convergent couplingparameters when scaled up.In statistical physics: blockspin; In quantum physics:renormalization equation∂g/∂ lnµ = β(g); In nonlineardynamics: the universal routeto chaos.
Block spin renormalizationgroup for a spin systemdescribed byH(T, J):(T, J) → (T ′, J ′) →(T ′′, J ′′). Resummation?
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Renormalization group in physics
RG investigates changes ofphysical laws at differentscales.RG and scale invariance: arenormalizable system at onescale consists of self-similarcopies of itself at a smallerscale, with convergent couplingparameters when scaled up.In statistical physics: blockspin; In quantum physics:renormalization equation∂g/∂ lnµ = β(g); In nonlineardynamics: the universal routeto chaos.
Block spin renormalizationgroup for a spin systemdescribed byH(T, J):(T, J) → (T ′, J ′) →(T ′′, J ′′). Resummation?
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Main contents
1 IntroductionPhysics and heteroclinic connectionsRenormalization groupThe RG and differential equations
2 An extension of the RG analysis
3 Several examplesThe Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
4 Summary
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
One example
The Van der Pol equation is
d2y
dt2+ y = ε[
dy
dt− 1
3(dy/dt)3] .
A naive expansion
y = y0 + εy1 + ε2y2 + · · ·gives
y(t) = R0 sin(t + Θ0) + ε[−R30
96cos(t + Θ0) +
R02 (1− R2
0
4)(t− t0) sin(t + Θ0) +
R30
96cos 3(t + Θ0)] + O(ε2) ,
where R0 ,Θ0 are determined by the initial conditions.The expansion breaks down when ε(t− t0) > 1. Thearbitrary initial time t0 may be treated as the ultravioletcutoff in the usual field theory.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
One example
The Van der Pol equation is
d2y
dt2+ y = ε[
dy
dt− 1
3(dy/dt)3] .
A naive expansion
y = y0 + εy1 + ε2y2 + · · ·gives
y(t) = R0 sin(t + Θ0) + ε[−R30
96cos(t + Θ0) +
R02 (1− R2
0
4)(t− t0) sin(t + Θ0) +
R30
96cos 3(t + Θ0)] + O(ε2) ,
where R0 ,Θ0 are determined by the initial conditions.The expansion breaks down when ε(t− t0) > 1. Thearbitrary initial time t0 may be treated as the ultravioletcutoff in the usual field theory.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
One example
The Van der Pol equation is
d2y
dt2+ y = ε[
dy
dt− 1
3(dy/dt)3] .
A naive expansion
y = y0 + εy1 + ε2y2 + · · ·gives
y(t) = R0 sin(t + Θ0) + ε[−R30
96cos(t + Θ0) +
R02 (1− R2
0
4)(t− t0) sin(t + Θ0) +
R30
96cos 3(t + Θ0)] + O(ε2) ,
where R0 ,Θ0 are determined by the initial conditions.The expansion breaks down when ε(t− t0) > 1. Thearbitrary initial time t0 may be treated as the ultravioletcutoff in the usual field theory.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Renormalization?
Split t− t0 as t− τ + τ − t0 and absorb the terms containingτ − t0 into the renormalized counterparts R ,Θ of R0 andΘ0.Assume R0(t0) = Z1(t0, τ)R(τ) ,Θ0(t0) = Θ(τ) + Z2(t0, τ)where Z1 = 1 +
∑∞1 anεn , Z2 =
∑∞1 bnεn. The choice
a1 = −(1/2)(1−R2/4)(τ − t0) , b1 = 0 removes the secularterm to order ε:
y(t) = [R + εR
2(−R2
4)(t− τ)] sin(t + Θ)−
εR3
96cos(t + Θ) + ε
R3
96cos 3(t + Θ) + O(ε2) ,
where R ,Θ are functions of τ .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Renormalization?
Split t− t0 as t− τ + τ − t0 and absorb the terms containingτ − t0 into the renormalized counterparts R ,Θ of R0 andΘ0.Assume R0(t0) = Z1(t0, τ)R(τ) ,Θ0(t0) = Θ(τ) + Z2(t0, τ)where Z1 = 1 +
∑∞1 anεn , Z2 =
∑∞1 bnεn. The choice
a1 = −(1/2)(1−R2/4)(τ − t0) , b1 = 0 removes the secularterm to order ε:
y(t) = [R + εR
2(−R2
4)(t− τ)] sin(t + Θ)−
εR3
96cos(t + Θ) + ε
R3
96cos 3(t + Θ) + O(ε2) ,
where R ,Θ are functions of τ .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Renormalization?
The solution should not depend on τ . Therefore(∂y/∂τ)t = 0:
dR
dτ= ε
R
2(1− R2
4) + O(ε2) ,
dΘdτ
= O(ε2) .
The initial condition R(0) = 2a , Θ(0) = 0 gives
y(t) = R(t) sin(t) +ε
96R(t)3[cos(3t)− cos(t)] + O(ε2) .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Renormalization?
The solution should not depend on τ . Therefore(∂y/∂τ)t = 0:
dR
dτ= ε
R
2(1− R2
4) + O(ε2) ,
dΘdτ
= O(ε2) .
The initial condition R(0) = 2a , Θ(0) = 0 gives
y(t) = R(t) sin(t) +ε
96R(t)3[cos(3t)− cos(t)] + O(ε2) .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Series expansion of a differential equation
Suppose that we have a set of n-dimensional ODEs
x = Lx + εN(x)
We may make the expansion
x = u0 + εu1 + ε2u2 + · · ·
which results in
u0 = Lu0
u1 = Lu1 + N(u0)u2 = Lu2 + N2(u0,u1)
...
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
Series expansion of a differential equation
Suppose that we have a set of n-dimensional ODEs
x = Lx + εN(x)
We may make the expansion
x = u0 + εu1 + ε2u2 + · · ·
which results in
u0 = Lu0
u1 = Lu1 + N(u0)u2 = Lu2 + N2(u0,u1)
...
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
From the naive solution to the RG equation
This series of equations can be solved as
u0(t, t0) = eL(t−t0)A(t0)
u1(t, t0) = eL(t−t0)
∫ t
t0
e−L(τ−t0)N(eL(t−t0)A)dτ
u2(t, t0) = eL(t−t0)
∫ t
t0
e−L(τ−t0)N2(eL(t−t0)A,u1(t, t0))dτ .
The series expansion gives x = x(t; t0,A(t0)).
The RG equation is a set of equations for dA(t0)/dt0derived from
dx(t; t0,A(t0))dt0
|t=t0 = 0
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
One simple examle
Consider the simple example
x = y , y = −x ,
which can be solved exactly with
x = R sin(t− t0 + θ) , y = R cos(t− t0 + θ) ,
where R = R(t0) , θ = θ(t0) specify the initial condition.In phase space, orbits of the equation are circles with radiusR and azimuth angle θ.
The RG equation derived from
∂x(t;R(t0), θ(t0), t0)∂t0
|t=t0 = 0,∂y(t;R(t0), θ(t0), t0)
∂t0|t=t0 = 0
is dR(t0)/dt0 = 0, dθ(t0)/dt0 = 1 as expected.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
One simple examle
Consider the simple example
x = y , y = −x ,
which can be solved exactly with
x = R sin(t− t0 + θ) , y = R cos(t− t0 + θ) ,
where R = R(t0) , θ = θ(t0) specify the initial condition.In phase space, orbits of the equation are circles with radiusR and azimuth angle θ.
The RG equation derived from
∂x(t;R(t0), θ(t0), t0)∂t0
|t=t0 = 0,∂y(t;R(t0), θ(t0), t0)
∂t0|t=t0 = 0
is dR(t0)/dt0 = 0, dθ(t0)/dt0 = 1 as expected.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
One simple examle
Consider the simple example
x = y , y = −x ,
which can be solved exactly with
x = R sin(t− t0 + θ) , y = R cos(t− t0 + θ) ,
where R = R(t0) , θ = θ(t0) specify the initial condition.In phase space, orbits of the equation are circles with radiusR and azimuth angle θ.
The RG equation derived from
∂x(t;R(t0), θ(t0), t0)∂t0
|t=t0 = 0,∂y(t;R(t0), θ(t0), t0)
∂t0|t=t0 = 0
is dR(t0)/dt0 = 0, dθ(t0)/dt0 = 1 as expected.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
The RG analysis as a coordinate transformation
Hamiltonian dynamics: action-angle variables. For aharmonic oscillator H = 1/2p2 + 1/2q2 = I.In a general nonlinear dynamical system,
x = f(x)
which has the general solution x(t) = φ(t;A0(t0), t0). Theequation
∂φ(t;A0(t0), t0)∂t0
|t=t0 = 0
gives an equation for dA0(t0)/dt0, which governs theevolution of the new coordinates A0.The RG analysis is equivalent to a coordinatetransformation in this sense, but often associated withapproximations in the nonlinear case.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
The RG analysis as a coordinate transformation
Hamiltonian dynamics: action-angle variables. For aharmonic oscillator H = 1/2p2 + 1/2q2 = I.In a general nonlinear dynamical system,
x = f(x)
which has the general solution x(t) = φ(t;A0(t0), t0). Theequation
∂φ(t;A0(t0), t0)∂t0
|t=t0 = 0
gives an equation for dA0(t0)/dt0, which governs theevolution of the new coordinates A0.The RG analysis is equivalent to a coordinatetransformation in this sense, but often associated withapproximations in the nonlinear case.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Physics and heteroclinic connectionsRenormalization groupThe RG and differential equations
The RG analysis as a coordinate transformation
Hamiltonian dynamics: action-angle variables. For aharmonic oscillator H = 1/2p2 + 1/2q2 = I.In a general nonlinear dynamical system,
x = f(x)
which has the general solution x(t) = φ(t;A0(t0), t0). Theequation
∂φ(t;A0(t0), t0)∂t0
|t=t0 = 0
gives an equation for dA0(t0)/dt0, which governs theevolution of the new coordinates A0.The RG analysis is equivalent to a coordinatetransformation in this sense, but often associated withapproximations in the nonlinear case.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Further development of the RG analysis
It can be used for nonlinear partial equations and is able toderive the phase or amplitude equations.
The invariance condition has been extended to the analysisof maps.
It is also used to determine the center manifold near abifurcation point.
Problem: for the dynamics on a submanifold, the ninvariance equations contain less than n unknowns!?
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Further development of the RG analysis
It can be used for nonlinear partial equations and is able toderive the phase or amplitude equations.
The invariance condition has been extended to the analysisof maps.
It is also used to determine the center manifold near abifurcation point.
Problem: for the dynamics on a submanifold, the ninvariance equations contain less than n unknowns!?
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Further development of the RG analysis
It can be used for nonlinear partial equations and is able toderive the phase or amplitude equations.
The invariance condition has been extended to the analysisof maps.
It is also used to determine the center manifold near abifurcation point.
Problem: for the dynamics on a submanifold, the ninvariance equations contain less than n unknowns!?
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Further development of the RG analysis
It can be used for nonlinear partial equations and is able toderive the phase or amplitude equations.
The invariance condition has been extended to the analysisof maps.
It is also used to determine the center manifold near abifurcation point.
Problem: for the dynamics on a submanifold, the ninvariance equations contain less than n unknowns!?
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Dynamics on a submanifold
Without loss of generality, we will concentrate on the 1-dsubmanifold. Higher dimensional ones can be treated in asimilar way.
The initial vector A should be taken asA = (A1 , 0 , 0 , · · · , 0)t.
The i-th (i 6= 1) component of u1 can be computed as
u1,i(t, t0) = eλi(t−t0)
∫ t
e−λi(τ−t0)N(eL(t−t0)A)dτ ,
where∫ t denotes integration without constant term.
The first componentdx1(t; t0, A1(t0))
dt0|t=t0 = 0 (1)
is enough to derive the RG equation for dA1(t0)/dt0, whichalso satisfies other component equations.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Dynamics on a submanifold
Without loss of generality, we will concentrate on the 1-dsubmanifold. Higher dimensional ones can be treated in asimilar way.
The initial vector A should be taken asA = (A1 , 0 , 0 , · · · , 0)t.
The i-th (i 6= 1) component of u1 can be computed as
u1,i(t, t0) = eλi(t−t0)
∫ t
e−λi(τ−t0)N(eL(t−t0)A)dτ ,
where∫ t denotes integration without constant term.
The first componentdx1(t; t0, A1(t0))
dt0|t=t0 = 0 (1)
is enough to derive the RG equation for dA1(t0)/dt0, whichalso satisfies other component equations.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Dynamics on a submanifold
Without loss of generality, we will concentrate on the 1-dsubmanifold. Higher dimensional ones can be treated in asimilar way.
The initial vector A should be taken asA = (A1 , 0 , 0 , · · · , 0)t.
The i-th (i 6= 1) component of u1 can be computed as
u1,i(t, t0) = eλi(t−t0)
∫ t
e−λi(τ−t0)N(eL(t−t0)A)dτ ,
where∫ t denotes integration without constant term.
The first componentdx1(t; t0, A1(t0))
dt0|t=t0 = 0 (1)
is enough to derive the RG equation for dA1(t0)/dt0, whichalso satisfies other component equations.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Dynamics on a submanifold
Without loss of generality, we will concentrate on the 1-dsubmanifold. Higher dimensional ones can be treated in asimilar way.
The initial vector A should be taken asA = (A1 , 0 , 0 , · · · , 0)t.
The i-th (i 6= 1) component of u1 can be computed as
u1,i(t, t0) = eλi(t−t0)
∫ t
e−λi(τ−t0)N(eL(t−t0)A)dτ ,
where∫ t denotes integration without constant term.
The first componentdx1(t; t0, A1(t0))
dt0|t=t0 = 0 (1)
is enough to derive the RG equation for dA1(t0)/dt0, whichalso satisfies other component equations.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
A proof by mathematical induction
it is easy to write down an integral equation from its i-th(i 6= 1) component
xi(t; t0, A1(t0)) = εeλi(t−t0)
∫ t
e−λi(τ−t0)N(x(t; t0, A1(t0)))dτ .
Take t0-derivatives on both sides and impose t → t0
∂xi(t; t0, A1(t0))∂t0
|t=t0 = ε
∫ t0
e−λi(τ−t0)∇N(x(t0; t0, A1(t0)))
· ∂x(t; t0, A1(t0))∂t0
|t=t0dτ ∼ O(εm+1) .
Our assertion is surely true for m = 0. By induction, it istrue for all values of m.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
A proof by mathematical induction
it is easy to write down an integral equation from its i-th(i 6= 1) component
xi(t; t0, A1(t0)) = εeλi(t−t0)
∫ t
e−λi(τ−t0)N(x(t; t0, A1(t0)))dτ .
Take t0-derivatives on both sides and impose t → t0
∂xi(t; t0, A1(t0))∂t0
|t=t0 = ε
∫ t0
e−λi(τ−t0)∇N(x(t0; t0, A1(t0)))
· ∂x(t; t0, A1(t0))∂t0
|t=t0dτ ∼ O(εm+1) .
Our assertion is surely true for m = 0. By induction, it istrue for all values of m.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
A proof by mathematical induction
it is easy to write down an integral equation from its i-th(i 6= 1) component
xi(t; t0, A1(t0)) = εeλi(t−t0)
∫ t
e−λi(τ−t0)N(x(t; t0, A1(t0)))dτ .
Take t0-derivatives on both sides and impose t → t0
∂xi(t; t0, A1(t0))∂t0
|t=t0 = ε
∫ t0
e−λi(τ−t0)∇N(x(t0; t0, A1(t0)))
· ∂x(t; t0, A1(t0))∂t0
|t=t0dτ ∼ O(εm+1) .
Our assertion is surely true for m = 0. By induction, it istrue for all values of m.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Main contents
1 IntroductionPhysics and heteroclinic connectionsRenormalization groupThe RG and differential equations
2 An extension of the RG analysis
3 Several examplesThe Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
4 Summary
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Lotka-Volterra model
The Lotka-Volterra model ofcompetition is
x = x(3− x− 2y)y = y(2− x− y)
The model describes thecompetition between therabbits and the sheep fed onthe grass of the same lawn.Vector field and trajectories
four equilibriaP1 = (0, 0) , P2 = (0, 2) , P3 = (1, 1) , P4 = (3, 0).Their approximation is(1, 1) , (2.908,−0.003) , (−0.113, 2.105).
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Lotka-Volterra model
The Lotka-Volterra model ofcompetition is
x = x(3− x− 2y)y = y(2− x− y)
The model describes thecompetition between therabbits and the sheep fed onthe grass of the same lawn.Vector field and trajectories
four equilibriaP1 = (0, 0) , P2 = (0, 2) , P3 = (1, 1) , P4 = (3, 0).Their approximation is(1, 1) , (2.908,−0.003) , (−0.113, 2.105).
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Lotka-Volterra model
The Lotka-Volterra model ofcompetition is
x = x(3− x− 2y)y = y(2− x− y)
The model describes thecompetition between therabbits and the sheep fed onthe grass of the same lawn.Vector field and trajectories
four equilibriaP1 = (0, 0) , P2 = (0, 2) , P3 = (1, 1) , P4 = (3, 0).Their approximation is(1, 1) , (2.908,−0.003) , (−0.113, 2.105).
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Lotka-Volterra model
The Lotka-Volterra model ofcompetition is
x = x(3− x− 2y)y = y(2− x− y)
The model describes thecompetition between therabbits and the sheep fed onthe grass of the same lawn.Vector field and trajectories
four equilibriaP1 = (0, 0) , P2 = (0, 2) , P3 = (1, 1) , P4 = (3, 0).Their approximation is(1, 1) , (2.908,−0.003) , (−0.113, 2.105).
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Lotka-Volterra model
The Lotka-Volterra model ofcompetition is
x = x(3− x− 2y)y = y(2− x− y)
The model describes thecompetition between therabbits and the sheep fed onthe grass of the same lawn.Vector field and trajectories
four equilibriaP1 = (0, 0) , P2 = (0, 2) , P3 = (1, 1) , P4 = (3, 0).Their approximation is(1, 1) , (2.908,−0.003) , (−0.113, 2.105).
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Series solution
Around the saddle P3, we take a coordinate transformation
x = 1−√
23z +
√23w , y = 1 +
√13z +
√13w .
Assume
z = εz1 + ε2z2 + ε3z3 + O(ε4) (2)w = εw1 + ε2w2 + ε3w3 + O(ε4) , (3)
we have
L ◦ z1 = 0 , M ◦ w1 = 0L ◦ z2 = F2(z1, w1) , M ◦ w2 = G2(z1, w1)L ◦ z3 = F3(z1, w1, z2, w2) , M ◦ w3 = G3(z1, w1, z2, w2) ,
where L ≡ 1−√
2 + ddt ,M ≡ 1 +
√2 + d
dt andF2 , G2 , F3 , G3 are polynomial functions of their arguments.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Series solution
Around the saddle P3, we take a coordinate transformation
x = 1−√
23z +
√23w , y = 1 +
√13z +
√13w .
Assume
z = εz1 + ε2z2 + ε3z3 + O(ε4) (2)w = εw1 + ε2w2 + ε3w3 + O(ε4) , (3)
we have
L ◦ z1 = 0 , M ◦ w1 = 0L ◦ z2 = F2(z1, w1) , M ◦ w2 = G2(z1, w1)L ◦ z3 = F3(z1, w1, z2, w2) , M ◦ w3 = G3(z1, w1, z2, w2) ,
where L ≡ 1−√
2 + ddt ,M ≡ 1 +
√2 + d
dt andF2 , G2 , F3 , G3 are polynomial functions of their arguments.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Series solution
Around the saddle P3, we take a coordinate transformation
x = 1−√
23z +
√23w , y = 1 +
√13z +
√13w .
Assume
z = εz1 + ε2z2 + ε3z3 + O(ε4) (2)w = εw1 + ε2w2 + ε3w3 + O(ε4) , (3)
we have
L ◦ z1 = 0 , M ◦ w1 = 0L ◦ z2 = F2(z1, w1) , M ◦ w2 = G2(z1, w1)L ◦ z3 = F3(z1, w1, z2, w2) , M ◦ w3 = G3(z1, w1, z2, w2) ,
where L ≡ 1−√
2 + ddt ,M ≡ 1 +
√2 + d
dt andF2 , G2 , F3 , G3 are polynomial functions of their arguments.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Series solution (continued)
The general solution is
z1(t) = a(t0)e(√
2−1)(t−t0) , w1(t) = b(t0)e−(1+√
2)(t−t0) .
Set b(t0) = 0 and the solution is
z = εa(t0)e(√
2−1)(t−t0) +√
3ε2a2(t0)6
(√
2− 1)
(e2(√
2−1)(t−t0) − e(√
2−1)(t−t0)) + O(ε3)
w =√
3ε2a2(t0)102
(1 + 3√
2)e2(√
2−1)(t−t0) + O(ε3) .
From ∂z(t, t0)/∂t0 = 0, we get
da(t0)dt0
= a
(√
2− 1− 17√
3(3− 2√
2)102
εa
).
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Series solution (continued)
The general solution is
z1(t) = a(t0)e(√
2−1)(t−t0) , w1(t) = b(t0)e−(1+√
2)(t−t0) .
Set b(t0) = 0 and the solution is
z = εa(t0)e(√
2−1)(t−t0) +√
3ε2a2(t0)6
(√
2− 1)
(e2(√
2−1)(t−t0) − e(√
2−1)(t−t0)) + O(ε3)
w =√
3ε2a2(t0)102
(1 + 3√
2)e2(√
2−1)(t−t0) + O(ε3) .
From ∂z(t, t0)/∂t0 = 0, we get
da(t0)dt0
= a
(√
2− 1− 17√
3(3− 2√
2)102
εa
).
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Series solution (continued)
The general solution is
z1(t) = a(t0)e(√
2−1)(t−t0) , w1(t) = b(t0)e−(1+√
2)(t−t0) .
Set b(t0) = 0 and the solution is
z = εa(t0)e(√
2−1)(t−t0) +√
3ε2a2(t0)6
(√
2− 1)
(e2(√
2−1)(t−t0) − e(√
2−1)(t−t0)) + O(ε3)
w =√
3ε2a2(t0)102
(1 + 3√
2)e2(√
2−1)(t−t0) + O(ε3) .
From ∂z(t, t0)/∂t0 = 0, we get
da(t0)dt0
= a
(√
2− 1− 17√
3(3− 2√
2)102
εa
).
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Main contents
1 IntroductionPhysics and heteroclinic connectionsRenormalization groupThe RG and differential equations
2 An extension of the RG analysis
3 Several examplesThe Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
4 Summary
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Dependence on β
The Gray-Scott model represents the cubic autocatalyticchemical reactions for two chemical species. The stationarypatterns are described by
u′′ = uv2 − λ(1− v)γv′′ = v − uv2 .
The equation is invariant under x → −x. Two heteroclinicorbits exist at γ = 2/9 and λ = 9/2, together with threeequilibria P1 = (1, 0) , P2 = (1/3, 3) , P3 = (2/3, 3/2)It can be converted to a 4-d dynamical system
u′ = p
p′ = uv2 − λ(1− v)v′ = w
w′ = v − uv2 .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Dependence on β
The Gray-Scott model represents the cubic autocatalyticchemical reactions for two chemical species. The stationarypatterns are described by
u′′ = uv2 − λ(1− v)γv′′ = v − uv2 .
The equation is invariant under x → −x. Two heteroclinicorbits exist at γ = 2/9 and λ = 9/2, together with threeequilibria P1 = (1, 0) , P2 = (1/3, 3) , P3 = (2/3, 3/2)It can be converted to a 4-d dynamical system
u′ = p
p′ = uv2 − λ(1− v)v′ = w
w′ = v − uv2 .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Dependence on β
The Gray-Scott model represents the cubic autocatalyticchemical reactions for two chemical species. The stationarypatterns are described by
u′′ = uv2 − λ(1− v)γv′′ = v − uv2 .
The equation is invariant under x → −x. Two heteroclinicorbits exist at γ = 2/9 and λ = 9/2, together with threeequilibria P1 = (1, 0) , P2 = (1/3, 3) , P3 = (2/3, 3/2)It can be converted to a 4-d dynamical system
u′ = p
p′ = uv2 − λ(1− v)v′ = w
w′ = v − uv2 .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Series solution
The stability exponents of P1 are ±3√
2/2, both beingdoubly degenerate. We have to use two parametersr0(x0) , r1 to parametrize the initial position.we obtain
u = 1 + ε(−√
2r0f(x, x0)3
) + ε24
243(f2(x , x0)− f(x , x0))r2
0r21 + · · ·
p = εf(x , x0)r0 − ε22√
281
(2f2(x , x0)− f(x , x0))r20r
21 + · · ·
v = ε(−√
2r0r1f(x, x0)3
)− ε2227
(f2(x , x0)− f(x , x0))r20r
21 + · · ·
w = εf(x , x0)r0r1 − ε2√
29
(2f2(x , x0)− f(x , x0))r20r
21 + · · · ,
where
f(x , x0) = exp(−3√
22
(x− x0)) .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Series solution
The stability exponents of P1 are ±3√
2/2, both beingdoubly degenerate. We have to use two parametersr0(x0) , r1 to parametrize the initial position.we obtain
u = 1 + ε(−√
2r0f(x, x0)3
) + ε24
243(f2(x , x0)− f(x , x0))r2
0r21 + · · ·
p = εf(x , x0)r0 − ε22√
281
(2f2(x , x0)− f(x , x0))r20r
21 + · · ·
v = ε(−√
2r0r1f(x, x0)3
)− ε2227
(f2(x , x0)− f(x , x0))r20r
21 + · · ·
w = εf(x , x0)r0r1 − ε2√
29
(2f2(x , x0)− f(x , x0))r20r
21 + · · · ,
where
f(x , x0) = exp(−3√
22
(x− x0)) .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The RG equation
The RG equation for r0(x0) is given by setting∂u(x, x0)/∂x0 = 0 followed by taking x → x0
dr0(x0)dx0
= −3r0√2
+227
εr20r
21 +
r0
21870(−45
√2r2
1(9 + 2r1)ε2r20
+ 8r31(9 + 2r1)ε3r3
0) + · · · .
By setting r1 = −9/2, we have
dr0(x0)dx0
= − 3√2r0 +
32r20 .
which has the solution
r0(x0) =√
22
(1− tanh3x0
2√
2) .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The RG equation
The RG equation for r0(x0) is given by setting∂u(x, x0)/∂x0 = 0 followed by taking x → x0
dr0(x0)dx0
= −3r0√2
+227
εr20r
21 +
r0
21870(−45
√2r2
1(9 + 2r1)ε2r20
+ 8r31(9 + 2r1)ε3r3
0) + · · · .
By setting r1 = −9/2, we have
dr0(x0)dx0
= − 3√2r0 +
32r20 .
which has the solution
r0(x0) =√
22
(1− tanh3x0
2√
2) .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The profile of the exact solution
The exact analytic solution of the original equation is thus
u(x) = 1−√
23
r0(x) =13(2 + tanh
3x
2√
2)
v(x) =3√2r0(x) =
32(1− tanh
3x
2√
2) .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Main contents
1 IntroductionPhysics and heteroclinic connectionsRenormalization groupThe RG and differential equations
2 An extension of the RG analysis
3 Several examplesThe Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
4 Summary
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Kuramoto-Sivashinsky equation
The Kuramoto-Sivashinsky equation is an importantphysics model
ut = (u2)x − uxx − νuxxxx ,
where ν > 0 is the hyper-viscosity parameter.With periodic boundary condition on [0 , 2π], we mayexpand
u(t, x) = i
∞∑k=−∞
akeikx .
For the antisymmetric solution u(t,−x) = −u(t, x), ak isreal and a−k = −ak. The equation becomes
ak = (k2 − νk4)ak − k
∞∑m=−∞
amak−m .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Kuramoto-Sivashinsky equation
The Kuramoto-Sivashinsky equation is an importantphysics model
ut = (u2)x − uxx − νuxxxx ,
where ν > 0 is the hyper-viscosity parameter.With periodic boundary condition on [0 , 2π], we mayexpand
u(t, x) = i
∞∑k=−∞
akeikx .
For the antisymmetric solution u(t,−x) = −u(t, x), ak isreal and a−k = −ak. The equation becomes
ak = (k2 − νk4)ak − k
∞∑m=−∞
amak−m .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The Kuramoto-Sivashinsky equation
The Kuramoto-Sivashinsky equation is an importantphysics model
ut = (u2)x − uxx − νuxxxx ,
where ν > 0 is the hyper-viscosity parameter.With periodic boundary condition on [0 , 2π], we mayexpand
u(t, x) = i
∞∑k=−∞
akeikx .
For the antisymmetric solution u(t,−x) = −u(t, x), ak isreal and a−k = −ak. The equation becomes
ak = (k2 − νk4)ak − k
∞∑m=−∞
amak−m .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Perturbation analysis
Assumeak = εak,1 + ε2ak,2 + ε3ak,3 + · · · .
For the 1− d unstable manifold of the origin at ν < 1, wemay get
a1,1(t, t0) = r(t0)e(1−ν)(t−t0) , ak,1 = 0 for k > 1 .
where r(t0) is the renormalization parameter.The RG equation for r(t0) is obtained fromda1(t, t0)/dt0|t=t0 = 0:
dr0
dt0= (1− v)r0 +
2r30
1− 7ν− 6r5
0
(1− 7ν)2(−1 + 13ν)+ · · · .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Perturbation analysis
Assumeak = εak,1 + ε2ak,2 + ε3ak,3 + · · · .
For the 1− d unstable manifold of the origin at ν < 1, wemay get
a1,1(t, t0) = r(t0)e(1−ν)(t−t0) , ak,1 = 0 for k > 1 .
where r(t0) is the renormalization parameter.The RG equation for r(t0) is obtained fromda1(t, t0)/dt0|t=t0 = 0:
dr0
dt0= (1− v)r0 +
2r30
1− 7ν− 6r5
0
(1− 7ν)2(−1 + 13ν)+ · · · .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
Perturbation analysis
Assumeak = εak,1 + ε2ak,2 + ε3ak,3 + · · · .
For the 1− d unstable manifold of the origin at ν < 1, wemay get
a1,1(t, t0) = r(t0)e(1−ν)(t−t0) , ak,1 = 0 for k > 1 .
where r(t0) is the renormalization parameter.The RG equation for r(t0) is obtained fromda1(t, t0)/dt0|t=t0 = 0:
dr0
dt0= (1− v)r0 +
2r30
1− 7ν− 6r5
0
(1− 7ν)2(−1 + 13ν)+ · · · .
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The time evolution on the connection
ν = 0.5
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The manifold and physical observable
ν = 0.5
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The time evolution on the connection
ν = 0.3
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
The Lotka-Volterra model of competitionThe Gray-Scott modelThe Kuramoto-Sivashinsky equation
The manifold and physical observable
ν = 0.3
[Y. Lan, Phys. Rev. E 87, 012914(2013)]
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Summary
An extension of the RG method has been proposed and wassuccessfully used for the determination of heteroclinic orbitsin the phase space.The method was applied to three typical physical systems:the Lotka-Volterra model of competition, the Gray-Scottmodel and the Kuramoto-Sivasshinsky equation.There seems no obstacles to generalize the currenttechnique to the treatment of dynamics on invariantsubmanifolds of dimension higher than one.Problems and challenges:(1) How to adapt the current scheme to the oscillatory caseis an interesting problem.(2) What if eigen-directions are not known.(3) How to treat more complex connections.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Summary
An extension of the RG method has been proposed and wassuccessfully used for the determination of heteroclinic orbitsin the phase space.The method was applied to three typical physical systems:the Lotka-Volterra model of competition, the Gray-Scottmodel and the Kuramoto-Sivasshinsky equation.There seems no obstacles to generalize the currenttechnique to the treatment of dynamics on invariantsubmanifolds of dimension higher than one.Problems and challenges:(1) How to adapt the current scheme to the oscillatory caseis an interesting problem.(2) What if eigen-directions are not known.(3) How to treat more complex connections.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Summary
An extension of the RG method has been proposed and wassuccessfully used for the determination of heteroclinic orbitsin the phase space.The method was applied to three typical physical systems:the Lotka-Volterra model of competition, the Gray-Scottmodel and the Kuramoto-Sivasshinsky equation.There seems no obstacles to generalize the currenttechnique to the treatment of dynamics on invariantsubmanifolds of dimension higher than one.Problems and challenges:(1) How to adapt the current scheme to the oscillatory caseis an interesting problem.(2) What if eigen-directions are not known.(3) How to treat more complex connections.
Yueheng Lan Bridging steady states with RG analysis
IntroductionAn extension of the RG analysis
Several examplesSummary
Summary
An extension of the RG method has been proposed and wassuccessfully used for the determination of heteroclinic orbitsin the phase space.The method was applied to three typical physical systems:the Lotka-Volterra model of competition, the Gray-Scottmodel and the Kuramoto-Sivasshinsky equation.There seems no obstacles to generalize the currenttechnique to the treatment of dynamics on invariantsubmanifolds of dimension higher than one.Problems and challenges:(1) How to adapt the current scheme to the oscillatory caseis an interesting problem.(2) What if eigen-directions are not known.(3) How to treat more complex connections.
Yueheng Lan Bridging steady states with RG analysis