quality of numerics: tides. application to the rössler...
TRANSCRIPT
Quality of numerics: TIDES.Application to the Rössler model.
R. Barrio, F. Blesa, M. Rodríguez, S. Serrano
GME – University of Zaragoza, SPAIN
A. ShilnikovInstitute of Neuroscience, University of Georgia, USA
Workshop on Bifurcation Analysis and its ApplicationsMontreal, Canada, July 7-10, 2010
R. Barrio (University of Zaragoza) TIDES & Rössler 1 / 33
Outline
I State-of-the-art numerical ODE integrator: TIDESI Study the parametric phase space of three-dimensional systems
(but also general dynamical systems and Hamiltonian systems)
1 Why TIDES?: Motivation
2 Taylor’s method: TIDES
3 The Rössler equations
R. Barrio (University of Zaragoza) TIDES & Rössler 2 / 33
Outline
I State-of-the-art numerical ODE integrator: TIDESI Study the parametric phase space of three-dimensional systems
(but also general dynamical systems and Hamiltonian systems)
1 Why TIDES?: Motivation
2 Taylor’s method: TIDES
3 The Rössler equations
R. Barrio (University of Zaragoza) TIDES & Rössler 2 / 33
Outline
I State-of-the-art numerical ODE integrator: TIDESI Study the parametric phase space of three-dimensional systems
(but also general dynamical systems and Hamiltonian systems)
1 Why TIDES?: Motivation
2 Taylor’s method: TIDES
3 The Rössler equations
R. Barrio (University of Zaragoza) TIDES & Rössler 2 / 33
Outline
1 Why TIDES?: Motivation
2 Taylor’s method: TIDES
3 The Rössler equations
R. Barrio (University of Zaragoza) TIDES & Rössler 3 / 33
Biparametric analysis in Lorenz: σ = 10
0 21 3
MLE
parameter r
para
met
er b
-1
0
1
2
3b=8/3
r=200
BD
-1MLE
146 166.1 214
R. Barrio (University of Zaragoza) TIDES & Rössler 4 / 33
The Hénon-Heiles Hamiltonian
−0.11 −0.105 −0.1 −0.095 −0.09 −0.085 −0.08 −0.075 −0.070.2529
0.253
0.2531
0.2532
0.2533
0.2534
0.2535
Coordinate y
Ene
rgy
E
−4 −2 0 2 40.2529
0.253
0.2531
0.2532
0.2533
0.2534
0.2535
Stability index k
1
2
m=1Saddle-node bifurcationgeneric
m=1Pitchfork bifurcationsymmetric
SN
P
e< 0 e> 0e= 0
1
2
m=3Touch-and-go bifurcationgeneric
TG1
1
11
1
1
P
SN
TG
SN
TG
P
m=1m=3
m=1
Safe regions: Stable and bounded regions far from the KAM tori
R. Barrio (University of Zaragoza) TIDES & Rössler 5 / 33
Numerical requirements
1 Periodic orbits, invariant tori→ Short integration times, sometimes with very highprecision and simultaneous solution of the variational equations
2 Stability of the systems→ Medium to large integration times and simultaneoussolution of the variational equations
3 Structure in the complex plane, location of complex singularities, . . .→ Very highprecision (500, 1000 digits)
I TAYLOR’s method: Automatic differentiation
y(t0) = y0,
y(ti) ' yi = yi−1 +dy(ti−1)
dthi +
12!
d2y(ti−1)
dt2 h2i + . . .+
1p!
dpy(ti−1)
dtp hpi .
Very “new” −→ EULER, G. Wanner, G. Corliss, ...
In Dynamical Systems −→ NEW LIFECarles Simó and collaborators
A. Jorba and M. ZouJohn Guckenheimer and collaborators
GME (Zaragoza, SPAIN) TIDESR. Barrio (University of Zaragoza) TIDES & Rössler 6 / 33
Numerical requirements
1 Periodic orbits, invariant tori→ Short integration times, sometimes with very highprecision and simultaneous solution of the variational equations
2 Stability of the systems→ Medium to large integration times and simultaneoussolution of the variational equations
3 Structure in the complex plane, location of complex singularities, . . .→ Very highprecision (500, 1000 digits)
I TAYLOR’s method: Automatic differentiation
y(t0) = y0,
y(ti) ' yi = yi−1 +dy(ti−1)
dthi +
12!
d2y(ti−1)
dt2 h2i + . . .+
1p!
dpy(ti−1)
dtp hpi .
Very “new” −→ EULER, G. Wanner, G. Corliss, ...
In Dynamical Systems −→ NEW LIFECarles Simó and collaborators
A. Jorba and M. ZouJohn Guckenheimer and collaborators
GME (Zaragoza, SPAIN) TIDESR. Barrio (University of Zaragoza) TIDES & Rössler 6 / 33
Numerical requirements
1 Periodic orbits, invariant tori→ Short integration times, sometimes with very highprecision and simultaneous solution of the variational equations
2 Stability of the systems→ Medium to large integration times and simultaneoussolution of the variational equations
3 Structure in the complex plane, location of complex singularities, . . .→ Very highprecision (500, 1000 digits)
I TAYLOR’s method: Automatic differentiation
y(t0) = y0,
y(ti) ' yi = yi−1 +dy(ti−1)
dthi +
12!
d2y(ti−1)
dt2 h2i + . . .+
1p!
dpy(ti−1)
dtp hpi .
Very “new” −→ EULER, G. Wanner, G. Corliss, ...
In Dynamical Systems −→ NEW LIFECarles Simó and collaborators
A. Jorba and M. ZouJohn Guckenheimer and collaborators
GME (Zaragoza, SPAIN) TIDESR. Barrio (University of Zaragoza) TIDES & Rössler 6 / 33
Numerical requirements
1 Periodic orbits, invariant tori→ Short integration times, sometimes with very highprecision and simultaneous solution of the variational equations
2 Stability of the systems→ Medium to large integration times and simultaneoussolution of the variational equations
3 Structure in the complex plane, location of complex singularities, . . .→ Very highprecision (500, 1000 digits)
I TAYLOR’s method: Automatic differentiation
y(t0) = y0,
y(ti) ' yi = yi−1 +dy(ti−1)
dthi +
12!
d2y(ti−1)
dt2 h2i + . . .+
1p!
dpy(ti−1)
dtp hpi .
Very “new” −→ EULER, G. Wanner, G. Corliss, ...
In Dynamical Systems −→ NEW LIFECarles Simó and collaborators
A. Jorba and M. ZouJohn Guckenheimer and collaborators
GME (Zaragoza, SPAIN) TIDESR. Barrio (University of Zaragoza) TIDES & Rössler 6 / 33
Numerical requirements
1 Periodic orbits, invariant tori→ Short integration times, sometimes with very highprecision and simultaneous solution of the variational equations
2 Stability of the systems→ Medium to large integration times and simultaneoussolution of the variational equations
3 Structure in the complex plane, location of complex singularities, . . .→ Very highprecision (500, 1000 digits)
I TAYLOR’s method: Automatic differentiation
y(t0) = y0,
y(ti) ' yi = yi−1 +dy(ti−1)
dthi +
12!
d2y(ti−1)
dt2 h2i + . . .+
1p!
dpy(ti−1)
dtp hpi .
Very “new” −→ EULER, G. Wanner, G. Corliss, ...
In Dynamical Systems −→ NEW LIFECarles Simó and collaborators
A. Jorba and M. ZouJohn Guckenheimer and collaborators
GME (Zaragoza, SPAIN) TIDESR. Barrio (University of Zaragoza) TIDES & Rössler 6 / 33
Outline
1 Why TIDES?: Motivation
2 Taylor’s method: TIDES
3 The Rössler equations
R. Barrio (University of Zaragoza) TIDES & Rössler 7 / 33
What is TIDES?TIDES: a Taylor series Integrator for Differential EquationS
Taylor series method using Variable-Stepsize Variable-Order formulationand extended formulas for the variational equations.
Free numerical software based on extended Taylor series method:TIDES (Abad, B., Blesa & Rodríguez ’09).
Extremely easy to use via a MATHEMATICA preprocessorminf-TIDES (Fortran), minc-TIDES (C)dp-TIDES (C-double precision), mp-TIDES (C-arbitrary precision)
Automatic construction of Fortran or C codes for solving ODEs
Automatic construction of C codes for solving solutions of ODEs andvariational equations up to any order (and sensitivities with respect toany parameter up to any order)
Easy to use arbitrary precision (do you need 500 digits?, 1000?)
Robust and stable numerical ODE solver (being an explicit method)
Solution as a power series (useful for events detection)
R. Barrio (University of Zaragoza) TIDES & Rössler 8 / 33
What is TIDES?TIDES: a Taylor series Integrator for Differential EquationS
Taylor series method using Variable-Stepsize Variable-Order formulationand extended formulas for the variational equations.
Free numerical software based on extended Taylor series method:TIDES (Abad, B., Blesa & Rodríguez ’09).
Extremely easy to use via a MATHEMATICA preprocessorminf-TIDES (Fortran), minc-TIDES (C)dp-TIDES (C-double precision), mp-TIDES (C-arbitrary precision)
Automatic construction of Fortran or C codes for solving ODEs
Automatic construction of C codes for solving solutions of ODEs andvariational equations up to any order (and sensitivities with respect toany parameter up to any order)
Easy to use arbitrary precision (do you need 500 digits?, 1000?)
Robust and stable numerical ODE solver (being an explicit method)
Solution as a power series (useful for events detection)
R. Barrio (University of Zaragoza) TIDES & Rössler 8 / 33
Join TIDES community
Where?: http://gme.unizar.es/software/tidesor Email: [email protected]
And ... IT IS FREE (freeware-software)!!!!!!
R. Barrio (University of Zaragoza) TIDES & Rössler 9 / 33
Join TIDES community
Where?: http://gme.unizar.es/software/tidesor Email: [email protected]
What is TIDES?
minf-tides basic TSM FORTRAN
minc-tides basic TSM C
dp-tides extended TSM C
mp-tides extended TSM + arbitrary precision C, MPFR
MathTIDES preprocessor MATHEMATICA
And ... IT IS FREE (freeware-software)!!!!!!
R. Barrio (University of Zaragoza) TIDES & Rössler 9 / 33
Join TIDES community
Where?: http://gme.unizar.es/software/tidesor Email: [email protected]
What is TIDES?
minf-tides basic TSM FORTRAN
minc-tides basic TSM C
dp-tides extended TSM C
mp-tides extended TSM + arbitrary precision C, MPFR
MathTIDES preprocessor MATHEMATICA
And ... IT IS FREE (freeware-software)!!!!!!
R. Barrio (University of Zaragoza) TIDES & Rössler 9 / 33
Proposition (Extended AD rules)If f (t , y(t)), g(t , y(t)) : (t , y) ∈ Rs+1 7→ R functions of class Cn, i = (i1, . . . is) ∈ Ns
0,i∗ = i− (0, . . . , 0, 1, 0, . . . , 0) = (i1, i2, . . . , ik − 1, 0, . . . , 0) and ‖i‖ =
∑sj=1 ij the total
order of derivation, we denote
f [j, i] :=1j!
∂‖i‖ f (j)(t)∂y i1
1 ∂y i22 · · · ∂y is
s
, f [j, 0] := f [j] =1j!
d j f (t)dt j ,
the jth Taylor coefficient of the partial derivative of f (t , y(t)) with respect to i and
h[j, v]n, i = h[j, v], (j 6= n or v 6= i), h[n, i]
n, i = 0.
Besides, given v = (v1, . . . , vs) ∈ Ns0 we define the multi-combinatorial number( i
v
)=( i1
v1
)·( i2
v2
)· · ·( is
vs
), and we consider the classical partial order in Ns
0. Then
(v) If h(t) = f (t)α with α ∈ R then h[0, 0] = (f [0](t))α and
h[0, i] =1
f [0]∑
v≤ i∗
(i∗
v
){α h[0, v] · f [0, i−v] − h[0, i−v]
0, i · f [0, v]}, i > 0,
h[n, i] =1
n f [0]
n∑j=0
(nα− j(α + 1)
) {∑v≤ i
( i
v
)h[j, v]
n, i · f [n−j, i−v]}, n > 0, i > 0.
R. Barrio (University of Zaragoza) TIDES & Rössler 10 / 33
Programming: WITHOUT VARIATIONAL EQUATIONS
Two body problem (Kepler)
x = − x(x2 + y2)3/2 , y = − y
(x2 + y2)3/2
KEPLER PROBLEM
for m = 0 to n − 2 doc = (1 + m)(2 + m)
s[m]1 = x × x
[m]+ y × y
[m]
s[m]2 = (s1)
−3/2[m]
x [m+2] = − x × s2[m]/c
y [m+2] = − y × s2[m]/c
end
R. Barrio (University of Zaragoza) TIDES & Rössler 11 / 33
Programming: WITHOUT VARIATIONAL EQUATIONS
Two body problem (Kepler)
x = − x(x2 + y2)3/2 , y = − y
(x2 + y2)3/2
KEPLER PROBLEM
for m = 0 to n − 2 doc = (1 + m)(2 + m)
s[m]1 = x × x
[m]+ y × y
[m]
s[m]2 = (s1)
−3/2[m]
x [m+2] = − x × s2[m]/c
y [m+2] = − y × s2[m]/c
end
KEPLER PROBLEM & SENSITIVITY VALUES
for m = 0 to n − 2 doc = (1 + m)(2 + m)for v = 0 to i do
s[m,v]1 = x × x
[m,v]+ y × y
[m,v]
s[m,v]2 = (s1)
−3/2[m,v]
x [m+2,v] = − x × s2[m,v]
/c
y [m+2,v] = − y × s2[m,v]
/cend
end
R. Barrio (University of Zaragoza) TIDES & Rössler 11 / 33
Standard integrator: USE
Lorenz Equations: x = σ(y − x), y = −xz + rx − y , z = xy − bz
Hénon-Heiles Hamiltonian: H = 12 (x
2 + y2) + 12 (x
2 + y2) + x2y − y3
3
MATHEMATICA code:lor=FirstOrderODE[{s*(y-x), -x*z+r*x-y, x*y-b*z},
t, {x, y, z}, {s, r, b}];CodeFiles[lor,"lorenz",MinTIDES->"Fortran",
Output->"lorOut.txt"]
HH=HamiltonianToODE[H, t, {x, y, X, Y}];CodeFiles[HH,"henon",MinTIDES->"Fortran",
Output->"henon.txt"]
R. Barrio (University of Zaragoza) TIDES & Rössler 12 / 33
Numerical test: minf-TIDES vs. DOP853, ODEXdouble precision + quadruple precision
10−10
10−5
10−3
10−2
CPU
time
dop853odexminftides
10−15
10−10
10−5
10−3
10−2
Relative error
CPU
time
dop853odexminftides
Henon-Heiles Lorenz problem
10−30
10−20
10−10
10−1
100
101
102
CPU
time
dop853odexminftides
Henon-Heiles
10−30
10−20
10−10
10−1
100
101
Relative error
CPU
time
dop853odexminftides
Lorenz problemRelative error
Relative error
DOUB
LE P
RECI
SION
QUAD
RUPL
E PR
ECIS
ION
R. Barrio (University of Zaragoza) TIDES & Rössler 13 / 33
PARTIAL DERIVATIVES: USE
Kepler potential (two body problem): V = −1/(x2 + y2)1/2
MATHEMATICA code:kep=PotentialToODE[V, t, {x, y}]CodeFiles[kep,"kepler1",addPartials->{{x},1},
Output->"kep1Out.txt"]CodeFiles[kep,"kepler2",addPartials->{{x,y,X,Y},7},
Output->"kep2Out.txt"]
CodeFiles[lor,"lorenz1.txt",addPartials->{{s},1},Output->"lor1Out.txt"]
CodeFiles[lor,"lorenz2.txt",addPartials->{{s,b},9},Output->"lor1Out.txt"]
∂x∂s,∂y∂s,∂z∂s, . . .
∂5x∂s2∂b3 , . . . ,
∂9z∂b9
R. Barrio (University of Zaragoza) TIDES & Rössler 14 / 33
PARTIAL DERIVATIVES: USE
Kepler potential (two body problem): V = −1/(x2 + y2)1/2
MATHEMATICA code:kep=PotentialToODE[V, t, {x, y}]CodeFiles[kep,"kepler1",addPartials->{{x},1},
Output->"kep1Out.txt"]CodeFiles[kep,"kepler2",addPartials->{{x,y,X,Y},7},
Output->"kep2Out.txt"]
CodeFiles[lor,"lorenz1.txt",addPartials->{{s},1},Output->"lor1Out.txt"]
CodeFiles[lor,"lorenz2.txt",addPartials->{{s,b},9},Output->"lor1Out.txt"]
∂x∂s,∂y∂s,∂z∂s, . . .
∂5x∂s2∂b3 , . . . ,
∂9z∂b9
R. Barrio (University of Zaragoza) TIDES & Rössler 14 / 33
PARTIAL DERIVATIVES: USE
Kepler potential (two body problem): V = −1/(x2 + y2)1/2
MATHEMATICA code:kep=PotentialToODE[V, t, {x, y}]CodeFiles[kep,"kepler1",addPartials->{{x},1},
Output->"kep1Out.txt"]CodeFiles[kep,"kepler2",addPartials->{{x,y,X,Y},7},
Output->"kep2Out.txt"]
CodeFiles[lor,"lorenz1.txt",addPartials->{{s},1},Output->"lor1Out.txt"]
CodeFiles[lor,"lorenz2.txt",addPartials->{{s,b},9},Output->"lor1Out.txt"]
∂x∂s,∂y∂s,∂z∂s, . . .
∂5x∂s2∂b3 , . . . ,
∂9z∂b9
R. Barrio (University of Zaragoza) TIDES & Rössler 14 / 33
PARTIAL DERIVATIVES: APPLICATIONS
1 Chaos indicatorsLyapunov exponents
↪→ Partial derivatives up to order 1.OFLI, OFLI2 (Barrio 2005)
↪→ Partial derivatives up to order 2.2 Uncertainty propagation
↪→ Partial derivatives up to order n.
R. Barrio (University of Zaragoza) TIDES & Rössler 15 / 33
PARTIAL DERIVATIVES: APPLICATIONS
1 Chaos indicatorsLyapunov exponents
↪→ Partial derivatives up to order 1.OFLI, OFLI2 (Barrio 2005)
↪→ Partial derivatives up to order 2.2 Uncertainty propagation
↪→ Partial derivatives up to order n.
R. Barrio (University of Zaragoza) TIDES & Rössler 15 / 33
TIDES and partial derivativesUncertainty propagation or box propagation: test on the two-body problem
Uncertainties on the initial conditions and parameters
y(t ; y0 + δy ,p + δp) ≈M∑
k=0
∑|i|=k
1i1! · · · is+l !
∂|i|y(t ; y0,p)
∂y i11 · · · ∂y is
s ∂pis+11 · · · ∂pis+l
l
δi11 · · · δ
is+ls+l
with the notation δy = (δ1, . . . , δs) and δp = (δs+1, . . . , δs+l).
- 1.5 -1.0 - 0.5 0.5
- 0.5
0.5
- 1.5 -1.0 - 0.5 0.5
order 1order 2order 3order 5-9points
AA
CC
B
order 9Monte-Carlo
R. Barrio (University of Zaragoza) TIDES & Rössler 16 / 33
HIGH PRECISION: USE
For Lorenz problem
MATHEMATICA code:lor=FirstOrderODE[eq, t, {x, y, z}, {s, r, b}]CodeFiles[lor,"lorenzMP",Precision->Multiple[300]]
Output->"lorOut.txt"]
Initial conditions:Must be given with 300 digitsObtained through Analytical studies in Lorenz Model(Prof. D. Viswanath)
R. Barrio (University of Zaragoza) TIDES & Rössler 17 / 33
HIGH PRECISION: USE
For Lorenz problem
MATHEMATICA code:lor=FirstOrderODE[eq, t, {x, y, z}, {s, r, b}]CodeFiles[lor,"lorenzMP",Precision->Multiple[300]]
Output->"lorOut.txt"]
Initial conditions:Must be given with 300 digitsObtained through Analytical studies in Lorenz Model(Prof. D. Viswanath)
R. Barrio (University of Zaragoza) TIDES & Rössler 17 / 33
HIGH PRECISION: USE
For Lorenz problem
MATHEMATICA code:lor=FirstOrderODE[eq, t, {x, y, z}, {s, r, b}]CodeFiles[lor,"lorenzMP",Precision->Multiple[300]]
Output->"lorOut.txt"]
Initial conditions:Must be given with 300 digitsObtained through Analytical studies in Lorenz Model(Prof. D. Viswanath)
R. Barrio (University of Zaragoza) TIDES & Rössler 17 / 33
Outline
1 Why TIDES?: Motivation
2 Taylor’s method: TIDES
3 The Rössler equations
R. Barrio (University of Zaragoza) TIDES & Rössler 18 / 33
The Rössler equations
The Rössler equationsdxdt
= −(y + z),dydt
= x + ay ,dzdt
= b + z(x − c),
Three dimensionless control parameters: a,b, c
♥ This model is a famous prototype of a continuousdynamical system exhibiting chaotic behavior withminimum ingredients.
• The fixed points: P1 and P2 (for c2 > 4ab) given byP1 = (−ap1, p1,−p1) and P2 = (−ap2, p2,−p2) with
p1 :=12
(−c
a−√
c2 − 4aba
), p2 :=
12
(−c
a+
√c2 − 4ab
a
).
R. Barrio (University of Zaragoza) TIDES & Rössler 19 / 33
The Rössler equations
The Rössler equationsdxdt
= −(y + z),dydt
= x + ay ,dzdt
= b + z(x − c),
Three dimensionless control parameters: a,b, c
♥ This model is a famous prototype of a continuousdynamical system exhibiting chaotic behavior withminimum ingredients.
• The fixed points: P1 and P2 (for c2 > 4ab) given byP1 = (−ap1, p1,−p1) and P2 = (−ap2, p2,−p2) with
p1 :=12
(−c
a−√
c2 − 4aba
), p2 :=
12
(−c
a+
√c2 − 4ab
a
).
R. Barrio (University of Zaragoza) TIDES & Rössler 19 / 33
MLE and OFLI2 plots in the (b, c) plane
c c c
c c c
bbb
bb b
a=0.2 a=0.1
a=0.4
a=0.35
a=0.3
a=0.6
Red: Chaotic orbits, Blue: regular orbits, White: escape orbits.
R. Barrio (University of Zaragoza) TIDES & Rössler 20 / 33
Bifurcation curves in the (b, c) plane
a=0.1
48
A
c
c
b
a=0.1, b=0.4
A1
a=0.1
Bifurcation diagram
MLE+OFLI2+AUTO(bifurcations)
Light: Chaotic orbits, Dark: regular orbits.Red: Period doubling curves.Blue: Fold curves.
R. Barrio (University of Zaragoza) TIDES & Rössler 21 / 33
Bifurcation curves in the (b, c) plane
a=0.2 a=0.3
When the parameter a grows, the structures are no more isolated.The period doubling and fold curves show a repeated pattern anddelimit the structures.
R. Barrio (University of Zaragoza) TIDES & Rössler 22 / 33
Structures in the (b, c) plane
a=0.1B
a=0.2
D
12482 4
Isolated structure LPCPD
Coupled structures
region of coexistence
cusp bifurcation
parameter c
par
amet
er b
parameter c
par
amet
er b
When thestructures arecoupled, acodimension2 cuspbifurcationappears.
R. Barrio (University of Zaragoza) TIDES & Rössler 23 / 33
Coexistence: two limit cycles (B3)
B1 B2
bb
a=0.2, (x,y,z)(0)=(1,1,1) a=0.2, (x,y,z)(0)=(-7,-7,-7)
B3
B4B7B6
B7B6B5
cc
c
a=0.2, b=9
−500
50−50
050
0
100
200
xyz
c=50
coexistence region
B3
R. Barrio (University of Zaragoza) TIDES & Rössler 24 / 33
Coexistence a limit cycle and a chaotic attractor (B4)
B1 B2
bb
a=0.2, (x,y,z)(0)=(1,1,1) a=0.2, (x,y,z)(0)=(-7,-7,-7)
B3
B4B7B6
B7B6B5
cc
a=0.2, b=4.5
coexistence region
c
−100−50
0 50
−100−50
050
0
200
400
xyz
c=60B4
R. Barrio (University of Zaragoza) TIDES & Rössler 25 / 33
Coexistence: two chaotic attractors (B5)
B1 B2
bb
a=0.2, (x,y,z)(0)=(1,1,1) a=0.2, (x,y,z)(0)=(-7,-7,-7)
B3
B4B7B6
B7B6B5
cc
c
a=0.2, b=1
coexistence region−50
050
−100−50
050
0
200
400
xy
z
c=48.5B5
R. Barrio (University of Zaragoza) TIDES & Rössler 26 / 33
Chaotic attractors: topological templates
14 16 18 20 22 24
14
16
18
20
22
24
10 15 20 25
10
15
20
25
10 15 20 25
10
15
20
25
y(n)
y(n+
1)
y(n)
y(n+
1)
y(n)
y(n+
1)
a=0.08969 a=0.2 a=0.25
−0.5 0 0.5 1 1.5 2−0.5
0
0.5
1
1.5
2
a=0.542, b=2, c=4
y(n)
y(n+
1)
spiral attractor screw attractor
01
2301 012
funnel attractor (4 branches)
Letellier et al.
Msp =
(0 −1−1 −1
), Msc =
0 −1 −1−1 −1 −2−1 −2 −2
, Mfu(n) =
0 −1 · · · −1 −1−1 −1 · · · −2 −2
.
.
....
. . ....
.
.
.−1 −2 · · · −(n − 1) −n−1 −2 · · · −n −n
R. Barrio (University of Zaragoza) TIDES & Rössler 27 / 33
And in the (a, c) plane ...
b=0.2
A
b=2
B
# The bifurcation curves in the (a, c) plane go to ...?
R. Barrio (University of Zaragoza) TIDES & Rössler 28 / 33
Global bifurcations: homoclinic bifurcations
A
b=0.2
Hopf-P1Hopf-P2
Hopf-P1
BT
ZH
1HB2PDLPC SN
Hopf-P2
Hopf-P1
sp-sc
BT: Bogdanov-Takens (or double-zero) bifurcation point
1: double real leading unstable eigenvalues λ2 = λ3 > 0 (Belyakov).
infinite number of bifurcation curves of homoclinic orbits and foldbifurcations of periodic orbits
R. Barrio (University of Zaragoza) TIDES & Rössler 29 / 33
Global bifurcations: homoclinic bifurcations
A
b=0.2
Hopf-P1Hopf-P2
Hopf-P1
BT
ZH
1HB2PDLPC SN
Hopf-P2
Hopf-P1
sp-sc
BT: Bogdanov-Takens (or double-zero) bifurcation point
1: double real leading unstable eigenvalues λ2 = λ3 > 0 (Belyakov).
infinite number of bifurcation curves of homoclinic orbits and foldbifurcations of periodic orbits
R. Barrio (University of Zaragoza) TIDES & Rössler 29 / 33
Global bifurcations: homoclinic bifurcations
A
b=0.2
Hopf-P1Hopf-P2
Hopf-P1
BT
ZH
1HB2PDLPC SN
Hopf-P2
Hopf-P1
sp-sc
BT: Bogdanov-Takens (or double-zero) bifurcation point
1: double real leading unstable eigenvalues λ2 = λ3 > 0 (Belyakov).
infinite number of bifurcation curves of homoclinic orbits and foldbifurcations of periodic orbits
R. Barrio (University of Zaragoza) TIDES & Rössler 29 / 33
Global bifurcations: homoclinic bifurcations
0 2 4 6
−3
−2
−1
0
x
y
A
b=0.2
Hopf-P1Hopf-P2
Hopf-P1
BT
ZH
1
P2
HB2PDLPC SN
Hopf-P2
Hopf-P1
a=2.39a=2.37a=2.24a=1.99
−5015
−1005
0
30
−10 010−20
0100
50A2
A0
P2 P2
a=0.90 a=0.39
sp-sc
0 5 10−6
020
10
20 A3
P2
a=1.46A1
sp-sc: spiral to screw attractor bifurcation
R. Barrio (University of Zaragoza) TIDES & Rössler 30 / 33
Global bifurcations: homoclinic bifurcations
0 2 4 6
−3
−2
−1
0
x
y
A
b=0.2
Hopf-P1Hopf-P2
Hopf-P1
BT
ZH
1
P2
HB2PDLPC SN
Hopf-P2
Hopf-P1
a=2.39a=2.37a=2.24a=1.99
−5015
−1005
0
30
−10 010−20
0100
50A2
A0
P2 P2
a=0.90 a=0.39
sp-sc
0 5 10−6
020
10
20 A3
P2
a=1.46A1
sp-sc: spiral to screw attractor bifurcation
R. Barrio (University of Zaragoza) TIDES & Rössler 30 / 33
Global bifurcations: homoclinic bifurcations b = 2
−10 0 10−1005
0
30
−10 0 10−20
0100
50
−10 020
−20010
0
50
B
B1 B2 B3
b=2
P2
P1
P2P2
P1P1
Hopf-P1
Hopf-P2
22
BT
HB1PDLPC SN
Hopf-P2
Hopf-P1
a=0.55c=5.79
a=0.51c=7.48
a=0.49c=8.61
BT: Bogdanov-Takens (or double-zero) bifurcation point2: neutral saddle-focus bifurcation (resonant eigenvalues λ3 = −α of P1, beingthe eigenvalues λ1,2 = α± iβ and λ3 > 0). (Belyakov).
R. Barrio (University of Zaragoza) TIDES & Rössler 31 / 33
Global bifurcations: homoclinic bifurcations b = 2
−10 0 10−1005
0
30
−10 0 10−20
0100
50
−10 020
−20010
0
50
B
B1 B2 B3
b=2
P2
P1
P2P2
P1P1
Hopf-P1
Hopf-P2
22
BT
HB1PDLPC SN
Hopf-P2
Hopf-P1
a=0.55c=5.79
a=0.51c=7.48
a=0.49c=8.61
BT: Bogdanov-Takens (or double-zero) bifurcation point2: neutral saddle-focus bifurcation (resonant eigenvalues λ3 = −α of P1, beingthe eigenvalues λ1,2 = α± iβ and λ3 > 0). (Belyakov).
R. Barrio (University of Zaragoza) TIDES & Rössler 31 / 33
Conclusions
♣ Taylor method gives an easy, powerful and stable implementation.
Use TIDES. It is GOOD, it is FAST, it is EASY and it is FREE.
♣ Results for the Rössler model:Coexistence of different kind of attractors and escape dynamics.
Three-parametric model explains the already observed behaviour.
Parametric changes.
Global homoclinic bifurcations explain the parametric structure.
Changes in the attractor’s topology (Perestroikas).
♣ New methods to compute T-points
R. Barrio (University of Zaragoza) TIDES & Rössler 32 / 33
Conclusions
♣ Taylor method gives an easy, powerful and stable implementation.
Use TIDES. It is GOOD, it is FAST, it is EASY and it is FREE.
♣ Results for the Rössler model:Coexistence of different kind of attractors and escape dynamics.
Three-parametric model explains the already observed behaviour.
Parametric changes.
Global homoclinic bifurcations explain the parametric structure.
Changes in the attractor’s topology (Perestroikas).
♣ New methods to compute T-points
R. Barrio (University of Zaragoza) TIDES & Rössler 32 / 33
Conclusions
♣ Taylor method gives an easy, powerful and stable implementation.
Use TIDES. It is GOOD, it is FAST, it is EASY and it is FREE.
♣ Results for the Rössler model:Coexistence of different kind of attractors and escape dynamics.
Three-parametric model explains the already observed behaviour.
Parametric changes.
Global homoclinic bifurcations explain the parametric structure.
Changes in the attractor’s topology (Perestroikas).
♣ New methods to compute T-points
R. Barrio (University of Zaragoza) TIDES & Rössler 32 / 33
Bibliography
A. Abad, R. Barrio, F. Blesa, M. Rodríguez, TIDES: A Taylor series Integrator forDifferential EquationS. Preprint (2010).
Where?: http://gme.unizar.es/software/tides
or Email: [email protected]
R. Barrio, Sensitivity analysis of ODE’s/DAE’s using the Taylor series method,SIAM J. Sci. Comput. 27 (6) (2006) 1929–1947.
R. Barrio, Sensitivity tools vs. Poincaré sections, Chaos Solitons Fractals 25 (3)(2005) 711–726.
R. Barrio, Painting chaos: a gallery of sensitivity plots of classical problems, Int.J. Bifurc. Chaos 16 (10) (2006) 2777–2798.
R. Barrio, F. Blesa, S. Serrano, Qualitative analysis of the Rössler equations:Bifurcations of limit cycles and chaotic attractors, Phys. D 238 (2009) 1087–1100.
R. Barrio, A. Shilnikov, preprint (2010).
R. Barrio (University of Zaragoza) TIDES & Rössler 33 / 33