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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 1
Universität Stuttgart 20. April 2004
AnT 4.669– a tool for simulatingand investigating dynamical systems
Dr. Michael Schanz
![Page 2: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/2.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 2
What is AnT 4.669? Introduction
1. Introduction
AnT 4.669– a simulation and Analysis Tool for dynamical systems
![Page 3: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/3.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 2
What is AnT 4.669? Introduction
1. Introduction
AnT 4.669– a simulation and Analysis Tool for dynamical systems
AnT 4.669application areas:
I science and education
![Page 4: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/4.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 2
What is AnT 4.669? Introduction
1. Introduction
AnT 4.669– a simulation and Analysis Tool for dynamical systems
AnT 4.669application areas:
I science and education
AnT 4.669capabilities:
I several classes of dynamical systemsI several investigation methodsI one-, two-, and higher dimensional scansI distributed computation (grid computing)
AnT 4.669properties:
I open software architectureI GNU public licenseI supported platforms
Solaris, Linux, FreeBSD, Windows (98, NT, 2000, XP)
![Page 5: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/5.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 3
History Introduction
AnT 4.669was designed, is maintained and will be further develo-ped by the Non–Linear Dynamics Group of the department ImageUnderstanding (Head: Prof. Dr. P. Levi) at the Institute of Par-allel and Distributed Systems (IPVS) of the University of Stuttgart.
Members of the group:
I Dr. Michael Schanz
I Dr. Viktor Avrutin
I Robert Lammert
I Georg Wackenhut
and about 25 students
History of the project:
1998: first prototypes(FORTRAN, C)
2000: AnT 4.66(C)
2001: AnT 4.669(C++)
Current state:≈ 120 000 lines of source code
![Page 6: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/6.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
![Page 7: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/7.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
mathematics
![Page 8: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/8.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
mathematics
physics engineering chemistry biology
![Page 9: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/9.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
mathematics
physics engineering chemistry biology
electronics medicine
![Page 10: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/10.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
mathematics
physics engineering chemistry biology
electronics medicinecomputer science
. . .
![Page 11: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/11.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
mathematics
physics engineering chemistry biology
electronics medicinecomputer science
. . .
modeling
⇓simulation
⇓analysis
⇓interpretation
![Page 12: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/12.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
mathematics
physics engineering chemistry biology
electronics medicinecomputer science
. . .
modeling
⇓simulation
⇓analysis
⇓interpretation
investigation of the dynamic behavior
![Page 13: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/13.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
mathematics
physics engineering chemistry biology
electronics medicinecomputer science
. . .
modeling
⇓simulation
⇓analysis
⇓interpretation
investigation of the dynamic behavior����
analytic
![Page 14: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/14.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
mathematics
physics engineering chemistry biology
electronics medicinecomputer science
. . .
modeling
⇓simulation
⇓analysis
⇓interpretation
investigation of the dynamic behavior����
analytic?
semi-analytic
![Page 15: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/15.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
mathematics
physics engineering chemistry biology
electronics medicinecomputer science
. . .
modeling
⇓simulation
⇓analysis
⇓interpretation
investigation of the dynamic behavior����
analytic?
semi-analytic
HHHj
numeric
![Page 16: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/16.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 4
Motivation Introduction
Application areas for dynamical systems:
mathematics
physics engineering chemistry biology
electronics medicinecomputer science
. . .
modeling
⇓simulation
⇓analysis
⇓interpretation
investigation of the dynamic behavior����
analytic?
semi-analytic
HHHj
numeric
⇓ ⇓numerical simulation and Analysis Tools
are required
![Page 17: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/17.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 5
Development of a simulation and analysis tool Introduction
Required knowledge and experience?Involved areas of computer science?
![Page 18: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/18.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 5
Development of a simulation and analysis tool Introduction
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics
![Page 19: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/19.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 5
Development of a simulation and analysis tool Introduction
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics
• numerics
• scientific computing
• . . .
![Page 20: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/20.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 5
Development of a simulation and analysis tool Introduction
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics
• numerics
• scientific computing
• . . .
?
![Page 21: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/21.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 6
Examples Dynamical systems
2. Dynamical systems
Attractors of the map–class
Generalized Hénon–Lozi map
xn+1 = 1− a|xn|γ + yn
yn+1 = bxn
a = 1.4, b = 0.3, γ = 2.0
yn
xn
a = 1.8, b = 0.3, γ = 1.0
yn
xn
![Page 22: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/22.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 7
Examples Dynamical systems
Attractors of the ODE–class
Lorenz 63 system
x = σ(y − x)
y = rx− y − xz
z = −bz + xy
σ = 16.0, r = 370.0, b = 4.0
x
y
z
σ = 16.0, r = 305.0, b = 4.0
xy
z
σ = 10, r = 146.7981, b = 83
xy
z
![Page 23: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/23.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 8
Examples Dynamical systems
Coexisting periodic attractors of the DDE–class
Phase Locked Loop (PLL)with time delay
x(t) = −R sin(x(t− τ))
τ = 1.0
four different constant initial functions
on the interval [−τ, 0]:
R = 3.5, R = 4.10, R = 4.10
x
x
R = 4.102
x
x
R = 4.11
x
x
![Page 24: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/24.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 9
Examples Dynamical systems
Dynamics of a stochastic system
Ornstein–Uhlenbeck process:
d x t = −M x dt + σ d W t
σ = 1 , M =
−10−4 0.1 −0.2−0.1 −10−4 0.20.5 −0.5 −10−4
z
x y
![Page 25: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/25.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 10
Examples Dynamical systems
Transient and asymptotic dynamics of a coupled map lattice(CML)
Coupled piecewise–linear maps:
xin+1 = f(κi
n) f(x) =
{x + a if x < 1
0 if x ≥ 1i = 1..N
κin =
γ1 x(i−1) mod Nn + γ2 xi
n + γ3 x(i+1) mod Nn
γ1 + γ2 + γ3
1
spac
ein
dex
i
238
6
n -
N = 238, γ1 = γ2 = γ3 = 1, a = 0.36
![Page 26: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/26.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 11
Examples Dynamical systems
Transient dynamics of a partial differential equation (PDE)
Heat conduction equation:
∂T (x, t)
∂t= κ
∂2T (x, t)
∂x2with κ = 0.01
T (x, t)
t
x
![Page 27: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/27.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 12
Classes of dynamical systems Dynamical systems
Classes of dynamical systems supported by AnT 4.669
I basic
• map
• ODE
• DDE
• FDE
I composite
• CML
• CODEL
• 1D–PDE
I hybrid• hybrid map• hybrid ODE• hybrid DDE
I stochastic• stochastic map• stochastic ODE• stochastic DDE
I etc.• recurrent map• external data
![Page 28: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/28.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 12
Classes of dynamical systems Dynamical systems
Classes of dynamical systems supported by AnT 4.669
I basic
• map
• ODE
• DDE
• FDE
I composite
• CML
• CODEL
• 1D–PDE
I hybrid• hybrid map• hybrid ODE• hybrid DDE
I stochastic• stochastic map• stochastic ODE• stochastic DDE
I etc.• recurrent map• external data
⇒ Support of 15 different classes of dynamical systems
![Page 29: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/29.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 13
Iterator concept Dynamical systems
Support of different types of dynamical systems is possible due tothe general concept of an abstract iterator, which is a special kindof an abstract transition:
previous state
next state
iterator
![Page 30: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/30.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 14
Iterator concept Dynamical systems
Support of different types of dynamical systems is possible due tothe general concept of an abstract iterator, which is a special kindof an abstract transition:
previous state
next state
iterator proxy
systemfunction
![Page 31: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/31.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 14
Iterator concept Dynamical systems
Support of different types of dynamical systems is possible due tothe general concept of an abstract iterator, which is a special kindof an abstract transition:
previous state
next state
iterator proxy
systemfunction
Dependent on the current type of the dynamical system the ab-stract iterator can be instantiated as:
I simple iterator(for maps, CMLs, Poincaré maps, external data input, etc.)
I ODE integrator (for ODEs, CODELs, 1D-PDEs)I DDE integrator (for DDEs, CDDELs)I FDE integrator (for FDEs)I ...
![Page 32: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/32.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 15
Integration methods Dynamical systems
Integration methods for ODEs, DDEs and FDEs,supported by AnT 4.669
one–step steppers
I Euler (expl., impl.)I Heun (expl., impl.)I MidpointI RadauI RalstonI Runge–KuttaI GillI Runge–Kutta–MersonI Runge–Kutta–FehlbergI Butcher
• pre–defined arrays• user–defined arrays
multi-step steppers
I Adams–Bashforth (6)I Adams–Moulton (6)I BDF (6)I PECE–AB–AM (6× 6)I PECE–AB–BDF (6× 6)
wrappers
I basic, backwardI step size adaption
• gradient based• halfstep• two steppers
⇒70
≈ 2000. . .
different integration methods
![Page 33: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/33.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 16
Integration methods Dynamical systems
Remarks on numerical integration I
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
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Page 16
Integration methods Dynamical systems
Remarks on numerical integration I
Two time series of the Lorenz 63 system for identical initial con-ditions calculated with two different integration methods:
blue: Gill’s method, green: Runge–Kutta method
t
x
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
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AnT 4.669 features
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Page 16
Integration methods Dynamical systems
Remarks on numerical integration I
Two time series of the Lorenz 63 system for identical initial con-ditions calculated with two different integration methods:
blue: Gill’s method, green: Runge–Kutta method
t
x
Two time series of the Lorenz 63 system for identical initial con-ditions calculated with the same integration method:
blue/red: Gill’s method, two different implementations
t
x
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
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Page 17
Integration methods Dynamical systems
Remarks on numerical integration II
Numerical solution of the circular co–planar restricted three bodyproblem:
x = x + 2 y − (1− µ)x + µ
r13− µ
x + µ− 1r2
3
y = y + 2 x− (1− µ)y
r13− µ
y
r23
r1 =[(x + µ)2 + y2
] 12
r2 =[(x + µ− 1)2 + y2
] 12
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
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AnT 4.669 features
Summary
Outlook
Page 17
Integration methods Dynamical systems
Remarks on numerical integration II
Numerical solution of the circular co–planar restricted three bodyproblem:
x = x + 2 y − (1− µ)x + µ
r13− µ
x + µ− 1r2
3
y = y + 2 x− (1− µ)y
r13− µ
y
r23
r1 =[(x + µ)2 + y2
] 12
r2 =[(x + µ− 1)2 + y2
] 12
x
y
![Page 38: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/38.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 17
Integration methods Dynamical systems
Remarks on numerical integration II
Numerical solution of the circular co–planar restricted three bodyproblem:
x = x + 2 y − (1− µ)x + µ
r13− µ
x + µ− 1r2
3
y = y + 2 x− (1− µ)y
r13− µ
y
r23
r1 =[(x + µ)2 + y2
] 12
r2 =[(x + µ− 1)2 + y2
] 12
x
y
x
y
![Page 39: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/39.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 18
Investigation methods Investigation
3. Investigation methods
Investigation methods supported by AnT 4.669
I General trajectory evaluations– orbits, velocities, extreme values, cobweb diagrams
I Basic statistics– mean values, standard deviations, cross–correlations
I Box counting methods– invariant measures, fractal dimensions
I Lyapunov exponents analysis– for maps, CMLs, ODEs, DDEs, FDEs, hybrid systems
I Extended Poincaré sections and Poincaré return mapsI Period analysis (systems discrete in time)I Region analysis (based on period analysis)I Spectral analysisI Condition checkerI Principal component analysisI Symbolic sequence analysis
– symbolic entropies for an arbitrary description levelI Symbolic image analysis
– detection of invariant sets, basins of attraction,– calculation of stable and unstable manifolds
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 19
Examples Investigation
Examples for several investigation methods
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 19
Examples Investigation
Examples for several investigation methods
Natural measure of a chaotic attractor
ρ(x, y)
xy
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 19
Examples Investigation
Examples for several investigation methods
Natural measure of a chaotic attractor
ρ(x, y)
xy
Cobweb diagram
xn+1
xn
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 19
Examples Investigation
Examples for several investigation methods
Natural measure of a chaotic attractor
ρ(x, y)
xy
Cobweb diagram
xn+1
xn
Poincaré section of a chaotic attractor
x
y
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 19
Examples Investigation
Examples for several investigation methods
Natural measure of a chaotic attractor
ρ(x, y)
xy
Cobweb diagram
xn+1
xn
Poincaré section of a chaotic attractor
x
y
Power spectrum of a limit cycle
lg P (f)
f
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 20
Examples Investigation
Duffing oscillator:
x = y
y = x− x3 − εy
ε = 0.15
stable and unstable manifolds of the fixed point at the origin
y
xIn cooperation with D. Fundinger and G.OsipenkoState Polytechnic University, St. Petersburg, Russia
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 21
Iteration Machine Investigation
Basic concept: Iteration Machine
iterationinitialization
methodsplug-ins proxy
iterator
methodsplug-ins
systemfunction
iteration stopcriterion (timer)
methodsplug-ins
Iteration Machine
Structure: pre-sequence, cyclic sequence, post-sequence
Contents: iterator, iteration method plug-ins
Setup: dynamically during initialization phase
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 22
Scans supported by AnT 4.669 Scans
4. Scans
Scans supported by AnT 4.669
Scannable objects:
I system parameters bifurcation scenarios,regions with different behavior
I initial values coexisting objects,basins of attraction
I method parameters method tuning
Scan types:
I real, integerI linear, logarithmicI using external dataI parametric (linear, elliptic)
Scan item sequences ⇒ N–dimensional scans.
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 23
Scan types Scans
Parameter scan (1D)
logistic map: xn+1 = αxn(1− xn)
ln (|α− α∞|)
ln(∣ ∣ x−
1 2
∣ ∣)bifurcation points
α8 α7 α6 α5 α4 α3 α2 α1
periodsT = 256T = 128
T = 64T = 32
T = 16T = 8
T = 4T = 2
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 24
Scan types Scans
Initial value scan (2D)
Gingerbreadman map:
xn+1 = 1− yn + |xn|yn+1 = xn
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 24
Scan types Scans
Initial value scan (2D)
Gingerbreadman map:
xn+1 = 1− yn + |xn|yn+1 = xn
period analysis
y0
x0
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 24
Scan types Scans
Initial value scan (2D)
Gingerbreadman map:
xn+1 = 1− yn + |xn|yn+1 = xn
period analysis
y0
x0
region analysis
y0
x0
![Page 52: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/52.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 25
Scan types Scans
Scans of investigation method parameters
Parameter tuning for an investigationmethod: Lyapunov Exponents.
I norm ε of deviation vectorsI number NGSO of steps between two
Gram Schmidt orthonormalizations
log|λ−
λ∗ |
ε
λ1,λ
2,λ
3
NGSO
log|λ−
λ∗ |
NGSO
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 26
Examples Scans
The three Lyapunov exponents of the Lorenz 63 system:
r
λ1,λ
2,λ
3
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 26
Examples Scans
The three Lyapunov exponents of the Lorenz 63 system:
r
λ1,λ
2,λ
3
The three Lyapunov exponents of the Aizawa system:
ε
λ1,λ
2,λ
3
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 27
Examples Scans
Example for a 2D parameter scan
Period Adding Big Bang Bifurcation in a piecewise-linear map
xn+1 =
{fl(xn) = bxn + c if xn < 1
2fr(xn) = xn − a if xn ≥ 1
2
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 27
Examples Scans
Example for a 2D parameter scan
Period Adding Big Bang Bifurcation in a piecewise-linear map
xn+1 =
{fl(xn) = bxn + c if xn < 1
2fr(xn) = xn − a if xn ≥ 1
2
b = 12 , 2D parameter space [a× c]
c
a
2
3
4
5
5
7
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 27
Examples Scans
Example for a 2D parameter scan
Period Adding Big Bang Bifurcation in a piecewise-linear map
xn+1 =
{fl(xn) = bxn + c if xn < 1
2fr(xn) = xn − a if xn ≥ 1
2
b = 12 , 2D parameter space [a× c]
c
a
2
3
4
5
5
7
Bifurcation diagram
x
φ
Period diagram
T
φ
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 28
Examples Scans
Example for a 2D parameter scan
2D bifurcation scenarios, induced by Big Bang Bifurcations
β
α
2
3
4
5
5
7
9
4
6
2D period adding scenario
β
α
2
345
5
2D period increment scenario
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 29
Examples Scans
Example of a 2D parameter scan
Largest Lyapunov exponent of the Lorenz 63 system:
λ
σ
R
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 30
Scan Machine Scans
Basic concept: Scan Machine
scaninitialization
methodsplug-ins
iterationmachine
methodsplug-ins
scan stopcriterionmethodsplug-ins
Scan Machine
iterationinitializationmethodsplug-ins proxyiterator
methodsplug-ins
systemfunction
iteration stopcriterion (timer)methodsplug-ins
Iteration Machine
Structure: pre-sequence, cyclic sequence, post-sequence
Contents: Iteration Machine, scan method plug-ins
Setup: dynamically during initialization phase
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 31
Scan Machine for Poincaré Maps Scans
scaninitialization
methodsplug-ins
iterationmachinemethodsplug-ins
scan stopcriterion
methodsplug-ins
Scan Machine
iterationinitialization
methodsplug-ins
mapproxy
map iterator
methodsplug-ins
iteration stopcriterion (timer)
methodsplug-ins
Iteration Machine
iterationinitialization
(e.g. ODE)proxy
(e.g. ODE)integrator
systemfunction
iteration stop criterion(Poincaré condition)
Iteration Machine (inside)
The system function of a Poincaré mapis given by a complete iteration machinecontaining a dynamical system inside.
The generalized Poincaré condition definesthe stop criterion of the iteration machine in-side.
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 32
Poincaré maps Scans
Examples of scans of a Poincaré map (ODE)
Rössler system:
x = −(x + z)
y = x + ay
z = b + z(x− c)
a = 0.15, b = 0.2Poincaré section using the fixed half–plane
{(x, y, z)T | y = 0, x > 0
}
zn
c
λ1,λ
2
c
xy
z
xy
z
x
z
x
z
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 33
Poincaré maps Scans
Examples of scans of a Poincaré map (DDE)
PLL system with delay: x(t) = −R sin(x(t− τ))Poincaré section using the condition defined by:x = 0 and x ∈ [1, 2]
x
R
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University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 34
Distribution AnT 4.669 features
5. AnT 4.669 features
Grid computing
Distribution of a scan among several nodes
I Client/Server architecture
I The server distributes tasks and manages calculation results
I An arbitrary number of clients perform the calculations
I Adding and removing of clients on-the-fly
I Data are sent/received via TCP/IP socket connections
I platform independence ⇒ running of server and clients in aheterogeneous environment
![Page 65: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/65.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 35
User Interface AnT 4.669 features
User Interface
I simplification of the complex initialization phase
I specification induced, automatic widget creation
I extendable for multi-lingual support
![Page 66: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/66.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 36
Visualization AnT 4.669 features
Visualization
• time series, space-time plots,phase portraits
• translation, scaling, rotation
• multiple views
• based on OpenGL standard
![Page 67: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/67.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 37
Web interface AnT 4.669 features
Web interface
Challenge: AnT 4.669 is designed as a desktop–application
Architecture:
• Separation between the computation engine and the graphi-cal user interface
Target solution:
• The computation engineis on the server side
• Configuration input andvisualization are on theclient side
Three new applications:
• Server• Configuration editor• Visualization client
![Page 68: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/68.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 38
Transition concept Summary
6. Summary
Transition concept
Reasons for building transition subclasses:
1. creating new transition structures2. implementing specific functionality
The transition concept is used in AnT4.669extensively.
![Page 69: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/69.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 38
Transition concept Summary
6. Summary
Transition concept
Reasons for building transition subclasses:
1. creating new transition structures2. implementing specific functionality
The transition concept is used in AnT4.669extensively.
![Page 70: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/70.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing
![Page 71: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/71.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing• software architecture
– transitions, machines, proxies, plug-ins
![Page 72: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/72.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing• software architecture
– transitions, machines, proxies, plug-ins• definition of user function interfaces (system functions)
![Page 73: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/73.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing• software architecture
– transitions, machines, proxies, plug-ins• definition of user function interfaces (system functions)• graphical user interface design
![Page 74: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/74.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing• software architecture
– transitions, machines, proxies, plug-ins• definition of user function interfaces (system functions)• graphical user interface design• design of description languages (initialization)
![Page 75: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/75.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing• software architecture
– transitions, machines, proxies, plug-ins• definition of user function interfaces (system functions)• graphical user interface design• design of description languages (initialization)• theoretical computer science
– algorithms and data structures
![Page 76: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/76.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing• software architecture
– transitions, machines, proxies, plug-ins• definition of user function interfaces (system functions)• graphical user interface design• design of description languages (initialization)• theoretical computer science
– algorithms and data structures• standards
– C++, TCP/IP, POSIX, OpenGL, GTK+, Web-programming
![Page 77: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/77.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing• software architecture
– transitions, machines, proxies, plug-ins• definition of user function interfaces (system functions)• graphical user interface design• design of description languages (initialization)• theoretical computer science
– algorithms and data structures• standards
– C++, TCP/IP, POSIX, OpenGL, GTK+, Web-programming• portability (Linux, Solaris, FreeBSD, Windows)
![Page 78: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/78.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing• software architecture
– transitions, machines, proxies, plug-ins• definition of user function interfaces (system functions)• graphical user interface design• design of description languages (initialization)• theoretical computer science
– algorithms and data structures• standards
– C++, TCP/IP, POSIX, OpenGL, GTK+, Web-programming• portability (Linux, Solaris, FreeBSD, Windows)• software engineering (project support: CVS, Doxygen)
![Page 79: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/79.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing• software architecture
– transitions, machines, proxies, plug-ins• definition of user function interfaces (system functions)• graphical user interface design• design of description languages (initialization)• theoretical computer science
– algorithms and data structures• standards
– C++, TCP/IP, POSIX, OpenGL, GTK+, Web-programming• portability (Linux, Solaris, FreeBSD, Windows)• software engineering (project support: CVS, Doxygen)• enhanced build mechanism (GNU autotools)
![Page 80: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/80.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 39
Development of a simulation and analysis tool Summary
Required knowledge and experience?Involved areas of computer science?
• nonlinear dynamics• numerics• scientific computing• software architecture
– transitions, machines, proxies, plug-ins• definition of user function interfaces (system functions)• graphical user interface design• design of description languages (initialization)• theoretical computer science
– algorithms and data structures• standards
– C++, TCP/IP, POSIX, OpenGL, GTK+, Web-programming• portability (Linux, Solaris, FreeBSD, Windows)• software engineering (project support: CVS, Doxygen)• enhanced build mechanism (GNU autotools)• ...
![Page 81: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/81.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 40
The next steps. . . Outlook
7. Outlook
I Extension of integration methods:
– usage of third party ODE and DDE integrators– implementation of symplectic integrators
![Page 82: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/82.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 40
The next steps. . . Outlook
7. Outlook
I Extension of integration methods:
– usage of third party ODE and DDE integrators– implementation of symplectic integrators
I Extension of PDE solvers
– Implementation of 2D–PDEs– Implementation of adaptive grid methods for PDEs
![Page 83: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/83.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 40
The next steps. . . Outlook
7. Outlook
I Extension of integration methods:
– usage of third party ODE and DDE integrators– implementation of symplectic integrators
I Extension of PDE solvers
– Implementation of 2D–PDEs– Implementation of adaptive grid methods for PDEs
I Extension of investigation methods:
– continuation method– local divergence rates– . . .
![Page 84: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/84.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 40
The next steps. . . Outlook
7. Outlook
I Extension of integration methods:
– usage of third party ODE and DDE integrators– implementation of symplectic integrators
I Extension of PDE solvers
– Implementation of 2D–PDEs– Implementation of adaptive grid methods for PDEs
I Extension of investigation methods:
– continuation method– local divergence rates– . . .
I Improvement of the visualization
– ’attractor flight’– more sophisticated coloring schemes
![Page 85: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/85.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 40
The next steps. . . Outlook
7. Outlook
I Extension of integration methods:
– usage of third party ODE and DDE integrators– implementation of symplectic integrators
I Extension of PDE solvers
– Implementation of 2D–PDEs– Implementation of adaptive grid methods for PDEs
I Extension of investigation methods:
– continuation method– local divergence rates– . . .
I Improvement of the visualization
– ’attractor flight’– more sophisticated coloring schemes
I Implementation of new system classes
– DAEs, ImDEs, IDEs, PIDEs and PDDEs
![Page 86: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/86.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 41
For more information see
www.AnT4669.de
![Page 87: a tool for simulating and investigating dynamical systems ...jancely/NLF/Texty/Ant/IPVS.pdf · Dynamical systems Investigation methods Scans AnT 4.669 features Summary Outlook Page](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec51506abc034532473ddaf/html5/thumbnails/87.jpg)
University of Stuttgart
Institute of Paralell andDistributed Systems
Department ofImage Understanding
www.AnT4669.de
Introduction
Dynamical systems
Investigation methods
Scans
AnT 4.669 features
Summary
Outlook
Page 42
System function implementation
Example of a system function for an ODE
#define sigma parameters[0] #define X currentState[0]#define r parameters[1] #define Y currentState[1]#define b parameters[2] #define Z currentState[2]
bool lorenz63(const Array<real_t>& currentState ,
const Array<real_t>& parameters ,Array<real_t>& rhs )
{
x = f(x, p)
rhs[0] = sigma * (Y - X);
rhs[1] = X * (r - Z) - Y;
rhs[2] = - b * Z + X * Y;
x = σ(y − x)y = rx− y − xz
z = −bz + xy
return true;}
extern "C"{ void connectSystem ()
{ ODE_Proxy::systemFunction = lorenz63 ; }}