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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Rigorous Computational Simulation of
Dynamical Systems
Walid Gomaa
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
1 Outline
2 Introduction
3 Chaos
4 Disaster Arising from Non-Rigorous Simulation
5 Models of Continuous Computation
6 Applications of Dynamical Systems
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
1 Outline
2 Introduction
3 Chaos
4 Disaster Arising from Non-Rigorous Simulation
5 Models of Continuous Computation
6 Applications of Dynamical Systems
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Dynamical Systems I
Representing the time evolution of any physical or
engineered system
Examples: digital computer, weather system, pendulum
movement, rocket motion, ball bouncing, solar system,
traffic,
stock market
Two entities: state and time
Dynamics can be determined by, e.g., system of differential
equations, recurrence equations
Walid Gomaa Rigorous Computational Simulation of Dynamical
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Dynamical Systems II
Four types of dynamical systems:
discrete-time discrete-space
discrete-time continuous-space
continuous-time discrete-space
continuous-time continuous-space
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
O l
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Discrete vs. Continuous Computation I
Traditional discrete computing: domain of computable spacecan
either be taken to be either of:
the natural numbers: N = {01 }
the set of finite strings over some alphabet (e.g., =
{01
})
Models of computation: Turing machine; well-founded theory
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
O tli
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Discrete vs. Continuous Computation II
In continuous computation an object in the computation
space does not have a finite representation
Examples: real numbers (2 ), the complex numbers, the
class of continuous functions over R
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
O tli
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
1 Outline
2 Introduction
3 Chaos
4 Disaster Arising from Non-Rigorous Simulation
5 Models of Continuous Computation
6 Applications of Dynamical Systems
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
Outline
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of Dynamical Systems
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Chaos
Figure : Blue: initial point, Red: equilibrium point
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
Outline
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Chaos
Figure : Although the exact physics may be known, tiny
errorscompound and trajectories that start similarly end
differently. Chaos!
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
Outline
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Logistic Map I
xn+1 = rxn(1 xn) (1)
0 xn 1 and parameter 0 < r 4
Discrete-time continuous space system
Typically used to model population growth
Linear term: population increases
Quadratic term: population decreases
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
Outline
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OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Logistic Map II
Figure : 1 < r < 2
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
Outline
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IntroductionChaos
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Logistic Map III
Figure : 1 < r < 2
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
Outline
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IntroductionChaos
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Logistic Map IV
Fixed points: xn+1 = F(xn) = xn
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
Outline
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IntroductionChaos
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Logistic Map V
0 < r < 1 : xn 0 regardless ofx0
1 < r < 3 : xn converges to one pointr1
r , regardless ofx0
3 < r < 357 : xn oscillates between several values;
samevalues regardless ofx0
357 < r : chaotic regime
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineI d i
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IntroductionChaos
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Logistic Map VI
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineI t d ti
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IntroductionChaos
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Lorenz Attractor I
Lorenz equations: mathematical model for atmospheric
convection
Thermal convection: hot air rises and cold air sinks
dx
dt= (y x)
dy
dt= x y xz
dz
dt= xy z
(2)
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
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IntroductionChaos
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Lorenz Attractor II
x: convective overturning on the plane, y: horizontal
temperature
variation, z: horizontal temperature variation
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
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IntroductionChaos
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Lorenz Attractor III
dx
dt= (y x)
dy
dt = x y xzdz
dt= xy z
(3)
: model parameters (physical quantities)
Best representation of earths atmosphere (Lorenz): = 10, = 28, =
8
3and initial condition: (x0 y0 z0) = (01 0)
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
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IntroductionChaos
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Lorenz Attractor IV
Chaotic behavior for some parameter values
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
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of Dynamical Systems
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IntroductionChaos
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Lorenz Attractor V
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
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of Dynamical Systems
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ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
1 Outline
2 Introduction
3 Chaos
4 Disaster Arising from Non-Rigorous Simulation
5
Models of Continuous Computation
6 Applications of Dynamical Systems
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
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of Dynamical Systems
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ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Disasters IPatriot missile failure
The Gulf War in 1991, on February 25th.
Patriot missile failed to intercept Iraqi Scud missile.
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
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of Dynamical Systems
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ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Disasters IIPatriot missile failure
Killig of 28 soldiers and injuring around 100 other people.
24-bit fixed point register
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
C
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ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Disasters IIIPatriot missile failure
1
10 represented in binary multiplied by a large number
An error of034 seconds enough for scud to travel half a
kilometer and so outside the range of Patriot
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
Ch
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ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Disasters IExplosion of Ariane 5
On June 4th 1996, an unmanned Ariane 5 rocket launched by
the ESA
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
Ch
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ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Disasters IIExplosion of Ariane 5
Explosion after 40 seconds
Cost: (1) rocket and cargo: $500M and (2) development:
$7Billion
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
Chaos
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ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Disasters IIIExplosion of Ariane 5
64-bit floating point number representing the horizontalvelocity
converted to 16-bit integer
Result larger than 32767, so conversion failed
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
Chaos
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of Dynamical Systems
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ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
1 Outline
2 Introduction
3 Chaos
4 Disaster Arising from Non-Rigorous Simulation
5
Models of Continuous Computation
6 Applications of Dynamical Systems
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
Chaos
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of Dynamical Systems
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ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Discrete vs. Continuous Computability
Unlike the discrete setting: no equivalence of theChurch-Turing
thesis exists
Several different kinds of models to define the notions of
computability, complexity, and numerical algorithms
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
Chaos
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ChaosDisaster Arising from Non-Rigorous Simulation
Models of Continuous ComputationApplications of Dynamical
Systems
Computable Analysis
The most physically realizable model
Extending standard Turing machine to either oracle TM or
Type II TM
Consider f: [0 1] RGiven: increasingly accurate representation
ofx
Output: increasingly accurate representation ofy = f(x)
More formally:
Given: r Q s.t. |r x| < Output: s Q s.t. |s f(x)| <
Complexity theory is extension of discrete complexity (time
and space resources)
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
Chaos
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Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Computable Analysis
The most physically realizable model
Extending standard Turing machine to either oracle TM or
Type II TM
Consider f: [0 1] RGiven: increasingly accurate representation
ofx
Output: increasingly accurate representation ofy = f(x)
More formally:
Given: r Q s.t. |r x| < Output: s Q s.t. |s f(x)| <
Complexity theory is extension of discrete complexity (time
and space resources)
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
Chaos
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of Dynamical Systems
33/44
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Computable Analysis
The most physically realizable model
Extending standard Turing machine to either oracle TM or
Type II TM
Consider f: [0 1] RGiven: increasingly accurate representation
ofx
Output: increasingly accurate representation ofy = f(x)
More formally:
Given: r Q s.t. |r x| < Output: s Q s.t. |s f(x)| <
Complexity theory is extension of discrete complexity (time
and space resources)
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
Chaos
http://find/
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of Dynamical Systems
34/44
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Computable Analysis
The most physically realizable model
Extending standard Turing machine to either oracle TM or
Type II TM
Consider f: [0 1] RGiven: increasingly accurate representation
ofx
Output: increasingly accurate representation ofy = f(x)
More formally:
Given: r Q s.t. |r x| < Output: s Q s.t. |s f(x)| <
Complexity theory is extension of discrete complexity (time
and space resources)
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
ChaosDi A i i f N Ri Si l i
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of Dynamical Systems
35/44
Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Computable Analysis
The most physically realizable model
Extending standard Turing machine to either oracle TM or
Type II TM
Consider f: [0 1] RGiven: increasingly accurate representation
ofx
Output: increasingly accurate representation ofy = f(x)
More formally:
Given: r Q s.t. |r x| < Output: s Q s.t. |s f(x)| <
Complexity theory is extension of discrete complexity (time
and space resources)
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
ChaosDi t A i i f N Ri Si l ti
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of Dynamical Systems
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Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Algebraic Models
A real number is represented as atomic entity (an alphabet
letter)
Started with the BSS (Blum-Shub-Smale) model in 1989
Good for studying algebraic complexity rather than machine
complexity
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
ChaosDisaster Arising from Non Rigorous Simulation
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of Dynamical Systems
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Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Algebraic Models
A real number is represented as atomic entity (an alphabet
letter)
Started with the BSS (Blum-Shub-Smale) model in 1989
Good for studying algebraic complexity rather than machine
complexity
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
ChaosDisaster Arising from Non Rigorous Simulation
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of Dynamical Systems
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Disaster Arising from Non-Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
Algebraic Models
A real number is represented as atomic entity (an alphabet
letter)
Started with the BSS (Blum-Shub-Smale) model in 1989
Good for studying algebraic complexity rather than machine
complexity
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
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Disaster Arising from Non Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
II. Analog Computation I
GPAC: General Purpose Analog Computer
Introduced by C. Shannon in 1941 as a mathematical model
of the differential analyzer
DA used from 1930s to 1960s to solve differential equations,
e.g., in ballistics problems
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroduction
ChaosDisaster Arising from Non-Rigorous Simulation
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Disaster Arising from Non Rigorous SimulationModels of
Continuous ComputationApplications of Dynamical Systems
II. Analog Computation II
Components of GPAC:
Walid Gomaa Rigorous Computational Simulation of Dynamical
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OutlineIntroductionChaos
Disaster Arising from Non-Rigorous Simulation
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g gModels of Continuous ComputationApplications of Dynamical
Systems
II. Analog Computation III
Example of GPAC:
Figure : Generating sin and cos via a GPAC
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroductionChaos
Disaster Arising from Non-Rigorous Simulation
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g gModels of Continuous ComputationApplications of Dynamical
Systems
1 Outline
2 Introduction
3 Chaos
4 Disaster Arising from Non-Rigorous Simulation
5 Models of Continuous Computation
6 Applications of Dynamical Systems
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroductionChaos
Disaster Arising from Non-Rigorous Simulation
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Models of Continuous ComputationApplications of Dynamical
Systems
Reachability Problems
Given a set of initial states X0
Apply dynamics rule
Set of reachable states as time goes to infinity
Can be used for verification of certain properties of programsor
design of control systems
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
OutlineIntroductionChaos
Disaster Arising from Non-Rigorous Simulationf C C
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Models of Continuous ComputationApplications of Dynamical
Systems
Walid Gomaa Rigorous Computational Simulation of Dynamical
Systems
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