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MAT332
Introduction to Nonlinear Dynamical Systems and Chaos
Lecture 1
Qun WANG
University of Toronto Mississauga
09-09-2021
(UTM)
MAT332 Intro Nonlinear Dyn and Chao
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Outline
1 Brief History of Nonlinear Dynamical Systems
2 First Encounter of Nonlinear Dynamical Systems
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From an Old Movie to an Older Theory
Figure: A Famous Movie Figure:
Have you seen this movie?
Who is this character (played by Jeff) in the movie?
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From an Old Movie to an Older Theory
Let’s listen to Jeff’s Explanation for Chaos Theory:
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From an Old Movie to an Older Theory
A wide range of evolution processes share the following properties: 1 deterministic
2 finite dimensional 3 differentiable
From your experience, which of the evolution processes below satisfies all these three properties?
Figure: Free fall object Figure: Heat propagation
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From an Old Movie to an Older Theory
The determinism has prevailed for more than 200 years since
deterministic
finite dimensional ⇒ Ordinary Differential Equations ⇒ No Free Will differentiable
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The 3 Body Problem
The universal gravity law of Newton
F = G m1m2 , (1)
r2
The 3 body problem describes the interaction of 3 point masses under
Newton’s universal gravity law.
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The 3 Body Problem
During a long period, the effort is put on the investigation of explicit formula of special solutions.
Euler: collinear periodic orbits of the 3BP (1767);
Lagrange: equilateral triangle periodic orbits of the 3BP (1772).
However the closed form solution does not exist in general for the 3 body problem.
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The Prize, the mistake, and the new discovery
Poincar ́e’s prize winning paper on restricted 3-body problem motivates many ideas in modern dynamical theory, including
The stable theory: Theorem of Kolmogorov-Arnold-Moser
The chaotic theory: Theorem of Birkhoff-Smale
Since then, the study of dynamical system has switched from
quantitative aspects to qualitative aspects.
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Fortunately and Unfortunately
Forunately: in the beginging of the 20th century, the advent of quantum mechanics has substantially deluded the belief of determinism since Newton’s era.
Unfortunately: this revolution has put the great discovery of Poincar ́e in shadow.
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Revival of the Great Idea
In the 1960s, the fast development of electronic computer has made the large scale iteration possible.
Ueda: Observed the random transitive phenomena
Lorentz: Shows that long term weather prediction does not exists.
: Connects the Chaos with Fractals.
Feigenbaum (1975) and Coullet & Tresser (1978): Discovered the
universality in chaos
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On the beach of Rio
“…I would spend my mornings on that wide beautiful sandy beach swimming and body surfing. Also I took a pen and paper and would work on mathematics.”
–
“…This blythe spirit leads mathematicians to seriously propose that the common man who pays the taxes ought to feel that mathematical creation should b e supported with public funds on the beaches of Rio”
–
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On the beach of Rio
The Smale horseshoe reveals the mechanism of chaos from a geometric way, becomes the essential phenomenon in the search of chao.
Figure: Horseshoe Map Figure: Intersection “Thus the mathematics created on the beach of Rio was the horse-
shoe and the higher dimensional Poincar ́e’s conjecture”
–
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Basic Concepts in Dynamical Systems Definition (Phase Space)
In the above system, the vector x = (x1, x2, …xn) ∈ Rn is called a state of the system. The set of all possible states is called the phase space.
The ordinary differential equation is a natural tool to describe the evolution of , , processes. More precisely, we are interested in studying the following system
x ̇1(t) = f1(t, x1(t), x2(t), …, xn(t)); x ̇2(t) = f2(t, x1(t), x2(t), …, xn(t)); x ̇3(t) = f3(t, x1(t), x2(t), …, xn(t));
.
x ̇n(t) = fn(t, x1(t), x2(t), …, xn(t)).
One can consider the right-hand side as a ”dynamo”, providing
velocity for the evolution of x.
(2)
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Basic Concepts in Dynamical Systems
Let the vector x = (x1, x2, …xn) ∈ Rn stand for a vector in Rn, t is a real number, and f = (f1, f2, .., fn), then the above system can be abbreviated as
x ̇(t) = f(t,x(t)) (3)
INPUT
t: instantaneous time
x(t): state at time t
OUTPUT
x ̇(t): instantaneous velocity
In this way one can interpret the system as governing the evolution of the process.
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Basic Concepts in Dynamical Systems
Next let’s discuss the definition of a vector field in an intuitive way
Definition (vector field)
A vector field is an assignment of a vector v(x) for each point x in the phase space, such that v(x) is a differentiable map of x.
Remark: Essentially the vector field consists of two parts: The point x
The vector v(x)
Warning: A vector field may vary along with time!
Example
The equation
x ̇(t)=t x(r) |x(r )|
defined on the phase space R2 {0} depends on explicit on t (UTM) MAT332 Intro Nonlinear Dyn and Chao
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Autonomous Systems vs. Non-Autonomous Systems Definition (Autonomous System)
Consider the dynamical system x ̇(t) = f(t,x(t)). If f does not depend on t explicitly, then the system is called autonomous, otherwise the system is non-autonomous.
More precisely, an autonomous system is of the following form:
x ̇1(t) = f1(x1(t), x2(t), …, xn(t)); x ̇2(t) = f2(x1(t), x2(t), …, xn(t)); x ̇3(t) = f3(x1(t), x2(t), …, xn(t));
.
x ̇n(t) = fn(x1(t), x2(t), …, xn(t)).
Autonomous System implies that the physical law that the system follow does not depend on time.
(4)
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Linear Systems vs. Nonlinear System Systems Definition (Linear System)
Consider the dynamical system x ̇(t) = f(x(t)). If f consists of only linear functions, then the system is called linear, otherwise the system is nonlinear. Let A be its (constant) coefficient matrix, the system can be denoted by
Example
x ̇(t) = Ax(t)
(5)
For example, let’s consider for n = 2 the following system x ̇(t)=2x +3x
This is a linear system. What is its coefficient matrix?
1 1√2 x ̇(t)=x+ 2x
212
(6)
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Harmonic Oscillator
Example (The Harmonic Oscillation)
Consider the equation x ̈(t) = −x, x ∈ R. This equation is equivalent to the following system
x ̇ = y ;
y ̇ = −x. (7)
One can explicitly write down the solution as x (t ) = A cos(t + φ). From the explicit formule, one verifies easily that
The flow starting from (0, 0) will remain there forever.
The flow starting from any point (x0, y0) ̸= (0, 0) is 2π- periodic.
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Pendulum
Example (The pendulum)
Consider the equation x ̈(t) = −sinx, x ∈ R. This equation is equivalent to the following system
x ̇ = y ;
y ̇ = −sinx. (8)
Can you still solve the equation explicitly?
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A Geometric Approach
We can instead draw the vector field in the phase space, in order to understand some of its qualitative dynamical behaviours.
Question: Identify the fixed points, the periodic orbits, and the homoclinic orbit for the pendulum according to the picture.
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Double Pendulum
Now let’s amuse ourselves by watching behaviours of the double pendulum, and feel the meaning of ”sensitive to the initial condition”…
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Lessons From the above Examples
Hopefully we agree on the following (not proved yet) conclusions:
1 Two sources for the complexity of the system:
The nonlinearity in the dynamics (from oscilltor to pendulum);
The degree of freedom (from pendulum to double pendulum).
2 For nonlinear systems, we can still use geometric approach to study them (draw the vector field)
3 To start with the geometric approach, some typical orbits are essential (fixed points, periodic orbits, homoclinic/heteoclinic orbits)
4 To go further, we need to understand the behavior of the system near these special orbits, and we may want to approximate the nonlinear system by a linear system (when x is near 0, sin(x) ∼ x ).
A long yet exciting journey to go for the rest of the semester!
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