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MAT332 Lecture Notes
Introduction to Nonlinear Dynamics and Chaos
Author: Qun WANG Institute: University of Toronto
Date: Sep-Dec 2021
Material for MAT332, 2020-2021 Fall semester
Chapter1 Introduction 1.1 BasicConcepts
Objectives
DifferentiableDynamicalSystems; Linear/Non-linearsystems;
QualitativeAnalysis;
1.1.1 PhaseSpace,VectorField,andFlow
PhaseSpace;
VectorfieldandFlow; Oscillator/Pendulum.
In many fields of science, one is required to study some evolutionary processes, i.e., phenomena that change with a parameter (often considered as time). Some of these processes have the following properties:
Deterministic : Reversible ; Differentiable :
The ordinary differential equations(ODEs) are natural tools to describe the evolution of such processes. More precisely, we are interested in studying the following first order ODE 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)). First, let’s define our battlefield in which lives the evolution process:
Definition 1.1. 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.
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)) (1.2)
One can consider the right-hand side as a “dynamo”, which takes the variable t (“instantaneous time”), and the vector x(t) (“state at time t”) as input, and in turn generates x ̇ (t) (“instantaneous
(1.1)
♣
1.1 Basic Concepts – 2 –
velocity”) as an output. In this way one can interpret the system as governing the evolution of the process. Given a dynamical system x ̇ (t) = f (t, x(t)), we consider this as giving a vector field on the phase space.
Definition 1.2. Vector Field
A vector field is an assignment of a vector v(t,x) for each t ∈ R,x ∈ U ⊂ R2n, such that v(x) is a differentiable map of x. It is denoted by V (x) = (x, v(t, x)) ∈ R2n.
Remark Essentially the vector field consists of two parts: The point x
The vector v(t, x)
By considering the integral curve along the vector field, we get the flow.
Definition 1.3. Flow
(t, x0) −→ x(t) the flow of the system, denoted as φt(x0).
♣
Remark Recall that the Lipschitz constant and maximal interval in which the flow exists depends on x0. In other words, for any given initial state x0 ∈ U, the flow is local in general, as it is only locally defined for t ∈ (−c, c). If the flow turns out to exist for any t ∈ R, we call such flow global.
The flow has an important property, namely it is a semi group.
Proposition 1.1. Semi Group Property
∀s1,s2 ∈Rand∀x∈U,onehasthat
1. φ0(x)=x;
2. φs2(φs1(x))=φs1(φs2(x))=φs1+s2(x),aslongastheflowiswell-defined.
♣
Let x0 ∈ U be the initial state and x(t) solves the system with initial condition x(0) = x0. We call the map
(1.3)
φ:R×U→U φ
1.1.2 Autonomous System vs. Non-Autonomous System
Definition 1.4. 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.
1.2 Linear System v.s. Nonlinear System: First Encounter
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)).
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Remark From now on, unless otherwise stated, we will always assume that we are dealing with autonomous systems. Such a restriction does not lose any generality, because one can easily switch from a non-autonomous system to an autonomous system, at the cost of introducing an extra variable (justify it.). Our preference of the autonomous system results from the fact that the vector field for an autonomous system does not vary along time, and is uniquely determined by its state.
Example1.1 Mechanical System Recall that in physics, Newton’s principle of determinacy implies that all motions of a system are uniquely determined by their initial positions and initial velocities. More precisely speaking, consider the motion of an object in Rd. The Newton’s equation expresses the acceleration as
x ̈(t) = F (t, x(t), x ̇ (t)) (1.5)
where F : R × Rd × Rd → Rd is interpreted as the force. Now this is a second order ordinary differential equation system. We can introduce an extra variable y = x ̇ (after all, in this setting the velocity and position are independent), which turns the Newton’s equation into the following form:
x ̇ = y
y ̇ =F(t,x(t),y(t)) (1.6)
As a consequence it becomes a dynamical system in the form of equations 1.1. If F does not depend on t explicitly, this is an autonomous system. Otherwise it is a non-autonomous system. Remark Suppose that F is smooth. The phase space of a mechanical system in Rd is R2d, rather then Rd. Because the phase space consists of position (x ∈ Rd) and velocity y ∈ Rd.
1.2 Linear System v.s. Nonlinear System: First Encounter
Definition 1.5. 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
x ̇ (t) = Ax(t) (1.7) ♣
(1.4)
1.2 Linear System v.s. Nonlinear System: First Encounter
Example1.2 For example, let’s consider for n = 2 the following system x ̇1(t) = 2×1 + 3×2
x ̇2(t) = x1 + √2×2 This is a linear system, with coefficient matrix
⎡⎣2 3⎤⎦ A= 1 √2
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Example1.3 Harmonic Oscillation Consider the harmonic oscillator. It experiences a restoring force F proportional to x, where x is the displacement from its equilibrium position. For instance, When a spring is stretched or compressed by a mass, Hooke’s law gives the relationship of the restoring force exerted by the spring when the spring is compressed (x > 0) or stretched (x < 0) a certain length.
Figure 1.1: Harmonic Oscillator
Consider the equation x ̈(t) = −x, x ∈ R. According to the remark, this equation is
equivalent to the following system
x ̇ = y ;
y ̇ = −x. (1.9)
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 periodic.
Without solving the equation explicitly, we can also draw the vector field in the phase
space.
Example1.4 Pendulum A simple pendulum is a weight suspended from a pivot through a
Figure 1.2: Pendulum
(1.8)
1.2 Linear System v.s. Nonlinear System: First Encounter – 5 –
Figure 1.3: vector fields of two systems
massless cord, so that it can swing freely. The restoring force is due to gravity and the motion
̈
is free from friction or air drag. Consider the equation θ(t) = − sin θ, θ ∈ R, where θ is angle
displaced from the vertical line. This equation is equivalent to the following system
θ ̇ = p;
p ̇ = − sin θ. (1.10)
This time it is much harder to solve the equation explicitly. But still we can instead draw the vector field in the phase space, in order to understand some of its dynamical behaviours.
Let’s compare the phase flows of these two systems.
The simple pendulum system has two fixed points (the green point and the red point), flow starting from these two points will remain there forever;
The flows starting from any point near the green point will be periodic orbits, and are quite similar to those of the harmonic oscillator;
There is a flow connecting the red point to itself with infinite time, namely the homoclinic
orbit.
Example1.5 Double Pendulum Now consider a double pendulum, which is a pendulum with
Figure 1.4: Double Pendulum
another pendulum attached to its end. By choosing an appropriate coordinate systems and after
1.2 Linear System v.s. Nonlinear System: First Encounter – 6 – some simplification, one can express the equations as the following (ignore the detail for the
moment...)
θ ̇1 =62p1 −3cos(θ1 −θ2)p2 16−9cos2(θ1 −θ2)
θ ̇2 =68p2 −3cos(θ1 −θ2)p1 16−9cos2(θ1 −θ2)
p ̇1 − 12 θ ̇1θ ̇2 sin(θ1 − θ2) + 3g sin θ1
p ̇ 2 = − 12 − θ ̇ 1 θ ̇ 2 s i n ( θ 1 − θ 2 ) + g s i n θ 2 .
As we see, the dimension of phase space is now 4, due to the supplementary joint which makes this system even more complicated than it looks, The sensitive dependence on the initial conditions can be shown by the following video. ( https://youtu.be/d0Z8wLLPNE0)
Summary Putting the technical details aside, the above examples at least give us some intuitions and general ideas for the rest of the course:
1.
The complexity of a dynamical system might come from two aspects:
1. A nonlinear system will in general be more difficult than a linear system. (oscillator vs. pendulum)
2. A system with more degree of freedom is in general more difficult than a system with fewer degree of freedom (pendulum vs. double pendulum)
When given a complex dynamical system, it is in general not a good idea to look for explicit solutions. Somehow, geometric approach still permits us to carry out qualitative analysis. In a system, some orbits of typical dynamical properties are important to understand the systems. For instance, in the example of the pendulum, we have identified:
fixed points: Points starting from where the system will stay at rest.
periodic orbits: Orbits that will come back to the initial points after certain (finite) time.
homoclinic/heterclinic orbits: Orbits that connect a fixed point to itself/ another fixed points in infinite time.
, to name but a few.
Near the fixed points, one can approximate system by using a simpler system (i.e., the linearised system), which could provide insights into the behaviors of the system near the fixed points. (What is sin θ when θ ∼ 0 ?)
K Chapter 1 Exercise k
(a). Consider a point moving on the plane under the following law: x ̈ = | x | 2 x
, where |x| represents the distance from x to the origin. What is the phase space? Is it autonomous or non-autonomous? Is it linear or nonlinear? What if the point is
1.2 Linear System v.s. Nonlinear System: First Encounter
moving in the 3-dimensional space?
(b). Consider a point moving on the plane under the following law:
x ̈ = 1 x |x|2
what is the phase space?
2. Recall the Taylor expansion of sin x at x = 0. From there convince yourself that near
θ = 0 the harmonic oscillator can be seen as an approximation of the pendulum.
3. Prove that (θ1, θ2, p1, p2) = (0, 0, 0, 0) is a fixed point of the double pendulum system.
Can you find another one? What do these fixed point look like in reality?
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