Mam1043H
2021
, Non-Linear
Dynamics
Introduction
1. Understanding what a dynamical system is.
2. Knowing how to write the maths of a dynamical system.
3. Knowing how to understand the interactions of dynamical
systems through quantitative and qualitative means.
Exponential Growth
𝑑𝑑𝑑𝑑 (𝑡𝑡)
= 𝑘𝑘𝑘𝑘(𝑡𝑡)
𝑑𝑑𝑑𝑑
o The rate of change of population (𝑃𝑃) with time is
proportional to the population that you have at that time.
Logistic Equation
𝑑𝑑𝑑𝑑(𝑡𝑡 ) 𝑃𝑃(𝑡𝑡)
= 𝑘𝑘𝑘𝑘(𝑡𝑡) �1 − �
𝑑𝑑𝑑𝑑 𝑀𝑀
o 𝑀𝑀 is a carrying capacity, related to the max. size of
population that the environment can sustain.
Pendulum
𝑑𝑑 2 𝜃𝜃(𝑡𝑡) 𝑔𝑔
= − sin (𝜃𝜃(𝑡𝑡))
𝑑𝑑𝑡𝑡 2 𝑙𝑙
o Relates the acceleration of the angle to the angle itself.
o The rate of change of angular velocity related to how far
from the lowest position the pendulum is.
, (A system is a group of interacting/interrelated elements that
act according to a set of rules to form a unified whole.)
Dynamical Systems are systems of interacting elements
which are described by differential equations.
All about how something changes in time.
All about the interaction of something, either with itself
or in some environment.
History of Dynamics
[Dynamical Systems can be written in the form of something
called a Difference Equation…
Has no derivatives.
Can be thought of as taking discrete time-steps.
Jumps by a single time interval, rather than moving along
continuously as it does in a differential equation.]
1. Gravity, and Newton’s and Kepler’s Laws.
2. Chinese Astronomy, the planet’s movements.
Two-Body Problem of Celestial Mechanics → Can’t be solved.
What do we mean by solved?
Write down a set of equations that encodes the
dynamics of this system, which takes the form of
differential equations.
Then, find solutions to these equations, which will
correspond to function(s) of time which satisfy the
equation(s).
Generally, when we say that someone “solved” a dynamical
system, we mean that they found all of the solutions.
2021
, Non-Linear
Dynamics
Introduction
1. Understanding what a dynamical system is.
2. Knowing how to write the maths of a dynamical system.
3. Knowing how to understand the interactions of dynamical
systems through quantitative and qualitative means.
Exponential Growth
𝑑𝑑𝑑𝑑 (𝑡𝑡)
= 𝑘𝑘𝑘𝑘(𝑡𝑡)
𝑑𝑑𝑑𝑑
o The rate of change of population (𝑃𝑃) with time is
proportional to the population that you have at that time.
Logistic Equation
𝑑𝑑𝑑𝑑(𝑡𝑡 ) 𝑃𝑃(𝑡𝑡)
= 𝑘𝑘𝑘𝑘(𝑡𝑡) �1 − �
𝑑𝑑𝑑𝑑 𝑀𝑀
o 𝑀𝑀 is a carrying capacity, related to the max. size of
population that the environment can sustain.
Pendulum
𝑑𝑑 2 𝜃𝜃(𝑡𝑡) 𝑔𝑔
= − sin (𝜃𝜃(𝑡𝑡))
𝑑𝑑𝑡𝑡 2 𝑙𝑙
o Relates the acceleration of the angle to the angle itself.
o The rate of change of angular velocity related to how far
from the lowest position the pendulum is.
, (A system is a group of interacting/interrelated elements that
act according to a set of rules to form a unified whole.)
Dynamical Systems are systems of interacting elements
which are described by differential equations.
All about how something changes in time.
All about the interaction of something, either with itself
or in some environment.
History of Dynamics
[Dynamical Systems can be written in the form of something
called a Difference Equation…
Has no derivatives.
Can be thought of as taking discrete time-steps.
Jumps by a single time interval, rather than moving along
continuously as it does in a differential equation.]
1. Gravity, and Newton’s and Kepler’s Laws.
2. Chinese Astronomy, the planet’s movements.
Two-Body Problem of Celestial Mechanics → Can’t be solved.
What do we mean by solved?
Write down a set of equations that encodes the
dynamics of this system, which takes the form of
differential equations.
Then, find solutions to these equations, which will
correspond to function(s) of time which satisfy the
equation(s).
Generally, when we say that someone “solved” a dynamical
system, we mean that they found all of the solutions.
