Welcome to a new section of Nonlinear Dynamics: Bifurcations! Bifurcations are points where a dynamical system (e.g. differential equation) undergoes a significant change in its dynamical behaviour when a certain parameter in the differential equation crosses a critical value.
In this video, I explain saddle node bifurcations. These are bifurcations in which varying a parameter causes the appearance of a half-stable fixed point, followed by two fixed points from nothing. I discuss bifurcation diagrams, bifurcation points, and describe the concept of normal forms.
Questions/requests? Let me know in the comments!
Pre-reqs: The videos before this one on this playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9C8iPDD5xW0jT-c3dtP4TR5
Lecture Notes: https://drive.google.com/open?id=1mt_5XJqUB6wtST-J0KBlRhSJY5v7lM7q
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- James Mark Wilson
- Yuan Gao
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair

Views: 3983
Faculty of Khan

Topics in Nonlinear Dynamics by Prof. V. Balakrishnan,Department of Physics,IIT Madras.For more details on NPTEL visit http://nptel.ac.in

Views: 5167
nptelhrd

Introduction to Dynamical Models in Biology: Module 1, Week 3

Views: 2053
NOC17 JAN-FEB BT05

See the full course: https://systemsacademy.io/courses/nonlinear-systems/
Twitter: http://bit.ly/2HobMld
A phase transition is the transformation of a system from one state to another through a period of rapid change. The classical example of this is the transition between solid, liquid and gaseous states that water passes through given some change in temperature, phase transitions are another hallmark of nonlinear systems. In this module we discuss the concept in tandem with its counterpart bifurcation theory.
Transcription excerpt:
Bifurcations & Phase transitions
As we have previously discussed the qualitative dynamic behavior of nonlinear systems is largely defined by the positive and negative feedback loops that regulate their development, with negative feedback working to dampen down or constrain change to a linear progression, while positive feedback works to amplify change typically in an super-linear fashion.
As opposed to negative feedback where we get a gradual and often stable development over a prolonged period of time, what we might call a normal or equilibrium state of development, positive feedback is characteristic of a system in a state of nonequilibrium. Positive feedback development is fundamentally unsustainable because all systems in reality exist in an environment that will ultimately place a limit on this grown.
From this we can see how the exponential grow enabled by positive feedback loops is what we might say special, it can only exist for a relatively brief period of time, when we look around us we see the vast majority of things are in a stable configuration constrained by some negative feedback loop whether this is the law of gravity, predator prey dynamics or the economic laws of having to get out of bed and go to work every day. These special periods of positive feedback development are characteristic and a key diver of what we call phase transitions.
A phase transition may be defined as some smooth, small change in a quantitative input variable that results in a qualitative change in the system’s state. The transition of ice to steam is one example of a phase transition. At some critical temperature a small change in the systems input temperature value results in a systemic change in the substance after which it is governed by a new set of parameters and properties, for example we can talk about cracking ice but not water, or we can talk about the viscosity of a liquid but not a gas as these are in different phases under different physical regimes and thus we describe them with respect to different parameters.
Another example of a phase transition may be the changes within a colony of bacteria that when we change the heat and nutrient input to the system we change the local interactions between the bacteria and get a new emergent structure to the colony, although this change in input value may only be a linear progression it resulted in a qualitatively different pattern emerging on the macro level of the colony. It is not simply that a new order or structure has emerged but the actual rules that govern the system change and thus we use the word regime and talk about it as a regime shift, as some small changes in a parameter that affected the system on the local level leads to different emergent structures that then feedback to define a different regime that the elements now have to operate under.
Another way of talking about this is in the language of bifurcation theory, whereas with phase transitions we are talking about qualitative changes in the properties of the system, bifurcation theory really talks about how a small change in parameter can causes a topological change in a system’s environment resulting in new attractor states emerging. A bifurcation means a branching, in this case we are talking about a point where the future trajectory of an element in the system divides or branches out, as new attractor states emerge, from this critical point it can go in two different trajectories which are the product of these attractors, each branch represents a trajectory into a new basin of attraction with a new regime and equilibrium.
Twitter: http://bit.ly/2TTjlDH
Facebook: http://bit.ly/2TXgrOo
LinkedIn: http://bit.ly/2TPqogN

Views: 13401
Systems Academy

Third Sperlonga Summer School on Mechanics and Engineering Sciences
Prof. Angelo Luongo (Università degli Studi dell’Aquila, M&MoCS, Italy)
Lecture on: “Multiple Scale Bifurcation Analysis for Discrete and Continuous Autonomous Dynamical Systems“

Views: 56
M&MoCS

UCI Math 113B: Intro to Mathematical Modeling in Biology (Fall 2014)
Lec 21. Intro to Mathematical Modeling in Biology: Hopf Bifurcations
View the complete course: http://ocw.uci.edu/courses/math_113b_intro_to_mathematical_modeling_in_biology.html
Instructor: German A. Enciso, Ph.D.
Textbook: Mathematical Models in Biology by Leah Edelstein-Keshet, SIAM, 2005
License: Creative Commons CC-BY-SA
Terms of Use: http://ocw.uci.edu/info
More courses at http://ocw.uci.edu
Description: UCI Math 113B is intended for both mathematics and biology undergrads with a basic mathematics background, and it consists of an introduction to modeling biological problems using continuous ODE methods (rather than discrete methods as used in 113A). We describe the basic qualitative behavior of dynamical systems in the context of a simple population model. As time allows, we will then discuss other types of models such as chemical reactions inside the cell, or excitable systems leading to oscillations and neuronal signals. The necessary linear algebra is also discussed to avoid including additional requirements for this course.
Recorded on February 28, 2014
Required attribution: Enciso, German A. Math 113B (UCI OpenCourseWare: University of California, Irvine), http://ocw.uci.edu/courses/math_113b_intro_to_mathematical_modeling_in_biology.html. [Access date]. License: Creative Commons Attribution-ShareAlike 3.0 United States License. (http://creativecommons.org/licenses/by-sa/3.0/deed.en_US)

Views: 12443
UCI Open

Lecture Series on Chaos, Fractals and Dynamical Systems by Prof.S.Banerjee,Department of Electrical Engineering, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in.

Views: 29083
nptelhrd

Multiphase flows:Analytical solutions and Stability Analysis by Prof. S.Pushpavanam,Department of Chemical Engineering,IIT Madras.For more details on NPTEL visit http://nptel.ac.in

Views: 4192
nptelhrd

These are videos from the Nonlinear Dynamics course offered on Complexity Explorer (complexity explorer.org) taught by Prof. Liz Bradley. These videos provide a broad introduction to the field of nonlinear dynamics, focusing both on the mathematics and the computational tools that are so important in the study of chaotic systems. The course is aimed at students who have had at least one semester of college-level calculus and physics, and who can program in at least one high-level language (C, Java, Matlab, R, ...).
After a quick overview of the field and its history, we review the basic background that students need in order to succeed in this course. We then dig deeper into the dynamics of maps—discrete-time dynamical systems—encountering and unpacking the notions of state space, trajectories, attractors and basins of attraction, stability and instability, bifurcations, and the Feigenbaum number. We then move to the study of flows, where we revisit many of the same notions in the context of continuous-time dynamical systems. Since chaotic systems cannot, by definition, be solved in closed form, we spend some time thinking about how to solve them numerically, and learning what challenges arise in that process. We then learn about techniques and tools for applying all of this theory to real-world data and close with a number of interesting applications: control of chaos, prediction of chaotic systems, chaos in the solar system, and uses of chaos in music and dance.
In each unit of this course, students will begin with paper-and-pencil exercises regarding the corresponding topics, and then write computer programs that operationalize the associated mathematical algorithms. This will not require expert programming skill, but you should be comfortable translating basic mathematical ideas into code. Any computer language that supports simple plotting—points on labelled axes—will suffice for these exercises. We will not ask you to turn in your code, but simply report and analyze the results that your code produces.

Views: 2286
Complexity Explorer

Mathematical modeling of physiological systems: Introduction to Dynamical Systems
Part 6: Bifurcations of fixed points

Views: 188
nbjanson

See the full course: https://systemsacademy.io/courses/nonlinear-systems/
Twitter: http://bit.ly/2HobMld
For many centuries the idea prevailed that if a system was governed by simple rules that were deterministic then with sufficient information and computation power we would be able to fully describe and predict its future trajectory, the revolution of chaos theory in the latter half of the 20th century put an end to this assumption showing how simple rules could, in fact, lead to complex behavior. In this module we will describe how this is possible when we have what is called sensitivity to initial conditions.
Transcription excerpt:
Isolated systems tend to evolve towards a single equilibrium, a special state that has been the focus of many-body research for centuries. But when we look around us we don't see simple periodic patterns everywhere, the world is a bit more complex than this and behind this complexity is the fact that the dynamics of a system maybe the product of multiple different interacting forces, have multiple attractor states and be able to change between different attractors over time. Before we get into the theory lets take a few examples to try and illustrate the nature of nonlinear dynamic systems.
A classical example given of this is a double pendulum; a simple pendulum with out a joint will follow the periodic and deterministic motion characteristic of linear systems with a single equilibrium that we discussed in the previous section. Now if we take this pendulum and put a joint in the middle of its arm so that it has two limbs instead of one, now the dynamical state of the system will be a product of these two parts interaction over time and we will get a nonlinear dynamic system.
To take a second example; in the previous section we looked at the dynamics of a planet orbiting another in a state of single equilibrium and attractor, but what would happen if we added another planet into this equation, physicists puzzled over this for a long time, we now have two equilibrium points creating a nonlinear dynamic system as our planet would be under the influence of two different gravitational fields of attraction.
Where as with our simple periodic motion it was not important where the system started out, there was only one basin of attraction and it would simply gravitate towards this equilibrium point and then continue in a periodic fashion. But when we have multiple interacting parts and basins of attraction, small changes in the initial state to the system can lead to very different long-term trajectories and this is what is called chaos.
Wikipedia has a good definition for chaos theory so lets take a quote from it. “Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions—a response popularly referred to as the butterfly effect. Small differences in initial conditions yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general”.
We should note that chaos theory really deals with deterministic systems
and more over it is primarily focuses on simple systems, in that it often deals with systems that have only a very few elements, as opposed to complex systems where we have very many components that are non deterministic, in these complex systems we would of cause expect all sorts of random, complex and chaotic behavior, but it is not something we would expect in simple deterministic systems.
Twitter: http://bit.ly/2TTjlDH
Facebook: http://bit.ly/2TXgrOo
LinkedIn: http://bit.ly/2TPqogN

Views: 37966
Systems Academy

Linearization for 1-D systems. Existence and uniqueness of solutions. Bifurcations. Saddle-node bifurcation. Bifurcation diagrams.
Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 3.0--3.2, 3.4.

Views: 62765
Cornell MAE

Join my Coursera course on differential equations:
https://www.coursera.org/learn/differential-equations-engineers

Views: 58765
Jeffrey Chasnov

See http://mathinsight.org/equilibria_discrete_dynamical_systems for context

Views: 5217
Duane Nykamp

Views: 5274
matsciencechannel

This video shows how simple it is to simulate discrete-time dynamical systems, such as the Logistic Map, in Matlab.
https://www.eigensteve.com/

Views: 2976
Steve Brunton

These are videos form the online course ‘Introduction to Dynamical Systems and Chaos’ hosted on Complexity Explorer. With these videos you'll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time.
Topics to be covered include: phase space, bifurcations, chaos, the butterfly effect, strange attractors, and pattern formation. The course will focus on some of the realizations from the study of dynamical systems that are of particular relevance to complex systems:
1. Dynamical systems undergo bifurcations, where a small change in a system parameter such as the temperature or the harvest rate in a fishery leads to a large and qualitative change in the system's behavior.
2. Deterministic dynamical systems can behave randomly. This property, known as sensitive dependence or the butterfly effect, places strong limits on our ability to predict some phenomena.
3. Disordered behavior can be stable. Non-periodic systems with the butterfly effect can have stable average properties. So the average or statistical properties of a system can be predictable, even if its details are not.
4. Complex behavior can arise from simple rules. Simple dynamical systems do not necessarily lead to simple results. In particular, we will see that simple rules can produce patterns and structures of surprising complexity.
Help us caption & translate this video!
http://amara.org/v/D2sp/

Views: 6040
Complexity Explorer

Greetings, Youtube!
This is the first video in my series on Nonlinear Dynamics. Comment below if you have any questions, and if you like the video, let me know.
Also, if you have any more requests for what subjects you want to see, tell me in the comments section!
Patreon Link: https://www.patreon.com/user?u=4354534

Views: 10555
Faculty of Khan

See http://mathinsight.org/equilibria_discrete_dynamical_systems_stability_examples for context.

Views: 8212
Duane Nykamp

See http://mathinsight.org/bifurcations_differential_equation_introduction for context.

Views: 41203
Duane Nykamp

Course: Systems Dynamics and Complexity (Fall 2011) from ETH Zurich.
Source: http://www.video.ethz.ch/lectures/d-mtec/2011/autumn/351-0541-00L.html

Views: 329
CosmoLearning

These are videos form the online course ‘Introduction to Dynamical Systems and Chaos’ hosted on Complexity Explorer. With these videos you'll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time.
Topics to be covered include: phase space, bifurcations, chaos, the butterfly effect, strange attractors, and pattern formation. The course will focus on some of the realizations from the study of dynamical systems that are of particular relevance to complex systems:
1. Dynamical systems undergo bifurcations, where a small change in a system parameter such as the temperature or the harvest rate in a fishery leads to a large and qualitative change in the system's behavior.
2. Deterministic dynamical systems can behave randomly. This property, known as sensitive dependence or the butterfly effect, places strong limits on our ability to predict some phenomena.
3. Disordered behavior can be stable. Non-periodic systems with the butterfly effect can have stable average properties. So the average or statistical properties of a system can be predictable, even if its details are not.
4. Complex behavior can arise from simple rules. Simple dynamical systems do not necessarily lead to simple results. In particular, we will see that simple rules can produce patterns and structures of surprising complexity.
Help us caption & translate this video!
http://amara.org/v/D40l/

Views: 2571
Complexity Explorer

See http://mathinsight.org/graphical_approach_equilibria_discrete_dynamical_systems

Views: 5780
Duane Nykamp

Lecture Series on Chaos, Fractals and Dynamical Systems by Prof.S.Banerjee,Department of Electrical Engineering, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in.

Views: 3290
nptelhrd

Lecture Series on Chaos, Fractals and Dynamical Systems by Prof.S.Banerjee,Department of Electrical Engineering, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in.

Views: 6763
nptelhrd

These are videos form the online course ‘Introduction to Dynamical Systems and Chaos’ hosted on Complexity Explorer. With these videos you'll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time.
Topics to be covered include: phase space, bifurcations, chaos, the butterfly effect, strange attractors, and pattern formation. The course will focus on some of the realizations from the study of dynamical systems that are of particular relevance to complex systems:
1. Dynamical systems undergo bifurcations, where a small change in a system parameter such as the temperature or the harvest rate in a fishery leads to a large and qualitative change in the system's behavior.
2. Deterministic dynamical systems can behave randomly. This property, known as sensitive dependence or the butterfly effect, places strong limits on our ability to predict some phenomena.
3. Disordered behavior can be stable. Non-periodic systems with the butterfly effect can have stable average properties. So the average or statistical properties of a system can be predictable, even if its details are not.
4. Complex behavior can arise from simple rules. Simple dynamical systems do not necessarily lead to simple results. In particular, we will see that simple rules can produce patterns and structures of surprising complexity.
Help us caption & translate this video!
http://amara.org/v/D2tK/

Views: 6588
Complexity Explorer

Views: 20267
nptelhrd

Views: 2177
nptelhrd

These are videos from the Nonlinear Dynamics course offered on Complexity Explorer (complexity explorer.org) taught by Prof. Liz Bradley. These videos provide a broad introduction to the field of nonlinear dynamics, focusing both on the mathematics and the computational tools that are so important in the study of chaotic systems. The course is aimed at students who have had at least one semester of college-level calculus and physics, and who can program in at least one high-level language (C, Java, Matlab, R, ...).
After a quick overview of the field and its history, we review the basic background that students need in order to succeed in this course. We then dig deeper into the dynamics of maps—discrete-time dynamical systems—encountering and unpacking the notions of state space, trajectories, attractors and basins of attraction, stability and instability, bifurcations, and the Feigenbaum number. We then move to the study of flows, where we revisit many of the same notions in the context of continuous-time dynamical systems. Since chaotic systems cannot, by definition, be solved in closed form, we spend some time thinking about how to solve them numerically, and learning what challenges arise in that process. We then learn about techniques and tools for applying all of this theory to real-world data and close with a number of interesting applications: control of chaos, prediction of chaotic systems, chaos in the solar system, and uses of chaos in music and dance.
In each unit of this course, students will begin with paper-and-pencil exercises regarding the corresponding topics, and then write computer programs that operationalize the associated mathematical algorithms. This will not require expert programming skill, but you should be comfortable translating basic mathematical ideas into code. Any computer language that supports simple plotting—points on labelled axes—will suffice for these exercises. We will not ask you to turn in your code, but simply report and analyze the results that your code produces.

Views: 592
Complexity Explorer

These are videos from the Nonlinear Dynamics course offered on Complexity Explorer (complexity explorer.org) taught by Prof. Liz Bradley. These videos provide a broad introduction to the field of nonlinear dynamics, focusing both on the mathematics and the computational tools that are so important in the study of chaotic systems. The course is aimed at students who have had at least one semester of college-level calculus and physics, and who can program in at least one high-level language (C, Java, Matlab, R, ...).
After a quick overview of the field and its history, we review the basic background that students need in order to succeed in this course. We then dig deeper into the dynamics of maps—discrete-time dynamical systems—encountering and unpacking the notions of state space, trajectories, attractors and basins of attraction, stability and instability, bifurcations, and the Feigenbaum number. We then move to the study of flows, where we revisit many of the same notions in the context of continuous-time dynamical systems. Since chaotic systems cannot, by definition, be solved in closed form, we spend some time thinking about how to solve them numerically, and learning what challenges arise in that process. We then learn about techniques and tools for applying all of this theory to real-world data and close with a number of interesting applications: control of chaos, prediction of chaotic systems, chaos in the solar system, and uses of chaos in music and dance.
In each unit of this course, students will begin with paper-and-pencil exercises regarding the corresponding topics, and then write computer programs that operationalize the associated mathematical algorithms. This will not require expert programming skill, but you should be comfortable translating basic mathematical ideas into code. Any computer language that supports simple plotting—points on labelled axes—will suffice for these exercises. We will not ask you to turn in your code, but simply report and analyze the results that your code produces.

Views: 906
Complexity Explorer

These are videos form the online course ‘Introduction to Dynamical Systems and Chaos’ hosted on Complexity Explorer. With these videos you'll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time.
Topics to be covered include: phase space, bifurcations, chaos, the butterfly effect, strange attractors, and pattern formation. The course will focus on some of the realizations from the study of dynamical systems that are of particular relevance to complex systems:
1. Dynamical systems undergo bifurcations, where a small change in a system parameter such as the temperature or the harvest rate in a fishery leads to a large and qualitative change in the system's behavior.
2. Deterministic dynamical systems can behave randomly. This property, known as sensitive dependence or the butterfly effect, places strong limits on our ability to predict some phenomena.
3. Disordered behavior can be stable. Non-periodic systems with the butterfly effect can have stable average properties. So the average or statistical properties of a system can be predictable, even if its details are not.
4. Complex behavior can arise from simple rules. Simple dynamical systems do not necessarily lead to simple results. In particular, we will see that simple rules can produce patterns and structures of surprising complexity.

Views: 7074
Complexity Explorer

Views: 12713
nptelhrd

Views: 931
Complexity Explorer

Views: 246
Hasan Nagiev

Multiphase flows:Analytical solutions and Stability Analysis by Prof. S.Pushpavanam,Department of Chemical Engineering,IIT Madras.For more details on NPTEL visit http://nptel.ac.in

Views: 5321
nptelhrd

Views: 1430
Complexity Explorer

The approach to Bifurcation Analysis implemented in CAAS QCA Bifurcation

Views: 711
PieMedicalImaging

Here i have used matcont3p4..You can download the latest version from http://sourceforge.net/projects/matcont/

Views: 4640
PaPaI G

These are videos form the online course ‘Introduction to Dynamical Systems and Chaos’ hosted on Complexity Explorer. With these videos you'll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time.
Topics to be covered include: phase space, bifurcations, chaos, the butterfly effect, strange attractors, and pattern formation. The course will focus on some of the realizations from the study of dynamical systems that are of particular relevance to complex systems:
1. Dynamical systems undergo bifurcations, where a small change in a system parameter such as the temperature or the harvest rate in a fishery leads to a large and qualitative change in the system's behavior.
2. Deterministic dynamical systems can behave randomly. This property, known as sensitive dependence or the butterfly effect, places strong limits on our ability to predict some phenomena.
3. Disordered behavior can be stable. Non-periodic systems with the butterfly effect can have stable average properties. So the average or statistical properties of a system can be predictable, even if its details are not.
4. Complex behavior can arise from simple rules. Simple dynamical systems do not necessarily lead to simple results. In particular, we will see that simple rules can produce patterns and structures of surprising complexity.
Help us caption & translate this video!
http://amara.org/v/DpFl/

Views: 2237
Complexity Explorer

http://www.uvm.edu/~cdanfort/main/266.html

Views: 686
Chaos, Fractals, & Dynamical Systems

Shows how to determine the fixed points and their linear stability of a first-order nonlinear differential equation.
Free books:
http://bookboon.com/en/differential-equations-with-youtube-examples-ebook
http://www.math.ust.hk/~machas/differential-equations.pdf

Views: 6030
Jeffrey Chasnov

(0:07) Overview (long & lame jokes). (1:15) Review nonlinear system of differential equations from Part 1, including nullclines and linearization. (9:21) Separatrices. (11:26) Making use of symmetry (across the y-axis) in the equations. (15:28) Analyze a related one-parameter family of nonlinear systems and find bifurcation values, making use of algebra, linearization, and the trace-determinant plane. (23:38) Discuss Hartman-Grobman Theorem (for hyperbolic equilibrium points). (30:00) Long and Lame Joke of the Day.

Views: 4426
Bill Kinney

These are videos form the online course ‘Introduction to Dynamical Systems and Chaos’ hosted on Complexity Explorer. With these videos you'll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time.
Topics to be covered include: phase space, bifurcations, chaos, the butterfly effect, strange attractors, and pattern formation. The course will focus on some of the realizations from the study of dynamical systems that are of particular relevance to complex systems:
1. Dynamical systems undergo bifurcations, where a small change in a system parameter such as the temperature or the harvest rate in a fishery leads to a large and qualitative change in the system's behavior.
2. Deterministic dynamical systems can behave randomly. This property, known as sensitive dependence or the butterfly effect, places strong limits on our ability to predict some phenomena.
3. Disordered behavior can be stable. Non-periodic systems with the butterfly effect can have stable average properties. So the average or statistical properties of a system can be predictable, even if its details are not.
4. Complex behavior can arise from simple rules. Simple dynamical systems do not necessarily lead to simple results. In particular, we will see that simple rules can produce patterns and structures of surprising complexity.
Help us caption & translate this video!
http://amara.org/v/DpG7/

Views: 2523
Complexity Explorer

These are videos form the online course ‘Introduction to Dynamical Systems and Chaos’ hosted on Complexity Explorer. With these videos you'll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time.
Topics to be covered include: phase space, bifurcations, chaos, the butterfly effect, strange attractors, and pattern formation. The course will focus on some of the realizations from the study of dynamical systems that are of particular relevance to complex systems:
1. Dynamical systems undergo bifurcations, where a small change in a system parameter such as the temperature or the harvest rate in a fishery leads to a large and qualitative change in the system's behavior.
2. Deterministic dynamical systems can behave randomly. This property, known as sensitive dependence or the butterfly effect, places strong limits on our ability to predict some phenomena.
3. Disordered behavior can be stable. Non-periodic systems with the butterfly effect can have stable average properties. So the average or statistical properties of a system can be predictable, even if its details are not.
4. Complex behavior can arise from simple rules. Simple dynamical systems do not necessarily lead to simple results. In particular, we will see that simple rules can produce patterns and structures of surprising complexity.

Views: 6102
Complexity Explorer