Home
Search results “Bifurcation analysis of dynamical systems”
Introducing Bifurcations: The Saddle Node Bifurcation
 
13:34
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: 1379 Faculty of Khan
Phase Transitions & Bifurcations
 
06:24
See the full course: https://goo.gl/9qB4CV Follow along with the course eBook: https://goo.gl/wQahvk 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. Produced by: http://complexitylabs.io Twitter: https://goo.gl/ZXCzK7 Facebook: https://goo.gl/P7EadV LinkedIn: https://goo.gl/3v1vwF 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.
Views: 12245 Complexity Labs
Mod-01 Lec-10 Elementary bifurcations
 
45:13
Topics in Nonlinear Dynamics by Prof. V. Balakrishnan,Department of Physics,IIT Madras.For more details on NPTEL visit http://nptel.ac.in
Views: 4539 nptelhrd
Linear Stability Analysis | Dynamical Systems 3
 
10:24
In this video (which happens to be my first ever 1080p video!), I discuss linear stability analysis, in which we consider small perturbations about the fixed point, and then analyze the local behavior of the differential equation/dynamical system around the fixed point. The derivation involves a simple linear expansion of the differential equation function around the fixed point. When the x derivative of the ODE function f(x) is positive, we have an unstable fixed point. When it's negative, we have a stable fixed point. I didn't discuss a special (but more rare case) in the video, but when f'(xf) is zero, we usually have a 'half-stable' fixed point (stable from one direction, unstable from the other). Questions/requests? Let me know in the comments below! Prereqs: The first 2 videos in my Nonlinear Dynamics/Dynamical Systems playlist: https://www.youtube.com/playlist?list=PLdgVBOaXkb9C8iPDD5xW0jT-c3dtP4TR5 Lecture Notes: https://drive.google.com/open?id=0BzC45hep01Q4UkZFbndqWWZMVHc Patreon: https://www.patreon.com/user?u=4354534 Twitter: https://twitter.com/FacultyOfKhan/ Special thanks to my Patrons: Jacob Soares
Views: 4669 Faculty of Khan
Multiple Scale Bifurcation Analysis for Discrete and Continuous Autonomous Dynamical Systems
 
02:06:18
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: 48 M&MoCS
Concepts of Bifurcation: Introduction
 
40:16
Introduction to Dynamical Models in Biology: Module 1, Week 3
Views: 1519 NOC17 JAN-FEB BT05
Mod-01 Lec-20 Introduction to stability of dynamical systems: ODEs
 
58:18
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: 3770 nptelhrd
Bifurcations of a differential equation
 
09:58
See http://mathinsight.org/bifurcations_differential_equation_introduction for context.
Views: 38337 Duane Nykamp
Lecture - 10 Flip and Tangent Bifurcations
 
56:20
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: 6627 nptelhrd
MAE5790-2 One dimensional Systems
 
01:16:44
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: 56608 Cornell MAE
Lecture - 36 Multiple Attractor Bifurcation and Dangerous
 
59:21
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: 4333 nptelhrd
Approach to Bifurcation Analysis
 
03:31
The approach to Bifurcation Analysis implemented in CAAS QCA Bifurcation
Views: 689 PieMedicalImaging
Complexicon: Dynamical System
 
02:47
Dr Sid Redner covers dynamical systems - an integral element of studying complex systems. Part of ComplexityeExplorer.org's Complexicon - concepts in complexity, explained for everyone. For more information visit santafe.edu
Views: 413 Complexity Explorer
Nonlinear Dynamics & Chaos
 
04:52
See the full course: https://goo.gl/9qB4CV Follow along with the course eBook: https://goo.gl/wQahvk 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. Produced by: http://complexitylabs.io Twitter: https://goo.gl/ZXCzK7 Facebook: https://goo.gl/P7EadV LinkedIn: https://goo.gl/3v1vwF 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.
Views: 31121 Complexity Labs
Solving linear discrete dynamical systems
 
08:05
See http://mathinsight.org/solving_linear_discrete_dynamical_systems for context.
Views: 17869 Duane Nykamp
Dynamical Systems, Part 6: Bifurcations of fixed points
 
26:16
Mathematical modeling of physiological systems: Introduction to Dynamical Systems Part 6: Bifurcations of fixed points
Views: 119 nbjanson
Examples of determining the stability of equilibria for discrete dynamical systems
 
11:19
See http://mathinsight.org/equilibria_discrete_dynamical_systems_stability_examples for context.
Views: 7591 Duane Nykamp
Lecture - 31 Non-Smooth Bifurcations
 
54:19
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: 3226 nptelhrd
Dynamical Systems Introduction
 
06:41
See the full course: https://goo.gl/9qB4CV Follow along with the course eBook: https://goo.gl/wQahvk Dynamical systems is a area of mathematics and science that studies how the state of systems change over time, in this module we will lay down the foundations to understanding dynamical systems as we talk about phase space and the simplest types of motion, transients and periodic motion, setting us up to approach the topic of nonlinear dynamical systems in the next module. Produced by: http://complexitylabs.io Twitter: https://goo.gl/ZXCzK7 Facebook: https://goo.gl/P7EadV LinkedIn: https://goo.gl/3v1vwF Transcription excerpt: Within science and mathematics, dynamics is the study of how things change with respect to time, as opposed to describing things simply in terms of their static properties the patterns we observe all around us in how the state of things change overtime is an alternative ways through which we can describe the phenomena we see in our world. A state space also called phase space is a model used within dynamic systems to capture this change in a system’s state overtime. A state space of a dynamical system is a two or possibly three-dimensional graph in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the state space. Now we can model the change in a system’s state in two ways, as continuous or discrete. Firstly as continues where the time interval between our measurements is negligibly small making it appear as one long continuum and this is done through the language of calculus. Calculus and differential equations have formed a key part of the language of modern science since the days of Newton and Leibniz. Differential equations are great for few elements they give us lots of information but they also become very complicated very quickly. On the other hand we can measure time as discrete meaning there is a discernable time interval between each measurement and we use what are called iterative maps to do this. Iterative maps give us less information but are much simpler and better suited to dealing with very many entities, where feedback is important. Where as differential equations are central to modern science iterative maps are central to the study of nonlinear systems and their dynamics as they allow us to take the output to the previous state of the system and feed it back into the next iteration, thus making them well designed to capture the feedback characteristic of nonlinear systems. The first type of motion we might encounter is simple transient motion, that is to say some system that gravitates towards a stable equilibrium and then stays there, such as putting a ball in a bowl it will role around for a short period before it settles at the point of least potential gravity, its so called equilibrium and then will just stay there until perturbed by some external force. Next we might see periodic motion, for example the motion of the planets around the sun is periodic. This type of periodic motion is of cause very predictable we can predict far out into the future and way back into the past when eclipses happen. In these systems small disturbances are often rectified and do not increase to alter the systems trajectory very much in the long run. The rising and receding motion of the tides or the change in traffic lights are also example of periodic motion. Whereas in our first type of motion the system simply moves towards its equilibrium point, in this second periodic motion it is more like it is cycling around some equilibrium. All dynamic systems require some input of energy to drive them, in physics they are referred to as dissipative systems as they are constantly dissipating the energy being inputted to the system in the form of motion or change. A system in this periodic motion is bound to its source of energy and its trajectory follows some periodic motion around it or towards and away from it. In our example of the planet’s orbit, it is following a periodic motion because of the gravitational force the sun exerts on it, if it were not for this driving force, the motion would cease to exist.
Views: 23811 Complexity Labs
Fixed points and stability of a nonlinear system
 
18:01
Find the fixed points and determines the linear stability of a system of two first-order nonlinear differential equations. Free books: http://bookboon.com/en/differential-equations-with-youtube-examples-ebook http://www.math.ust.hk/~machas/differential-equations.pdf
Views: 54371 Jeffrey Chasnov
Concepts of Bifurcation: Bifurcation in Biological Systems
 
42:50
Introduction to Dynamical Models in Biology: Module 2, Week 3
Views: 397 NOC17 JAN-FEB BT05
Stability and Bifurcation of Dynamical Systems - A. Luongo
 
01:47:29
Third Sperlonga Summer School on Mechanics and Engineering Sciences Prof. Angelo Luongo (Università degli Studi dell’Aquila, M&MoCS, Italy) Lecture on: “Stability and Bifurcation of Dynamical Systems “
Views: 155 M&MoCS
Lecture - 9 The Logistic Map and Period doubling
 
55:25
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: 19192 nptelhrd
Analysis of Nonlinear Systems, Part 2 (nullclines, linearization, bifurcations)
 
34:49
(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: 3931 Bill Kinney
Introduction to Nonlinear Dynamics
 
09:56
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: 8096 Faculty of Khan
Mathematical Biology. 21: Hopf Bifurcations
 
49:41
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: 11088 UCI Open
Mod-01 Lec-02 Critical points of a dynamical system
 
01:00:19
Topics in Nonlinear Dynamics by Prof. V. Balakrishnan,Department of Physics,IIT Madras.For more details on NPTEL visit http://nptel.ac.in
Views: 8918 nptelhrd
Dynamical systems
 
12:27
Qualitative description of (possibly) nonlinear systems of ODEs. Vector fields. Phase space. Potential energy. Physical interpretations.
Views: 191 Marcus Berg
Mathematical Biology. 14: Predator Prey Model
 
47:07
UCI Math 113B: Intro to Mathematical Modeling in Biology (Fall 2014) Lec 14. Intro to Mathematical Modeling in Biology: Predator Prey Model 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 10, 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: 16816 UCI Open
Love the Bifurcation You're With
 
02:47
Bifurcations of the dynamical system determined by: X = Y + A(t) * sin( B(t) * X ) mod one For more information, please see: http://poibella.org/viz Music: "Love the One You're With" as performed by the Gaylettes
Views: 726 poibella
Lecture - 8 Discrete Time Dynamical Systems
 
55:37
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: 12496 nptelhrd
Lecture - 32 Non-Smooth Bifurcations
 
54:51
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: 2142 nptelhrd
Lecture - 3 Limit Cycles
 
56:22
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: 27769 nptelhrd
Lecture - 26 Dynamics on a Torus
 
54:41
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: 3235 nptelhrd
Mathematical modeling of physiological systems: Dynamical systems (Part 6)
 
26:16
Introduction to Dynamical Systems Part 6: Bifurcations of fixed points My colleagues and I recorded lectures for the course on "Mathematical modeling of physiological systems" given at the University of Copenhagen. The goal of the course is to equip students with tools for biosimulation. The course is focused on dynamical aspects of regulatory mechanisms at different levels of biological organization, i. e. cellular and systems physiology. The lectures can be used in other related courses.
Views: 406 Olga Sosnovtseva
Lecture 3.2: Concepts of Bifurcation: Bifurcation in Biological Systems
 
42:50
NPTEL online course on Introduction to Dynamical Models in Biology: Module 2, Week 3
Views: 59 NOC18-BT07 IITG
Lecture - 35 Bifurcations in Piecewise Linear 2D Maps
 
52:59
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: 2467 nptelhrd
Effect of a Perturbation on the Stable Points of a Dynamical System
 
00:13
http://demonstrations.wolfram.com/EffectOfAPerturbationOnTheStablePointsOfADynamicalSystem The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. A bifurcation diagram illustrates how the fixed points in a first-order dynamical system x evolve as a function parameter r changes. Take any vertical line upwards (along the x dimension) and note the fixed points at the color boundaries. From brown to ... Contributed by: Andrew Read Audio created with WolframTones: http://tones.wolfram.com
Views: 121 wolframmathematica
Dynamical Systems, Part 5: Analyzing stability of fixed points: 2- dimensional case
 
43:14
Mathematical modeling of physiological systems: Introduction to Dynamical Systems. Part 5: Analyzing stability of fixed points: 2- dimensional case
Views: 41 nbjanson
Simulating the Logistic Map in Matlab
 
16:29
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: 1377 Steve Brunton
Exponential growth and decay in discrete dynamical systems
 
06:56
See http://mathinsight.org/exponential_growth_decay_discrete for context.
Views: 2857 Duane Nykamp
Lecture - 34 Bifurcations in Piecewise Linear 2D Maps
 
55:32
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: 2687 nptelhrd
MAE5790-1 Course introduction and overview
 
01:16:31
Historical and logical overview of nonlinear dynamics. The structure of the course: work our way up from one to two to three-dimensional systems. Simple examples of linear vs. nonlinear systems. 1-D systems. Why pictures are more powerful than formulas for analyzing nonlinear systems. Fixed points. Stable and unstable fixed points. Example: Logistic equation in population biology. Reading: Strogatz, "Nonlinear Dynamics and Chaos", Chapter 1 and Section 2.0--2.3.
Views: 146883 Cornell MAE

Cover letter document specialist
The cube 2012 application letters
Electricians cv cover letter
Job cover letter email title for job
Dialysis nurse resume cover letter