Difference between revisions of "CDS 140, Winter 2015"
(Created page with "{ width=100%   colspan=2 align=center  <font color='blue' size='+2'>Differential Equations and Dynamical Systems</font>__NOTOC__  valign=top  width=50%  '''Instructor...") 
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−  <font color='blue' size='+2'>  +  <font color='blue' size='+2'>Introduction to Dynamics</font>__NOTOC__ 
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'''Instructors'''  '''Instructors'''  
* Richard Murray (CDS/BE), murray@cds.caltech.edu  * Richard Murray (CDS/BE), murray@cds.caltech.edu  
−  * Lectures:  +  * Lectures: MWF, 1112:30, 312 ANB 
* Office hours: Wed 23 pm (please email to confirm)  * Office hours: Wed 23 pm (please email to confirm)  
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'''Teaching Assistants'''  '''Teaching Assistants'''  
−  *  +  * Vanessa Jonsson (CDS) 
* Contact: cds140tas@cds.caltech.edu  * Contact: cds140tas@cds.caltech.edu  
−  * Office hours: Tue,  +  * Office hours: Tue, 45 pm, Ann 106 
}  }  
=== Course Description ===  === Course Description ===  
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, PoincaréBendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.  Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, PoincaréBendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.  
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=== Lecture Schedule ===  === Lecture Schedule ===  
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−    +   5 Jan <br> 7 Jan 
−   Linear Differential Equations  +   Linear Differential Equations 
* Course overview and administration  * Course overview and administration  
* Linear differential equations  * Linear differential equations  
* Matrix exponential, diagonalization  * Matrix exponential, diagonalization  
* Stable and unstable spaces  * Stable and unstable spaces  
−  *  +  * S + N decomposition, Jordan form 
    
−  Perko, 1.11.  +  Perko, 1.11.10<br> 
−   [[CDS 140a Winter  +   [[CDS 140a Winter 2015 Homework 1HW 1]] <br> Due: 14 Jan (Wed) 
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−    +   12 Jan <br> 16 Jan 
−  +  
−  +  
−  +  
−  +  
−  +  
−  +  
−  +  
−  +  
 Nonlinear differential equations   Nonlinear differential equations  
* Existence and uniqueness  * Existence and uniqueness  
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* Linearization  * Linearization  
 Perko, 2.12.6   Perko, 2.12.6  
−   [[CDS 140a Winter  +   [[CDS 140a Winter 2015 Homework 2HW 2]] <br> Due: 21 Jan (Wed) 
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−    +   21 Jan <br> 23 Jan 
 Behavior of differential equations   Behavior of differential equations  
* Stable and unstable manifolds  * Stable and unstable manifolds  
* Stability of equilibrium points for planar systems  * Stability of equilibrium points for planar systems  
 Perko, 2.72.10   Perko, 2.72.10  
−  +   [[CDS 140a Winter 2015 Homework 3HW 3]] <br> Due: 28 Jan (Wed)  
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−    +   26 Jan <br> 30 Jan 
 Nonhyperbolic differential equations   Nonhyperbolic differential equations  
* Lyapunov functions  * Lyapunov functions  
* Center manifold theorem  * Center manifold theorem  
 Perko, 2.112.13   Perko, 2.112.13  
−   [[CDS 140a Winter  +   [[CDS 140a Winter 2015 Homework 4HW 4]] <br> Due: 4 Feb (Wed) 
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−    +   4 Feb <br> 6 Feb 
 Global behavior   Global behavior  
* Limit sets and attractors  * Limit sets and attractors  
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* Periodic orbits and limit cycles  * Periodic orbits and limit cycles  
 Perko, 3.13.3   Perko, 3.13.3  
−   [[CDS 140a Winter  +   [[CDS 140a Winter 2015 Homework 5HW 5]] <br> Due: 11 Feb (Wed) 
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−    +   9 Feb <br> 11 Feb 
 Limit cycles   Limit cycles  
* Poincare' map  * Poincare' map  
* Bendixson criterion for limit cycles in the plane  * Bendixson criterion for limit cycles in the plane  
+  * Describing functions (if time)  
 Perko, 3.43.5, 3.7   Perko, 3.43.5, 3.7  
−   [[CDS 140a Winter  +   [[CDS 140a Winter 2015 Homework 6HW 6]] <br> Due: 18 Feb (Wed) 
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−    +   18 Feb <br> 20 Feb 
 Bifurcations   Bifurcations  
* Sensitivity analysis  * Sensitivity analysis  
* Structural stability  * Structural stability  
* Bifurcation of equilibrium points  * Bifurcation of equilibrium points  
−   Perko 4.14.2  +   Perko 4.14.2 <br> 
−  +  [[http:www.cds.caltech.edu/~murray/BFSwiki/index.php/Main_PageBFS]] 3.2 and 5.4 ([[Media:cds140wi15_bfssensitivity.pdfPDF]])  
−   [[CDS 140a Winter  +   [[CDS 140a Winter 2015 Homework 7HW 7]] <br> Due: 25 Feb (Wed) 
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−    +   23 Feb <br> 25 Feb? <br> 27 Feb? 
 Bifurcations   Bifurcations  
* Hopf bifurcation  * Hopf bifurcation  
* Application example: rotating stall and surge in turbomachinery  * Application example: rotating stall and surge in turbomachinery  
 Perko 4.34.5 + notes   Perko 4.34.5 + notes  
−   [[CDS 140a Winter  +   [[CDS 140a Winter 2015 Homework 8HW 8]] <br> Due: 4 Mar (Wed) 
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−   11 Mar  +   2 Mar <br> 6 Mar 
+   Nonlinear control systems  
+   {{obc08OBC}}, Chapter 1  
+   OBC 1.3, 1.4ab, 1.5<br> Due: 11 Mar (Wed)  
+   valign=top  
+   9 Mar <br>  
 Course review   Course review  
 <! Reading >   <! Reading >  
−   Final exam <br> Due:  +   Final exam <br> Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab 
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−  The following additional texts may be useful for some students  +  The following additional texts may be useful for some students: 
{  {  
+   valign=top  
+   align=right [G&H]  
+   J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. SpringerVerlag, 1990.  
+   valign=top  
+   align=right [H&S]  
+   M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. SpringerVerlag, 1990.  
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 align=right  [J&S]   align=right  [J&S]  
−   D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007.  +   D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL) 
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 align=right  [Ver]   align=right  [Ver]  
−   F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006.  +   F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL) 
}  }  
=== Grading ===  === Grading ===  
The ﬁnal grade will be based on homework and a ﬁnal exam:  The ﬁnal grade will be based on homework and a ﬁnal exam:  
−  * Homework (75%)  There will be  +  * Homework (75%)  There will be 9 oneweek problem sets, due ''in class'' approximately one week after they are assigned. ''Late homework will not be accepted without <u>prior</u> permission from the instructor.'' 
* Final exam (25%)  The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 48N hour period).  * Final exam (25%)  The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 48N hour period).  
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No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.  No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.  
−  
[[Category:Courses]]  [[Category:Courses]] 
Latest revision as of 21:46, 1 March 2015
Introduction to Dynamics  
Instructors

Teaching Assistants

Course Description
Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, PoincaréBendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.
Lecture Schedule
Date  Topic  Reading  Homework 
5 Jan 7 Jan 
Linear Differential Equations

Perko, 1.11.10 
HW 1 Due: 14 Jan (Wed) 
12 Jan 16 Jan 
Nonlinear differential equations

Perko, 2.12.6  HW 2 Due: 21 Jan (Wed) 
21 Jan 23 Jan 
Behavior of differential equations

Perko, 2.72.10  HW 3 Due: 28 Jan (Wed) 
26 Jan 30 Jan 
Nonhyperbolic differential equations

Perko, 2.112.13  HW 4 Due: 4 Feb (Wed) 
4 Feb 6 Feb 
Global behavior

Perko, 3.13.3  HW 5 Due: 11 Feb (Wed) 
9 Feb 11 Feb 
Limit cycles

Perko, 3.43.5, 3.7  HW 6 Due: 18 Feb (Wed) 
18 Feb 20 Feb 
Bifurcations

Perko 4.14.2 
HW 7 Due: 25 Feb (Wed) 
23 Feb 25 Feb? 27 Feb? 
Bifurcations

Perko 4.34.5 + notes  HW 8 Due: 4 Mar (Wed) 
2 Mar 6 Mar 
Nonlinear control systems  OBC, Chapter 1  OBC 1.3, 1.4ab, 1.5 Due: 11 Mar (Wed) 
9 Mar 
Course review  Final exam Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab 
Textbook
The primary text for the course (available via the online bookstore) is
[Perko]  L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006. 
The following additional texts may be useful for some students:
[G&H]  J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. SpringerVerlag, 1990. 
[H&S]  M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. SpringerVerlag, 1990. 
[J&S]  D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL) 
[Ver]  F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL) 
Grading
The ﬁnal grade will be based on homework and a ﬁnal exam:
 Homework (75%)  There will be 9 oneweek problem sets, due in class approximately one week after they are assigned. Late homework will not be accepted without prior permission from the instructor.
 Final exam (25%)  The ﬁnal will be handed out the last day of class and is due back at the end of ﬁnals week. Open book, time limit to be decided (likely N hours over a 48N hour period).
The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the ﬁnal is higher than the weighted average of your homework and ﬁnal, your ﬁnal will be used to determine your course grade.
Collaboration Policy
Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reﬂect your understanding of the subject matter at the time of writing.
You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).
No collaboration is allowed on the ﬁnal exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.