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Instructor: Professor Igor Mezic,

2339 Engineering II,

Tel: 893-7603,

e-mail: mezic@engineering.ucsb.edu.

**Homework 4**

Posted: Fri, Feb 3

Due: Thu, Feb 9, 5pm in the ME 163 homework box

Assignment (pdf) - Solutions (pdf)**Homework 3**

Posted: Thu, Jan 26

Due: Thu, Feb 2, 5pm in the ME 163 homework box

Assignment (pdf) - Solutions (pdf)**Homework 2**

Posted: Fri, Jan 20

Due: Thu, Jan 26, 5pm in the ME 163 homework box

Assignment (pdf) - Solutions (pdf)**Homework 1**(this is a bonus homework; instructions are in the pdf)

Posted: Thu, Jan 12

Due: Thu, Jan 19, ME 163 homework box

Assignment (pdf) - Solutions (pdf)

This course will cover dynamical systems theory, and the application of dynamical systems techniques to mathematical, physical, biological, and technological systems described by ordinary differential equations or maps. The primary focus will be on dissipative systems, so that the course is complementary to the Advanced Dynamics sequence (ME 201) which primarily focus on conservative systems.

Igor Mezic

Engineering II 2339

Tel: 893-7603

e-mail: mezic@engineering.ucsb.edu

Lecture Hours: Mon – Wed 11:00-12:15 in Buchanan 1934.

Office Hours: Mon -Wed 12:30-2:00, Engineering II 2339.

Homeworks: Weekly [link].

Office Hours: Mon -Wed 12:30-2:00, Engineering II 2339.

Homeworks: Weekly [link].

- fixed points for vector fields and maps, and their stability properties (Ch. 1, Liapunov functions (Ch. 2)
- invariant manifolds for linear and nonlinear systems (Ch. 3)
- periodic orbits (Ch. 4)
- index theory (Ch. 6)
- asymptotic behavior, attractors (Ch. 8)
- Poincaré-Bendixson Theorem (Ch. 9)
- Poincaré maps (Ch. 10 and 11)
- structural stability (Ch. 12)
- center manifolds (Ch. 18)
- normal forms (Ch. 19)
- bifurcations of fixed points of vector fields (Ch. 20, 22)
- Melnikov’s method (Ch. 28)
- the Smale horseshoe (Ch. 23)
- symbolic dynamics (Ch. 24)
- chaos and strange attractors (Ch. 30).

- J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields”
- S. H. Strogatz, “Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering”
- P. Glendinning, “Stability, Instability, and Chaos”, http://www.scholarpedia.org

**Lecture 1:**Posted: Jan 11- Lecture 2: Posted: Jan 19
- Lecture 3: Posted: Jan 20
- Lecture 4: Posted: Jan 20
**Lecture 5:**Posted: Jan 26**Lecture 6:**Posted: Jan 26**Lecture 7:**Posted: Jan 31**Lecture 8:**Posted: Feb 2

This course will cover dynamical systems theory, and the application of dynamical systems techniques to mathematical, physical, biological, and technological systems described by ordinary differential equations or maps. The primary focus will be on dissipative systems, so that the course is complementary to the Advanced Dynamics sequence (ME 201) which primarily focus on conservative systems.

Igor Mezic

Engineering II 2339

Tel: 893-7603

e-mail: mezic@engineering.ucsb.edu

Lecture Hours: Mon – Wed 11:00-12:15 in Buchanan 1934.

Office Hours: Mon -Wed 12:30-2:00, Engineering II 2339.

Homeworks: Weekly [link].

Office Hours: Mon -Wed 12:30-2:00, Engineering II 2339.

Homeworks: Weekly [link].

- fixed points for vector fields and maps, and their stability properties (Ch. 1, Liapunov functions (Ch. 2)
- invariant manifolds for linear and nonlinear systems (Ch. 3)
- periodic orbits (Ch. 4)
- index theory (Ch. 6)
- asymptotic behavior, attractors (Ch. 8)
- Poincaré-Bendixson Theorem (Ch. 9)
- Poincaré maps (Ch. 10 and 11)
- structural stability (Ch. 12)
- center manifolds (Ch. 18)
- normal forms (Ch. 19)
- bifurcations of fixed points of vector fields (Ch. 20, 22)
- Melnikov’s method (Ch. 28)
- the Smale horseshoe (Ch. 23)
- symbolic dynamics (Ch. 24)
- chaos and strange attractors (Ch. 30).

- J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields”
- S. H. Strogatz, “Nonlinear Dynamics and Chaos: With Applications in Physics, Biology, Chemistry, and Engineering”
- P. Glendinning, “Stability, Instability, and Chaos”, http://www.scholarpedia.org

Posted: Jan 5, Due: Jan 12

Assignment (pdf) - Solutions (pdf)

Jan 19: Minor notation changes, and corrected moment of inertia for rotating rod

Assignment (pdf) - Solutions (pdf)

Jan 19: Minor notation changes, and corrected moment of inertia for rotating rod

**Homework 10 (Optional)**

Posted: Thu, Mar 11

Due: Thu, Mar 18, in class before the final

Assignment (pdf) - Solutions (pdf)**Homework 9 (Optional)**

Posted: Thu, Mar 4

Due: Tue, Mar 16, in dropbox for full credit

Assignment (zip) - Solutions (pdf)

**Update Mar 12**Minor changes in fourtransform.m function, new version is available above.**Homework 8**

Posted: Thu, Feb 25

Due: Thu, Mar 4, in class

25% off: by Fri, Mar 5, noon, ME163 dropbox (EngrII 2nd floor)

50% off: by Mon, Mar 8, noon, ME163 dropbox (EngrII 2nd floor)

Assignment (pdf) - Solutions (pdf)

**Update Mar 2**The last problem (Tongue 4.38) is for**extra credit**

**Update Mar 3**Typo: the coupling spring in**Problem 3**is torsional, so its units should be Nm/rad. In**Problem 1**, assume the beam is of negligible stiffness.**Homework 7**

Posted: Thu, Feb 18

Due: Thu, Feb 25, in class

25% off: by Fri, Feb 26, noon, ME163 dropbox (EngrII 2nd floor)

50% off: by Mon, Mar 1, noon, ME163 dropbox (EngrII 2nd floor)

Assignment (pdf) - Solutions (pdf) (Coming soon: Laplace transform solution to the first problem.)

**Update Feb 22**Problem 1 equilibrium height is x=0. Problem 3, torsional stiffnesses of roots added.**Homework 6**

Posted: Thu, Feb 11

Due: Thu, Feb 18, in class

25% off: by Fri, Feb 19, noon, ME163 dropbox (EngrII 2nd floor)

50% off: by Mon, Feb 22, noon, ME163 dropbox (EngrII 2nd floor)

Assignment (pdf) - Solutions (pdf)**Homework 5**

Posted: Thu, Feb 4

Due: Thu, Feb 11, in class

25% off: by Fri, Feb 12, noon, ME163 dropbox (EngrII 2nd floor)

50% off: by Mon, Feb 15, noon, ME163 dropbox (EngrII 2nd floor)

Assignment (zip) - Solutions (zip)**Homework 4**

Posted: Thu, Jan 28

Due: Thu, Feb 4, in class

25% off: by Fri, Feb 5, noon, ME163 dropbox (EngrII 2nd floor)

50% off: by Mon, Feb 8, noon, ME163 dropbox (EngrII 2nd floor)

Assignment (pdf) - Solutions (pdf)

**Update Feb 1**In Problem 3, L_eq = H/2. Notation updated to unambiguous form - 1/s means rad/s, not Hz.

**Update Feb 2**In Problem 3, a typo was corrected: the text should refer to "zeta" as damping RATIO, not as damping coefficient.**Homework 3**

Posted: Fri, Jan 22

Due: Thu, Jan 28, in class

25% off: by Fri, Jan 29, noon, ME163 dropbox (EngrII 2nd floor)

50% off: by Mon, Feb 1, noon, ME163 dropbox (EngrII 2nd floor)

Assignment (pdf) - Solutions (zip)

Answers to FAQ: In problem 3, you can treat the slider as a point-mass.

Damped frequency is defined only for underdamped systems, since for overdamped systems there are no oscillations in the output.**Homework 2**

Posted: Thu, Jan 14

Due: Thu, Jan 21, in class

25% off: by Fri, Jan 22, noon, ME163 dropbox (EngrII 2nd floor)

50% off: by Mon, Jan 25, noon, ME163 dropbox (EngrII 2nd floor)

Assignment (pdf) - Solutions (pdf)**Homework 1**

Posted: Thu, Jan 7

Due: Thu, Jan 14, in class

25% off: by Fri, Jan 15, noon, ME163 dropbox (EngrII 2nd floor)

50% off: by Mon, Jan 18, noon, ME163 dropbox (EngrII 2nd floor)

Assignment (pdf) - Solutions (pdf)

Jan 8:**Problem 2 update**: Unstretched rope length L = 0.4 m

Jan 12:**Problem 2 update**: You can assume that the rope stays horizontal during the motion.

Jan 13:**Problem 2 update**: Use t_final = 3 in your simulation.

Jan 13:**Problem 3 update**: Typo, instead M_eff it should read M=200 kg, i.e., the total mass of the cantilever is specified. The oscillating mass is then 200/3 kg.

Lecture notes link.

(please first log into the website using the fields to your right)

(please first log into the website using the fields to your right)

Updates:

- May 10, 2010
- May 12, 2009

I. Mezic: Lecture Notes

Mar 8, 2010 - correction pg 119, beating frequency epsilon = (w1 - w2)/2

Mar 4, 2010 - correction pg 117, eqn 3.10

Feb 25, 2010 - update

Feb 4, 2010 - correction to Example on page 75

Jan 21, 2010 - update to pages 53, 57, 60

Jan 12, 2010 - update

Mar 8, 2010 - correction pg 119, beating frequency epsilon = (w1 - w2)/2

Mar 4, 2010 - correction pg 117, eqn 3.10

Feb 25, 2010 - update

Feb 4, 2010 - correction to Example on page 75

Jan 21, 2010 - update to pages 53, 57, 60

Jan 12, 2010 - update

Solutions to the Midterm 2

**Note:** You are allowed to have TWO sheets of handwritten notes as a reminder during the final.

The Sections in the book we covered so far:

- Chapter 1- all except 1.5
- Chapter 2 - all
- Chapter 3 - all

In addition, we covered laplace transform method that is not covered in the book. The Midterm covers material **up to** **lecture 12** in lecture notes.

You can find a selection of practice problems and solutions from Tongue at the following links [Tongue, pdf], [Rayleigh, pdf].

You can find a selection of practice problems and solutions from Tongue at the following links [Tongue, pdf], [Rayleigh, pdf].

System diagonalization Matlab script [m-file]

The part shown in class starts at 4:00

Matlab script for generating a sine wave. [link]

A talk on Ocean Energy: