**Course Title:** EE 418/518 Nonlinear Dynamics & Chaos

**ABET Course Description:** This course provides students with an introduction to nonlinear dynamics and chaos. Topics will address ordinary differential equations and difference equations in terms of system stability, linearization, equilibrium/steady-state solutions, bifurcations, periodic solutions, limit cycles, oscillators, chaos, iterated maps and chaos control/synchronization. Various tools and methods used for analysis and design of real-world, nonlinear circuits and systems will be covered. When possible, systems will be treated analytically, however, general treatment of systems using numerical methods will be shown with MATLAB and/or Python. Ultimately, this course will provide experience for students beyond the limited behavior of low-degree, linear circuits and systems towards complex behaviors where determinism doesn’t always imply predictability.

**Objectives:** Upon completion of this course, the student will be able to:

- Master the fundamentals of nonlinear dynamics: linearization, stability, bifurcation, chaos, iterated maps
- Identification of sources of nonlinearity in electronic systems
- Analyze and interpret data from nonlinear & chaotic systems
- Realistic design and analysis needed for practical oscillator design
- Analysis and design of chaotic systems
- Apply knowledge of mathematics, science, and engineering in dynamics related research areas
- A knowledge of contemporary issues regarding nonlinear dynamics & chaos.

**Required Text:**

- Strogatz, S. H. (2018). Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering. CRC press.

**Suggested Resources:** (I use these materials/resources frequently…)

- Alligood, K. T., Sauer, T. D., & Yorke, J. A. (1996). Chaos (pp. 105-147). Springer New York.
- Hirsch, M. W., Smale, S., & Devaney, R. L. (1974). Differential equations, dynamical systems, and linear algebra (Vol. 60). Academic press.
- Gilmore, R., & Lefranc, M. (2002). The topology of chaos: Alice in stretch and squeezelan. Hoboken: Wiley & sons inc, 518.
- TI-89 Calculator (Or equivalent ACT-approved calculator i.e. TI-Nspire CAS)
- MATLAB, Octave, SciPy/NumPy
- LTSPICE or similar SPICE software

**Important Papers & Articles:**

- Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of atmospheric sciences, 20(2), 130-141.
- Crutchfield, J., Farmer, J., Packard, N., & Shaw, R. (1986). Chaos Scientific American 225 (6): 46-57. Scientific American.
- Shaw, R. (1981). Strange attractors, chaotic behavior, and information flow. Zeitschrift für Naturforschung A, 36(1), 80-112.

**Prerequisites:** Proficiency with Linear Signals & Systems, Differential Equations, Electronics

**Dr. Beal’s Course Statement:** Coming soon…

**Some exciting questions we can answer:**

What is chaos? When does it happen? Is it always bad? Can I build something using chaos?

How do order and complexity emerge from simple rule sets?

How can I dissect the most famous chaotic system into “information primitives”?

Can a deep neural network be replaced with a dynamical system?

How can fireflies help me build better phase locked loops?

How do I model a pendulum beyond small angles?

How can a system be deterministic and random at the same time?

Can oscillations come from linear systems?

When can I trust my simulations? (Solvers as iterated maps, shadowing theorem)

Did Barkhausen lie to me?

Why do “amplifiers tend to oscillate and oscillators tend to amplify”?

Why is the weather unpredictable?

Where does information come from?

What is randomness? How can randomness be reliably engineered?

Can signals and systems produce unique fingerprints (electronic signatures)?

How can ‘fancy’ random number generators like the Mersenne Twister be dangerous?

**Threads and themes for my chaos course:**

- Determinism $\neq$ Predictability
- Building numerical solvers: Euler $\rightarrow$ Runge-Kutta $\rightarrow$ Adams-Bashforth
- Numerical solvers translate ‘flows’ into ‘maps’.
- Heirachy of dynamical systems: Fixed points $\rightarrow$ Oscillations $\rightarrow$ Chaos $\rightarrow$ Undecidable
- Progressing theoretical dimensionality to design practical oscillators

**Notional Schedule of Topics & Assignments:**

Meeting | Topics Covered | Course Objectives |
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Mod. 01: Introduction to Nonlinearity & Chaos (Strogatz Ch. 1) |
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Lec. 01: | Introduction, Methods, Motivation & Nomenclature | |

Lec. 02: | History, Landscape & Examples | |

Mod. 02: One-dimensional Maps (Strogatz Ch. 10) |
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Lec. 03: | Map Definition, Properties, Fixed Points & Stability | |

Lec. 04: | Periodic Orbits & Stability | |

Mod. 03: Bifurcations & Chaos in Maps (Ch. 10) |
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Lec. 05: | Hierarchy of Attractors & Bifurcation | |

Lec. 06: | Chaos, Lyapunov Exponents & Partitions | |

Mod. 04: Dynamics in One Dimension (Ch. 2) |
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Lec. 07: | Dynamics on a Line & Linearization | |

Lec. 08: | Uniqueness, Potential Functions & Euler’s Method | |

Mod. 05: Bifurcation on Lines & Oscillation on Circles (Ch. 3-4) |
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Lec. 09: | One Dimensional Bifurcations | |

Lec. 10: | Dynamics on a Circle | |

Mod. 06: Dynamics in Two Dimensions (Strogatz Ch. 5 & 6) |
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Lec. 11: | Dynamics on a Plane & Linear Behavior | |

Lec. 12: | Nonlinear Planar Dynamics & Runge-Kutta Method | |

Midterm Exam (around week 7) |
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Mod. 07: Conservative & Reversible Systems (Ch. 6) |
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Lec. 13: | Bad Linearization & Double Well Potential | |

Lec. 14: | Reversible Systems & Nonlinear Pendulum | |

Mod. 08: Limit Cycles (Ch. 7) |
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Lec. 15: | Limit Cycles, Tests & Van der Pol Oscillator | |

Lec. 16: | Poincaré-Bendixon Theorem, Liénard Systems | |

One-week Break (Thanksgiving/Spring) |
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No Lecture | ||

No Lecture | ||

Mod. 09: Nonlinear Oscillators (Ch. 7) |
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Lec. 17: | Relaxation Oscillations, Multiple Scale Dynamics | |

Lec. 18: | Weakly Nonlinear Oscillators & Duffing Systems | |

Mod. 10: Bifurcations on Planes & Electronic Oscillators (Ch. 8) |
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Lec. 19: | Two Dimensional Bifurcations | |

Lec. 20: | Electronic Oscillators & Describing Functions | |

Mod. 11: Chaos (Strogatz Ch. 8 & Ch. 9) |
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Lec. 21: | Poincaré Maps & Lorenz Equations | |

Lec. 22: | Chaotic Flows & Analysis | |

Mod. 12: Deconsctructing the Lorenz System |
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Lec. 23: | Information Primitives of the Lorenz System | |

Lec. 24: | Second-order, Solvable Chaos | |

Mod. 13: Frontiers & Applications of Chaos |
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Lec. 25: | Connection to Communication Systems, Ranging & Randomness | |

Lec. 26: | Unique Fingerprints, Power Electroincs, Reservoir Computers | |

Lec. 27: | Advanced Topics & Review (as time permits) | |

Final Exam/Project (comprehensive) |