Lecture 1 - Tue 2020 09 01 - Virtual
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Students presentation.
Mechanics of class:
 (1) Go through Syllabus. (2) Discuss working groups. (3) Virtual lectures
 versus physical. The importance of writing and notes.
Questions and answers.

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Definition of dynamical system: there is a "phase space" describing the system,
and a rule for evolving in time [More on this below].
Examples: point mass in a potential in 3D, pendulum, point mass on the sphere.
          
Brief history; from Newton to the present. Important points
--- Development of the language of differential equations.
--- ODE [mechanics]; PDE [fluids, elasticity, EM, etc.]; difference equations
    [discrete time].
--- Linearity versus nonlinearity.
--- Nonlinearity spotty history. But not "neglected". People in fluids and
    astronomy had to face it from the start. Example: solitons and water waves.
--- Early tool in nonlinearity: oscillations and perturbations. Stokes ---> 
    1920 to now: oscillators basis of electronic industry.
--- Boltzman and Ergodic. 1st taste of chaos.
--- Poincare's GEOMETRIC APPROACH.
--- "3rd integral" of motion.
--- Lorenz and Henon-Heiles.

Back to definition of dynamical systems:
--- Discrete and continuous time. Maps and D.E.
--- Linear and nonlinear. Autonomous, non-autonomous.
--- ODE, reduction to 1st order and autonomous.
--- Dimension.
--- Trajectory/orbit.
--- Critical points, linearization, stability. 		<<=== Next lecture(s)
--- Bifurcations. 					<<=== Next lecture(s)

Examples 1-D systems 					<<=== Next lecture(s)
--- \dot{x} = sin(x). Exact solution vs phase portrait
--- \dot{x} = f(x) = - V'(x) and motivation [wire and goo].
--- Population grow, logistic equation.
--- No oscillations in 1-D.

EXISTENCE AND UNIQUENESS THEOREM for ode. Lipschitz.
--- Example of non-uniqueness.
    Chaos is extreme example, when bound for the
    continuity constant, e^{K*T}, actually happens.	<<=== Next lecture(s)
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