455 Semester Review: Models of Dynamical Phenomena

State described by 1 (real) number, time discrete: 1D maps

geometric growth/decay

logistic model

steady states, periodic states
stability (|slope| < 1)
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chaos, sensitive dependence on initial conditions (butterfly effect)
"conjugacy" of logistic and tent maps (mathematical equivalence of superficially distinct systems)

State specified by several (real) numbers, time discrete: linear multi-D maps

age-structured population model (linear, normalized)


stable age distribution ... a single persistent mode (others decay)!

Stochastic models

discreteness of populations
probability computations
death process (radioactive decay)
birth process (stochastic geometric growth)

agreement of deterministic and stochastic in large-numbers limit
(disagreement in continuous-time SIR infection model)

Epidemic models

nonlinear discrete-time multi-D maps


surprising conclusion (usefulness of mathematical modelling!)

Heat

state at each time described by a real-valued function (temperature)
(linear) partial differential evolution equation
fundamental plane source solution, combinations thereof

Chemical reactions

principle of mass action -> (nonlinear) ODEs (need solution techniques from MTH 306!)
add space, diffusion -> (nonlinear) reaction-diffusion PDEs
some special solutions: USSs
linearization
"normal modes"

stability: Re(&omega) < 0
growth for only some wave numbers
nonlinear saturation and length-scale selection

Numerical solution of nonlinear PDEs

Euler time-stepping
Finite difference formulas for 1st and 2nd derivative approximation

Abstract 2D discrete-time maps

Conway's 23/3 universe

discovery vs. design
emergence of infinite complexity
musings on reality, existence

Waves

wave equation
excitable dynamics
nonlinear travelling waves in excitable media

What we didn't get to:

Vibration

From pendulum to cochlea in your inner ear
modes of vibration: "shape" interesting, growth rates all negative