038786 – Introduction to Nonlinear And Chaotic Dynamical Systems

038786 – Introduction to Nonlinear And Chaotic Dynamical Systems

Time & Place: Winter 2016/17, Monday (1230-1530), Lady Davis building room 440, Technion.

Chaotic dynamical systems that are sensitive to initial conditions have been known to exist for over a century. This sensitivity in deterministic nonlinear systems results in response that is
unpredictable. Chaotic dynamics are observed in low-order systems that exhibit multiple coexisting (stable and unstable) solutions, and in the past three decades have been shown to govern the onset of
a-periodic pattern formation and complex non-stationary and spatio-temporal phenomena.
Examples include rigid body dynamics (gyroscopic effects, friction and contact phenomena), machine tool chatter, control systems (feedback and delay), elastic continua (buckling and whirling),
turbulence, fluid-structure-interaction (vortex-induced-vibration), nonlinear waves, electromagnetic systems and fields (smart structures, nano- and micro- electromechanical systems), chemical reactions and heart fibrillations.
Understanding and identification of chaotic system response and its role in bifurcation theory is essential for advanced research of nonlinear physical phenomena in general, and nonlinear
engineering systems in particular.
The objectives of this course are to introduce, develop and apply both the analytical and numerical tools required for understanding and investigation of nonlinear deterministic dynamical systems that
exhibit chaotic dynamics.

Topics
* geometric description and characterization of nonlinear systems:
multiple coexisting periodic and a-periodic solutions, spectral analyses, horseshoes and Poincare’ maps.
* local bifurcations: normal forms, co-dimension, stability of equilibrium (fold/Hopf), orbital stability (period doubling),
and explosions (crisis).
* global bifurcations: integrability/stochasticity in conservative systems (KAM), homoclinicity (solitons), heteroclinicity
(fronts) and Melnikov functions in non-conservative systems.
* classification and investigation of a-periodic response:
quasiperiodic tori, intermittency, strange attractors (vs. stochastic layers), Liapunov exponents and fractal dimensions.
* superstructure in the bifurcation set: degeneracy, exotic bifurcations, influence of stochastic noise, domains of attraction (robustness), control of chaos (adaptive, OGY schemes) , and sensitivity of numerical solvers.

Literature
applied: Moon, F.C., Chaotic and Fractal Dynamics, 1992; Chaotic Vibrations,2004.
Nayfeh, A.H. and Balachandran, B, Applied Nonlinear Dynamics, 1995 [eBook].
Strogatz, S., Nonlinear Dynamics and Chaos, (1994) 2015.
advanced: Guckenheimer J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems and
Bifurcations of Vector Fields, 1983 [eBook].
Kuznetsov, Y.A., Elements of Applied Bifurcation Theory, (1998) 2004 [eBook].
Wiggins, S., Int. to Applied Nonlinear Dynamical Systems and Chaos, (1990) 2003 [eBook].
selected journal papers.

Procedure

Homework problems, final project/term paper.