Technion – Israel Institute of Technology
Department of Mechanical Engineering

TChaotic 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 two decades have been shown to govern the onset of a-periodic pattern formation and complex non-stationary spatio-temporal phenomena.
Examples include rigid body dynamics (gyroscopic effects, friction and contact phenomena), machine tool chatter, nonlinear feedback and delay in control systems, buckling and whirling of elastic continua, turbulence, flow induced vibrations, 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
crucial 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.


  • 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 equilibria (fold/Hopf) and orbits (period doubling/Floquet), and explosions (crisis).
  • global bifurcations: integrability/stochasticity in conservative systems (KAM), homoclinicity (solitons), heteroclinicity (fronts) and Melnikov functions in non-conservative flows.
  • 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 (blue-sky), influence of stochastic noise, domains of attraction, and sensitivity of numerical solvers.


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


Homework problems, final project/term paper.

Time & Place

Fall 2008/09
Monday (0830-1130)
Lady Davis Bldg. Rm. 283, Technion
Oded Gottlieb