Outline

1. The concept of hyperbolicity; uniformly hyperbolic (Anosov) systems and their ergodic and topological properties.
2. Nonuniform hyperbolicity and Lyapunov exponents.
3. An example of a system with nonzero Lyapunov exponents. Existence of systems with nonzero Lyapunov exponents on smooth manifolds.
4. Stability theory of non-automous differential equations.
5. Lyapunov-Perron regularity and the Tempered Lemma. Mutiplicative. Ergodic Theorem of Oseledets. Hyperbolic invariant measures.
6. Stable Manifold Theorem; basic properties of stable (unstable) local (global) manifolds. Absolute continuity of local stable manifolds.
7. Ergodic properties of smooth hyperbolic measures: ergodicity and the entropy formula.

Description

Smooth ergodic theory studies the ergodic properties of smooth dynamical systems on Riemannian manifolds with respect to “natural" invariant measures such as smooth measures that are equivalent to volume. In these lectures I will discuss systems whose trajectories are hyperbolic in some sense.Roughly speaking, this means that the behavior of trajectories near a given orbit resembles the behavior of trajectories near a saddle point. Hyperbolic theory provides a mathematical foundation for the paradigm that is widely known as “deterministic chaos" - the appearance of irregular chaotic motions in purely deterministic dynamical systems. This paradigm asserts that conclusions about global properties of a nonlinear dynamical system with hyperbolic behavior can be deduced from studying the linearized systems along its orbits.

References

The following book can be used as a textbook for the course:

L. Barreira and Ya. Pesin, Lyapunov exponents and smooth ergodic theory, Univ. Lect. Series, Amer. Math. Soc. 23 2001.

The following book can be used as an optional reading:

L. Barreira and Ya. Pesin, Nonuniform hyperbolicity: dynamics of systems with nonzero Lyapunov exponents, CUP, 2007.

Outline

1. Introduction to delay effects in mechanical systems: linear stability and bifurcations in second order delay-differential equations
2. Position and force control of robots
3. Human and robotic balancing
4. Shimmying wheels
5. Machine tool vibrations

References

The following references constitute the skeleton of each one of the topics of the course:

1. T. Insperger and G. Stépán, Stability chart for the delayed Mathieu equation, Proc. R. Soc. Lond. A (2002) 458, 1989-1998
2. T. Insperger, L. Kovács, P.Galambos and G.Stépán, Increasing the Accuracy of Digital Force Control Process Using the Act-and-Wait Concept, IEEE/ASME Trans. Mechatronics (2010) 15, 291-2983.
3. G. Stépán, Delay effects in the human sensory system during balancing, Phil. Trans. R. Soc. A (2009) 367, 1195–1212
4. D. Takács, G. Orosz andG.Stépán, Delay effects in shimmy dynamics of wheels with stretched string-like tyres, European Journal of Mechanics A/Solids (2009) 28, 516–525
5. G. Stépán, Modelling nonlinear regenerative effects in metal cutting,Phil. Trans. R. Soc. Lon. A (2001) 359, 739-757

The following book can be used as an optional reading:

G. Stépán, Retarded dynamical systems: stability and characteristic functions,Longman Scientific and Technical, 1989.

Description

This short course will give an introduction to multi-frequency phenomena in dynamical systems, with particular attentions paid to those arising in differential equations. It will be divided into five sessions containing the following topics:

• Session 1: Multi-frequency Phenomena.
  • Physical motivations.
  • Dynamical motivations.
  • Dynamical complexity due to interactions of multi-frequencies.
• Session 2: Basic Theory of Multi-frequency Functions and Dynamics.
  • Quasi-periodicity.
  • Almost periodicity.
  • Almost automorphy .
• Session 3: Multi-frequency Dynamics in Monotone Systems.
  • Totally monotone systems.
  • Strongly monotone systems.
• Session 4: Multi-frequency Dynamics in Damped Oscillatory Systems.
  • Integral manifolds reduction.
  • Forced circle flows.
• Session 5: Multi-frequency Dynamics in Conservative Systems.
  • KAM Theory.
  • Aubry-Mather Theory.