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Lecture

Chaos Theory: Maps and Lyapunov Exponents

In course

ME-467: Turbulence

This course provides an introduction to the physical phenomenon of turbulence, its probabilistic description and modeling approaches including RANS and LES. Students are equipped with the basic knowledge to tackle complex flow problems in science and engin ...

Description

This lecture covers chaotic maps, fix points, linear stability, and Lyapunov exponents in discrete dynamical systems. It explores the transition to chaos, fractal dimensions, and the Arnold's Cat Map in 2D. The importance of Lyapunov exponents in determining chaos is highlighted.

Instructor

Tobias Schneider

Tobias Schneider is an assistant professor in the School of Engineering at EPFL, the Swiss Federal Institute of Technology Lausanne. He received his doctoral degree in theoretical physics in 2007 from the University of Marburg in Germany working on the transition to turbulence in pipe flow. He then joined Harvard University as a postdoctoral fellow. In 2012 Tobias Schneider returned to Europe to establish an independent Max-Planck research group at the Max-Planck Institute for Dynamics and Self-Organization in Goettingen. Since 2014, he is working at EPFL, where he teaches fluid mechanics and heads the 'Emergent Complexity in Physical Systems' laboratory. Tobias Schneider's research is focused on nonlinear mechanics with specific emphasis on spatial turbulent-laminar patterns in fluid flows transitioning to turbulence. His lab combines dynamical systems and pattern-formation theory with large-scale computer simulations. Together with his team, Schneider develops computational tools and continuation methods for studying the bifurcation structure of nonlinear differential equations such as those describing the flow of a fluid. These tools are published as open-source software at channelflow.ch. Publications: Google Scholar

Official source

Ontological neighbourhood

Systems science and engineering

Systems theory: Chaos theory

Related concepts (32)

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis).

Chaos theory

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization.

Stability theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

Ergodicity

In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process.

Hénon map

In mathematics, the Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (xn, yn) in the plane and maps it to a new point The map depends on two parameters, a and b, which for the classical Hénon map have values of a = 1.4 and b = 0.3. For the classical values the Hénon map is chaotic. For other values of a and b the map may be chaotic, intermittent, or converge to a periodic orbit.

Related lectures (31)

Chaos Theory: Discrete Dynamical Systems

Explores Chaos Theory through Discrete Dynamical Systems and the Arnold's Cat Map.

Chaos Theory: Turbulence and Double Pendulum

Delves into Chaos Theory, exploring chaotic systems, sensitivity to initial conditions, and the dynamics of the double pendulum.

Chaos Theory: Logistic Map and Periodic Orbits

Explores Chaos Theory, focusing on the logistic map, periodic orbits, and stability conditions.

Chaos Theory: Chaotic Maps and Logistic Map

Explores chaotic maps, fix points, periodic orbits, and intermittent chaos.

Fractals and Chaos Theory