Related Experiment Video
Updated: Aug 6, 2026
13:44
Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
Published on: August 30, 2013
Iterative techniques for computing the linearized manifolds of quasiperiodic tori
Derin B Wysham1, James D Meiss
1University of Colorado at Boulder, Applied Mathematics, Boulder, Colorado 80309-0526, USA.
Chaos (Woodbury, N.Y.)
|July 11, 2006
Summary
We developed an efficient iterative technique to compute invariant tori eigenfunctions for diffeomorphisms. This method uses a generalized eigenvalue problem for faster approximation of invariant manifolds, confirming prior Melnikov calculations.
Area of Science:
- Dynamical Systems and Chaos Theory
- Numerical Analysis and Scientific Computing
Background:
- Invariant tori are fundamental structures in dynamical systems, governing long-term behavior.
- Computing their stable and unstable manifolds is crucial for understanding system dynamics.
- Previous methods for approximating these manifolds can be computationally intensive.
Purpose of the Study:
- To develop an efficient iterative technique for computing eigenfunctions of invariant tori.
- To leverage generalized eigenvalue problems for faster manifold approximation.
- To validate the method by confirming existing theoretical predictions.
Main Methods:
- Reformulating linearized equations as a generalized eigenvalue problem.
- Employing an iterative power method for numerical computation.
- Applying the technique to a volume-preserving mapping example.
Main Results:
- An efficient computational method for approximating invariant manifolds of tori was developed.
- The technique significantly benefits from the speed of modern eigenvalue solvers.
- Numerical results confirmed qualitative predictions from Melnikov calculations.
Conclusions:
- The iterative eigenvalue approach provides an efficient means to study invariant tori.
- This method enhances the understanding of normal behavior to invariant tori.
- The technique offers a robust tool for analyzing dynamical systems.
More Related Videos
Related Concept Videos
01:23Linear time-invariant Systems
A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated...
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated...
01:26Linear Approximation in Frequency Domain
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
01:21Linear Approximation in Time Domain
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
01:26Linearization and Approximation
Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
01:29Application of Linearization and Approximation
A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...
01:20Torsion in Vector Calculus
A toy train ascending a winding track that curves and tilts offers an intuitive view of torsion, a key geometric concept in the study of space curves. While curvature measures how sharply a path bends, torsion captures how the path twists out of the plane of bending. This twisting behavior is crucial in understanding three-dimensional motion and is precisely described using the Frenet–Serret framework.At each point along a space curve, the Frenet–Serret frame consists of three orthogonal unit...