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It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea is that momentum is a vector. Like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when the statement of conservation of momentum is written for a...
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In multiple dimensions, the conservation of momentum applies in each direction independently. Hence, to solve collisions in multiple dimensions, we should write down the momentum conservation in each direction separately. To help understand collisions in multiple dimensions, consider an example.
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The chair conformation is the most stable form of cyclohexane due to the absence of angle and torsional strain. The absence of angle strain is a result of cyclohexane’s bond angle being very close to the ideal tetrahedral bond angle of 109.5° in its chair conformer. Similarly, the torsional strain is also absent owing to the perfectly staggered arrangement of bonds.
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Cartesian vector notation is a valuable tool in mechanical engineering for representing vectors in three-dimensional space, performing vector operations such as determining the gradient, divergence, and curl, and expressing physical quantities such as the displacement, velocity, acceleration, and force. By using Cartesian vector notation, engineers can more easily analyze and solve problems in various areas of mechanical engineering, including dynamics, kinematics, and fluid mechanics. This...
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A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
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Searching chaotic saddles in high dimensions.

M Sala1, J C Leitão1, E G Altmann1

  • 1Max Planck Institute for the Physics of Complex Systems, Nöthnizer Straße 38, 01187 Dresden, Germany.

Chaos (Woodbury, N.Y.)
|January 2, 2017
PubMed
Summary

We developed new numerical methods to find long-lasting trajectories in chaotic systems. These techniques improve upon existing methods for approximating non-attracting sets, aiding the study of transient chaos.

Area of Science:

  • Dynamical Systems and Chaos Theory
  • Numerical Analysis
  • Computational Physics

Background:

  • Transiently chaotic systems exhibit trajectories that escape a bounded region in finite time.
  • Characterizing non-attracting sets is crucial for understanding the dynamics of these systems.
  • Existing methods for approximating these sets face challenges, especially in high dimensions.

Purpose of the Study:

  • To propose novel numerical methods for approximating non-attracting sets in transiently chaotic systems.
  • To enhance the efficiency and accuracy of finding initial conditions with long escape times.
  • To investigate the performance of these methods in high-dimensional systems.

Main Methods:

  • Developed an exponential search domain method, scaling with the largest Lyapunov exponent (λ₁).

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  • Introduced an anisotropic search domain method utilizing singular values of the Jacobian matrix.
  • Compared performance against the Stagger-and-Step method using coupled Hénon maps.

    • Main Results:

    • Both proposed methods outperform the Stagger-and-Step method.
    • The anisotropic method demonstrates efficiency independent of escape time (τ) in high dimensions.
    • Simulations were conducted in systems up to 24 dimensions with multiple positive Lyapunov exponents.

    Conclusions:

    • The new numerical methods provide significant improvements for approximating non-attracting sets.
    • The anisotropic method offers a robust approach for high-dimensional transient chaos.
    • These findings suggest potential for characterizing non-attracting sets in complex spatio-temporal systems.