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Boundary Conditions: Lossless Lines01:21

Boundary Conditions: Lossless Lines

Consider a single-phase, two-wire, lossless transmission line terminated by an impedance at the receiving end and a source with Thevenin voltage and impedance at the sending end. The line, with length, has a surge impedance and wave velocity determined by the line's inductance and capacitance.
At the receiving end, the boundary condition states that the voltage equals the product of the receiving-end impedance and current. This relationship is expressed as a function of the incident and...
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Line Section Model
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Related Experiment Video

Updated: May 10, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

Dynamic boundary crisis in the Lorenz-type map.

Oleg V Maslennikov1, Vladimir I Nekorkin

  • 1Institute of Applied Physics of RAS, 603950, 46 Ulyanov Str., Nizhny Novgorod, Russia. olmaov@neuron.appl.sci-nnov.ru

Chaos (Woodbury, N.Y.)
|July 5, 2013
PubMed
Summary

This study extends dynamic bifurcations to chaos, revealing delayed exit from chaotic regions in Lorenz-type maps. Increasing control parameter rates enhance this delay, impacting trajectory survival.

Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Dynamical Systems

Background:

  • Previous research detailed bifurcations of equilibria and limit cycles under slowly varying control parameters.
  • Dynamic bifurcations, a concept analyzing system changes, have not been extensively applied to chaotic phenomena.

Purpose of the Study:

  • To extend the concept of dynamic bifurcations to chaotic systems.
  • To investigate the dynamic boundary crisis of a chaotic attractor in a Lorenz-type map.
  • To analyze the effects of control parameter change rate on chaotic dynamics.

Main Methods:

  • Analysis of a Lorenz-type map.
  • Investigation of dynamic bifurcations under parameter variations.
  • Examination of chaotic attractor boundary crisis.

Related Experiment Videos

Last Updated: May 10, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

Main Results:

  • Discovery of delayed exit from the chaotic region as a control parameter crosses a critical value.
  • Observation and analysis of non-exponential decay in surviving trajectory points.
  • Demonstration and explanation of increased delay with a higher rate of control parameter change.

Conclusions:

  • Dynamic bifurcations can be effectively extended to chaotic phenomena.
  • The rate of control parameter change significantly influences the dynamics of chaotic attractors, specifically the exit time and trajectory survival.
  • The findings provide new insights into the behavior of complex systems near crisis events.