Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

134
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,...
134
Construction of Root Locus01:15

Construction of Root Locus

189
The construction of a root locus involves several key steps to analyze and visualize the behavior of a system's poles with varying gain. The number of branches in the root locus equals the number of closed-loop poles and is symmetrical about the real axis.
For positive gain values, the root locus exists on the real axis to the left of an odd number of finite open-loop poles or zeros. The root locus starts at the open-loop poles and traces the paths of the closed-loop poles as the gain...
189
Second Order systems II01:18

Second Order systems II

195
In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
195
Second Order systems I01:20

Second Order systems I

271
A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which...
271
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

112
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
112
Oscillations about an Equilibrium Position01:04

Oscillations about an Equilibrium Position

5.7K
Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. For an unstable equilibrium point, if the object is disturbed slightly, it will not return to the equilibrium point. There are three conditions for equilibrium points—stable, unstable, and half-stable. A half-stable equilibrium point is also unstable, but is named so...
5.7K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Nonlinear periodic orbit solutions and their bifurcation structure at the origin of soliton hopping in coupled microresonators.

Communications physics·2026
Same author

High sensitivity differential microwave sensor based on SIW technology for measuring the permittivity and thickness of the solid materials.

Scientific reports·2026
Same author

High sensitivity sensor for differential measurement of thickness of thin substrates using double T-shaped resonators.

Scientific reports·2025
Same author

Ghost states underlying spatial and temporal patterns: How nonexistent invariant solutions control nonlinear dynamics.

Physical review. E·2025
Same author

Data-driven guessing and gluing of unstable periodic orbits.

Physical review. E·2025
Same author

The topology of a chaotic attractor in the Kuramoto-Sivashinsky equation.

Chaos (Woodbury, N.Y.)·2025

Related Experiment Video

Updated: Oct 2, 2025

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

9.1K

Constructing periodic orbits of high-dimensional chaotic systems by an adjoint-based variational method.

Sajjad Azimi1, Omid Ashtari1, Tobias M Schneider1

  • 1Emergent Complexity in Physical Systems Laboratory (ECPS), École Polythechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland.

Physical Review. E
|February 23, 2022
PubMed
Summary

Researchers developed a novel matrix-free method to compute unstable periodic orbits in chaotic systems. This approach offers robust convergence for high-dimensional spatiotemporal chaos, aiding in predicting fluid turbulence dynamics.

More Related Videos

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions
11:51

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Published on: February 22, 2018

8.8K
The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
12:14

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Published on: August 12, 2013

22.0K

Related Experiment Videos

Last Updated: Oct 2, 2025

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

9.1K
Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions
11:51

Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions

Published on: February 22, 2018

8.8K
The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
12:14

The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

Published on: August 12, 2013

22.0K

Area of Science:

  • Applied Mathematics
  • Computational Physics
  • Dynamical Systems Theory

Background:

  • Chaotic dynamics in complex systems are underpinned by unstable periodic orbits (UPOs).
  • Accurate computation of UPOs is crucial for understanding and predicting chaotic phenomena like fluid turbulence.
  • Existing methods for computing UPOs in high-dimensional systems suffer from poor convergence or high computational cost.

Purpose of the Study:

  • To introduce a new, efficient, and robust method for computing UPOs in high-dimensional spatiotemporally chaotic systems.
  • To overcome the limitations of existing computational techniques, specifically addressing poor convergence and computational expense.

Main Methods:

  • A novel matrix-free, adjoint-based variational method is proposed.
  • This method constructs an initial value problem in the space of closed loops, making UPOs attracting fixed points.
  • The approach is demonstrated on the one-dimensional Kuramoto-Sivashinsky equation, a model for spatiotemporal chaos.

Main Results:

  • The proposed method exhibits robust convergence for computing UPOs.
  • It is unaffected by exponential error amplification common in time-marching methods.
  • Convergence is independent of the orbit's period and does not require accurate initial guesses.

Conclusions:

  • The new matrix-free method provides a powerful framework for analyzing chaotic dynamics in high-dimensional systems.
  • This advancement facilitates the prediction of statistics for chaotic flows, including fluid turbulence.
  • The method's global convergence and efficiency make it applicable to a wide range of complex dynamical systems.