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

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving01:30

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...
Hyperbolic Functions01:26

Hyperbolic Functions

A flexible cable suspended between two points at the same height naturally forms a curve known as a catenary. This shape results from the balance between the cable’s weight and the tension acting along its length, representing a state of mechanical equilibrium. Unlike simpler approximations, the true shape of a hanging cable is described using hyperbolic functions.Hyperbolic functions are closely related to exponential functions and are named for their connection to the geometry of the...
The Squeeze Theorem01:30

The Squeeze Theorem

Certain mathematical functions exhibit unpredictable or highly variable behavior near specific input values, making direct evaluation of their limits challenging. This complexity may arise from rapid oscillations or irregular patterns that obscure the function’s trend. In such cases, the Squeeze Theorem offers a reliable method for determining limits.According to the Squeeze Theorem, if a function is confined between two other functions near a particular point, and both outer functions approach...
Turbulent Flow01:24

Turbulent Flow

Turbulent flow is characterized by unpredictable fluctuations in velocity and pressure, which result in a chaotic fluid movement distinct from the orderly patterns of laminar flow. While laminar flow is governed by smooth, parallel layers with minimal mixing, turbulent flow exhibits highly irregular, three-dimensional patterns. This behavior arises due to instabilities in the fluid's velocity profile, and amplifies as the flow velocity increases. Minor disturbances, known as turbulent spots,...
Limits with Oscillating Discontinuities01:19

Limits with Oscillating Discontinuities

An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the most...
The Entropy as a State Function01:14

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...

You might also read

Related Articles

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

Sort by
Same author

Effect of Ion Channel Randomness on Sensitivity of Neurons to External Electromagnetic Fields: Computational Study.

Entropy (Basel, Switzerland)·2026
Same author

Coherence properties of collective modes in ensembles of oscillators.

Physical review. E·2026
Same author

Neural-network maps for two-parameter modeling of bistability and codimension-two bifurcations in two-dimensional flow dynamical systems.

Chaos (Woodbury, N.Y.)·2026
Same author

Comment on "Hydrodynamic Equations for Space-Inhomogeneous Aggregating Fluids with First-Principle Kinetic Coefficients".

Physical review letters·2025
Same author

Internal reliability and antireliability in dynamical networks.

Physical review. E·2025
Same author

Multiscale energy spreading in hard-particle chains.

Physical review. E·2025
Same journal

Erratum: Bacterial Turbulence at Compressible Fluid Interfaces [Phys. Rev. Lett. 136, 138301 (2026)].

Physical review letters·2026
Same journal

Unveiling Light-Quark Yukawa Flavor Structure via Dihadron Fragmentation at Lepton Colliders.

Physical review letters·2026
Same journal

Adaptable Route to Fast Coherent State Transport via Bang-Bang-Bang Protocols.

Physical review letters·2026
Same journal

Topological Transition and Emergence of Elasticity of Dislocation in Skyrmion Lattice: Beyond Kittel's Magnetic-Polar Analogy.

Physical review letters·2026
Same journal

Pound-Drever-Hall Method for Superconducting-Qubit Readout.

Physical review letters·2026
Same journal

Coupling a ^{73}Ge Nuclear Spin to an Electrostatically Defined Quantum Dot in Silicon.

Physical review letters·2026
See all related articles

Related Experiment Video

Updated: May 18, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Hyperbolic chaos of Turing patterns.

Pavel V Kuptsov1, Sergey P Kuznetsov, Arkady Pikovsky

  • 1Department of Instrumentation Engineering, Saratov State Technical University, Politekhnicheskaya 77, Saratov 410054, Russia. p.kuptsov@rambler.ru

Physical Review Letters
|September 26, 2012
PubMed
Summary
This summary is machine-generated.

We reveal that external modulation of Turing patterns leads to chaotic dynamics. This chaos, governed by an expanding circle map, is robust and exhibits a strange attractor.

More Related Videos

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

Related Experiment Videos

Last Updated: May 18, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

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

Area of Science:

  • Nonlinear dynamics
  • Pattern formation
  • Mathematical biology

Background:

  • Turing patterns are fundamental to understanding pattern formation in biological and chemical systems.
  • External periodic parameter modulation can significantly alter the dynamics of these patterns.

Purpose of the Study:

  • To investigate the time evolution of Turing patterns under periodic parameter modulation.
  • To characterize the emergent chaotic dynamics and their underlying mechanisms.

Main Methods:

  • Theoretical analysis using an extended Swift-Hohenberg equation.
  • Numerical simulations to observe pattern evolution.
  • Stroboscopic observation of spatial phases.

Main Results:

  • Successive emergence of longwave and shortwave Turing patterns with a 1:3 length scale ratio.
  • Spatial phases are governed by an expanding circle map, indicating hyperbolic chaos.
  • The chaotic dynamics are associated with a Smale-Williams solenoid strange attractor.

Conclusions:

  • Periodic modulation induces robust hyperbolic chaos in Turing patterns.
  • The expanding circle map provides a theoretical framework for understanding this chaos.
  • The findings offer insights into complex pattern dynamics in extended systems.