Related Experiment Video
Updated: Jul 7, 2026
06:42
Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
Multiresonant forcing of the complex Ginzburg-Landau equation: pattern selection
Jessica M Conway1, Hermann Riecke
1Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 1, 2008
Summary
We investigated spatial patterns in systems near oscillations. For small amplitudes, stripes and hexagons are stable, but larger amplitudes lead to complex, bistable patterns not predicted by simple models.
Area of Science:
- Nonlinear dynamics
- Pattern formation
- Bifurcation theory
Background:
- Systems near a Hopf bifurcation can exhibit spatially homogeneous oscillations.
- Multifrequency forcing can excite complex spatial patterns.
Purpose of the Study:
- To analyze spatial patterns arising from resonant, multifrequency forcing near a Hopf bifurcation.
- To investigate the stability of different spatial patterns (stripes, hexagons, rectangles, super-hexagons) under varying forcing amplitudes.
Main Methods:
- Third-order weakly nonlinear analysis.
- Numerical simulations.
Main Results:
- Linear stability analysis predicts stripe and hexagon patterns for small amplitudes.
- For larger amplitudes, weakly nonlinear analysis suggests rectangles and super-hexagons, but simulations show super-hexagons are unstable.
- Numerical simulations reveal large-amplitude hexagons can emerge and coexist (be bistable) with weakly nonlinear hexagons.
Conclusions:
- Weakly nonlinear analysis is insufficient for describing pattern formation at larger amplitudes.
- Complex spatio-temporal dynamics and bistability arise in systems driven by multifrequency forcing near a Hopf bifurcation.
Related Concept Videos
01:12Double Resonance Techniques: Overview
Double resonance techniques in Nuclear Magnetic Resonance (NMR) spectroscopy involve the simultaneous application of two different frequencies or radiofrequency pulses to manipulate and observe two distinct nuclear spins. One important application of double resonance is spin decoupling, which selectively suppresses coupling with one type of nucleus while observing the NMR signal from another nucleus, simplifying the spectrum and enhancing resolution.
Spin decoupling is usually achieved by...
Spin decoupling is usually achieved by...
01:06Forced Oscillations
When an oscillator is forced with a periodic driving force, the motion may seem chaotic. The motions of such oscillators are known as transients. After the transients die out, the oscillator reaches a steady state, where the motion is periodic, and the displacement is determined.
02:16Resonance and Hybrid Structures
According to the theory of resonance, if two or more Lewis structures with the same arrangement of atoms can be written for a molecule, ion, or radical, the actual distribution of electrons is an average of that shown by the various Lewis structures.
Resonance Structures and Resonance Hybrids
The Lewis structure of a nitrite anion (NO2−) may actually be drawn in two different ways, distinguished by the locations of the N–O and N=O bonds.
Resonance Structures and Resonance Hybrids
The Lewis structure of a nitrite anion (NO2−) may actually be drawn in two different ways, distinguished by the locations of the N–O and N=O bonds.
01:23Parallel Resonance
The parallel RLC circuit is an arrangement where the resistor (R), inductor (L), and capacitor (C) are all connected to the same nodes and, as a result, share the same voltage across them. The parallel RLC circuit is analyzed in terms of admittance (Y), which reflects the ease with which current can flow. The admittance is given by:
01:12Fermi Level Dynamics
The vacuum level denotes the energy threshold required for an electron to escape from a material surface. It is usually positioned above the conduction band of a semiconductor and acts as a benchmark for comparing electron energies within various materials.
Electron affinity in semiconductors refers to the energy gap between the minimum of its conduction band and the vacuum level and it is a critical parameter in determining how easily a semiconductor can accept additional electrons.
The work...
Electron affinity in semiconductors refers to the energy gap between the minimum of its conduction band and the vacuum level and it is a critical parameter in determining how easily a semiconductor can accept additional electrons.
The work...
02:42Valence Bond Theory
Coordination compounds and complexes exhibit different colors, geometries, and magnetic behavior, depending on the metal atom/ion and ligands from which they are composed. In an attempt to explain the bonding and structure of coordination complexes, Linus Pauling proposed the valence bond theory, or VBT, using the concepts of hybridization and the overlapping of the atomic orbitals. According to VBT, the central metal atom or ion (Lewis acid) hybridizes to provide empty orbitals of suitable...