Related Experiment Video
Updated: Jul 18, 2026
08:32
Assembly and Characterization of an External Driver for the Generation of Sub-Kilohertz Oscillatory Flow in Microchannels
Published on: January 28, 2022
Archetypal oscillator for smooth and discontinuous dynamics.
Qingjie Cao1, Marian Wiercigroch, Ekaterina E Pavlovskaia
1Centre for Applied Dynamics Research, Department of Engineering, University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotland, United Kingdom.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 13, 2006
Summary
We introduce a model system to study smooth to discontinuous dynamics. At the discontinuous limit, velocity flow jumps due to lost hyperbolicity, leading to unique attractors and chaotic behavior.
Area of Science:
- Nonlinear Dynamics
- Chaos Theory
- Mathematical Physics
Background:
- The Duffing oscillator models systems with double-well potentials and hyperbolic stationary states.
- Understanding transitions between smooth and discontinuous dynamics is crucial in nonlinear systems.
- Local hyperbolicity and manifold structures govern system behavior near stationary states.
Purpose of the Study:
- To develop an archetypal system for investigating transitions from smooth to discontinuous dynamics.
- To analyze the departure from standard dynamics at the discontinuous limit.
- To explore the impact of damping and external excitation on system attractors.
Main Methods:
- Analysis of an archetypal system exhibiting a transition from smooth to discontinuous dynamics.
- Investigation of the hyperbolic structure associated with the double-well stationary state.
- Examination of velocity flow behavior and local hyperbolicity loss.
Main Results:
- The system exhibits standard dynamics similar to the Duffing oscillator in the smooth regime.
- At the discontinuous limit, velocity flow jumps occur due to the loss of local hyperbolicity.
- Coexisting attractors, chaotic saddles, and chaotic attractors emerge with damping and excitation.
- The chaotic attractor can bifurcate into periodic or quasiperiodic attractors depending on damping strength.
Conclusions:
- The proposed system effectively models transitions to discontinuous dynamics.
- Loss of local hyperbolicity is the key mechanism driving discontinuous behavior.
- Damping and excitation significantly influence the complex dynamics and attractor structures observed.
Related Concept Videos
01:19Limits 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...
01:07Damped Oscillations
In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...
Although friction and other non-conservative...
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.
01:04Oscillations about an Equilibrium Position
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 because...
01:31Oscillations In An LC Circuit
An idealized LC circuit of zero resistance can oscillate without any source of emf by shifting the energy stored in the circuit between the electric and magnetic fields. In such an LC circuit, if the capacitor contains a charge q before the switch is closed, then all the energy of the circuit is initially stored in the electric field of the capacitor. This energy is given by
01:20Types of Damping
If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...