Construct · 3 fields · 4 citations

limit cycle: one construct, many systems

One name. One construct. Different systems. One map.

● invariant · one construct, three systems Nearest construct by meaning·not indexed
the nerve impulse as a self-sustained oscillationpredator-prey cycles as an isolated periodic orbitlimit cycle3 SYSTEMSLimit cycleVAN DER POL · 1926Neural limit cycleFITZHUGH · 1961Population limit cycleMAY · 1972

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1926Limit cycle1961Neural limit cycle1972Population limit cycle
2026

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Nonlinear dynamics

An isolated closed orbit a system settles onto on its own

In nonlinear dynamics a limit cycle is an isolated closed trajectory in a system's state space onto which nearby trajectories spiral, producing a self-sustained oscillation whose amplitude and period are set by the system itself rather than by initial conditions or external forcing. Van der Pol's 1926 On relaxation-oscillations gave the archetype in a vacuum-tube circuit, and Strogatz's Nonlinear Dynamics and Chaos sets the construct in the modern theory of stable periodic orbits. The referent is a stable periodic solution of a dissipative nonlinear system, the identical mathematical object that reappears in firing neurons and oscillating populations.[1][2]

Unit
The amplitude and period of the isolated stable periodic orbit
Recovery means
Not applicable
Measured by
Established by a closed orbit that attracts neighbouring trajectories, with amplitude and period fixed by the system's parameters

Neurophysiology

The repetitive nerve impulse as a stable oscillation

In neurophysiology, repetitive firing of an excitable cell is a limit cycle: once a stimulus pushes the membrane past threshold, the sodium and potassium dynamics carry it around a closed orbit of depolarisation and recovery, generating a train of impulses at a self-set rate. FitzHugh's 1961 Impulses and Physiological States in Theoretical Models of Nerve Membrane reduced the Hodgkin-Huxley system to two variables and exhibited exactly this orbit in the phase plane. It is the same isolated-periodic-orbit construct as the van der Pol oscillator, realised in a nerve membrane and measured by firing rate rather than circuit amplitude.[3]

Unit
The firing rate and trajectory of the repetitive-firing orbit
Recovery means
Not applicable
Measured by
Seen as a closed orbit in the membrane phase plane, characterised by firing frequency and the stimulus threshold for sustained firing

Population ecology

Predator and prey locked in sustained cycles

In population ecology a limit cycle is a stable, self-sustained oscillation in the abundances of interacting species (classically predator and prey) toward which the system returns after disturbance, as distinct from the neutral cycles of the simplest Lotka-Volterra model. May's 1972 Limit Cycles in Predator-Prey Communities showed that realistic density dependence produces genuine attracting cycles. It is the same isolated-periodic-orbit construct as in a circuit or a neuron, here carried by birth-death interactions and measured by the amplitude and period of abundance over time.[4]

Unit
The amplitude and period of the sustained abundance oscillation
Recovery means
Not applicable
Measured by
Detected as an attracting oscillation in species abundances that persists and recovers after perturbation

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Nearby constructs

equilibrium4 fields · 4 citationsjingle · one word, four constructsamplitude4 fields · 4 citationsjingle · one word, four constructscarry1 fields · 1 citationsanchor · a standalone construct

References

  1. Van der Pol, B. (1926). On “relaxation-oscillations”. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11), 978–992. https://doi.org/10.1080/14786442608564127
  2. Strogatz, S. H. (2015). Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering (2nd ed.). Westview Press. ISBN 9780813349107
  3. FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445–466. https://doi.org/10.1016/S0006-3495(61)86902-6
  4. May, R. M. (1972). Limit cycles in predator-prey communities. Science, 177(4052), 900–902. https://doi.org/10.1126/science.177.4052.900

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