In 1974, Arthur T. Winfree introduced the notion of isochrons when studying biological oscillators' response to perturbations [1]. Oscillators are physical or abstract systems that evolve in time periodically passing through or nearby a reference state. The time needed to browse the circle of states is the period of the oscillator. Therefore, if you take a reference state, you can define another state by a fraction of the period. This number is called the phase of the oscillator. This is the case of the point which belongs to
the cycle but the system can have other states, perturbed states. For example, one can pull a maintained
spring till reaching a point B greater than the spring maximal elongation in normal conditions.
This new point B can also be described by a period fraction if we extend the notion of phase to latent phase.
The latent phase of the perturbed point B is equal to the phase of a point A on the cycle, if two identical springs one starting from A and the other from B tend to synchronize after some time.
Oscillator 1 (black) starts from A and oscillator 2 (red) starts from B.
After, some time the two springs synchronize.
Original link: http://www.scholarpedia.org/article/Isochron
The set of all the perturbed points having the same latent phase is called an isochron. This set
describes a hyper-surface : a curve in the case of 2D system and a suface in the case of 3D system. This was proved in [2] by Guckenheimer...
Bibliography
[1] Winfree, A.T. : Patterns of phase compromise in biological cycles. J
Math Biol 1 (1974) 73–95
[2] Guckenheimer, J. : Isochrons and phaseless sets. J Math Biol 1 (1975)
259–273
[3] The isochronal fibration : Characterization and impli-
cation in biology. Hedi Ben Amor, Nicolas Glade, Claudio Lobos,
Jacques Demongeot, Acta Biotheoretica (2010) 58, 121142.
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