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Unformatted text preview: 1 Phase Oscillators Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 2 Phase Oscillators Oscillations occur everywhere in biology, from the level of gene expression to the level of animal populations. This ubiq uity motivated Art Winfree to describe an oscillator as simply as possible (1967). His work was further developed by others, most notably by Kuramoto (1976). To set the stage, consider the system of equations _ x = ( r ) x ( r ) y (1) _ y = ( r ) x + ( r ) y (2) where r 2 = x 2 + y 2 . Systems of this form are called lambda omega systems . They are special, since if we change to polar coordinates, x = r cos , y = r sin , the ODEs can be written as _ r = r ( r ) (3) _ = ( r ) . (4) This system has a circular limit cycle at any radius r > 0 for which ( r * ) = 0, since _ r = 0 in this case. Is this a stable limit 3 cycle? d dr r ( r ) = ( r ) + r (5) and at r * , ( r * ) + r * ( r * ) = 0 + r * ( r * ) < (6) so ( r * ) < 0. Hence, the limit cycle at r = r * is stable if and only if ( r * ) < 0. x y r * The periodic solution travels around the circle with angu lar velocity of ( r * ). If the limit cycle is stable, then starting from any initial conditions (except the origin), the solution will eventually settle onto a regular oscillation with xed amplitude and period 2 ( r * ) . Hence, in the limit as t the system is described completely by its angular velocity around a circle. In this limit the system is called a phase oscillator , and it is 4 a reduction of the planar system to a 1dimensional ow on a circle. 5 Entrainment of Fire ies The following example comes from Strogatz (1994). In one species of southeast Asian re ies the males gather in trees at night and begin ashing. Di erent ies ash at di erent fre quencies when in isolation, and if they did this as a group then ashing in the trees would be continuous. However, periodic ashing is what is actually observed. This indicates that the re ies synchronize their activity. This observation motivated experiments in the 1970's in which a ashlight was used to entrain ies to the ashing period of the ashlight. Represent the ashing of the ashlight as a uniform phase oscillator : _ = (7) where is a constant. Represent the re y ashing as a nonuniform phase oscillator : _ = + A sin(  ) (8) where is the natural frequency of the re y and A is the coupling strength (assume that A > 0). If > , then 6 A sin(  ) > 0 and the re y speeds up, trying to catch up to the ashlight. flash De ne the phase difference as  . Then _ = _  _ (9) =   A sin . (10) The 3 parameters ( , , and A ) can be replaced by a single parameter by nondimensionalizing the system: t has units of seconds , , and A have units of radians per second....
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 Fall '07
 Zhang
 Calculus

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