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# multiscale - 1 Multiscale Oscillations Richard Bertram...

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Unformatted text preview: 1 Multiscale Oscillations Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 2 Relaxation Oscillations Relaxation oscillations are observed in biochemical re- actions, in cardiac tissue (very wide action potentials), and models of excitable membranes often produce this type of os- cillation. It is characterized by a very slow buildup in the pri- mary variable (like voltage), followed by a sudden discharge, repeated periodically. The name is used because the stress ac- cunulated during the slow buildup is relaxed during the sudden discharge. A well-analyzed model that can produce relaxation oscilla- tions is the van der Pol equation , x + μ ( x 2- 1) _ x + x = 0 (1) which describes a harmonic oscillator with a nonlinear friction term, μ ( x 2- 1) _ x , where μ is the friction coe cient. We will look at the strongly nonlinear case μ 0. We begin the analysis by converting to a system of rst- order ODEs. Note that x + μ _ x ( x 2- 1) = d dt _ x + μ ( 1 3 x 3- x ) . (2) 3 Let F ( x ) = 1 3 x 3- x , and de ne w as w ≡ _ x + μF ( x ). Then from Eq. 1, _ w + x = 0 . (3) Altogether, the system becomes _ x = w- μF ( x ) (4) _ w =- x . (5) Now make the variable change y ≡ w μ , obtaining the van der Pol system in Li enard coordinates : _ x = μ [ y- F ( x )] (6) _ y =- x μ . (7) The x-nullcline is y = F ( x ) = 1 3 x 3- x , a cubic curve, and the y-nullcline is x = 0. The trajectory follows the right and left branches of the cubic x-nullcline. It moves slowly while on these branches and jumps quickly between branches. The slow movement along the nullcline branches re ects the slow y time scale ( μ is large so _ y is small), while the fast jumps re ect the time scale of the faster x variable ( μ is large so _ x is large). 4 The x time course is a square wave, while the y time course is a saw tooth. x y B A Figure 1: The x-nullcline (red), y-nullcline (green), and limit cycle trajectory (blue) for the van der Pol relaxation oscillation. To be more precise, suppose that the initial condition is not too close to the cubic nullcline. That is, suppose that y- F ( x ) = O (1). Then | _ x | = O ( μ ) (8) | _ y | = O ( μ- 1 ) . (9) 5 t x y t 2/3-4 4-2/3 Hence, the velocity in the horizontal direction is much greater than that in the vertical direction, and the phase point moves almost horizontally toward the x-nullcline. The time required for this jump is the inverse of the rate, so jump time = O ( μ- 1 ). Once the trajectory gets close to the nullcline it reaches a point where y- F ( x ) = O ( μ- 2 ), and then | _ x | = O ( μ- 1 ) (10) | _ y | = O ( μ- 1 ) . (11) The phase point crosses the nullcline and moves slowly down 6 the back side of the right branch with a velocity of magnitude O ( μ- 1 ), which is slow. This continues until the knee is reached and then a jump occurs to the left branch of the x-nullcline....
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## This note was uploaded on 11/27/2011 for the course MAC 2312 taught by Professor Zhang during the Fall '07 term at FSU.

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multiscale - 1 Multiscale Oscillations Richard Bertram...

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