Ordinary Differential Equations: Basics and Beyond by David G. Schaeffer & John W. Cain

Author:David G. Schaeffer & John W. Cain
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(7.7)

then there are “phase-locked” or “entrained” solutions of (7.6) of the form

(7.8)

where are constants. Note that . Thus, (7.6) is satisfied if and only if

(7.9)

Subtracting these equations and rearranging, we obtain

Condition (7.7) guarantees that this equation has real solutions. Ignoring possible translation by 2π, we pick two of these, say α c and α d , where

(7.10)

(The subscript “c” is a mnemonic for “close,” since in this case, the runners are closer to each other; similarly, “d” is a mnemonic for “distant.”) Given the value of sinα, we substitute into (7.9) to obtain

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