Abstract
In this thesis, we will give the detail study of the Van-der Pol equation. The existence of the unique asymptotically stable limit cycle will be carefully carried out. Then the method of singular perturbation is used when small parameter is involved. Some examples were given to show how the method can be applied. Also, the coupling of two nearly identical Van-der Pol equations with Neumann boundary condition was studied. We found that when the noise is large the limit cycles of each system are not synchronized. However, when one increases the coupling strengths the system process synchronized phenomena.
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