A periodic function also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a cycle. For example, the trigonometric functions, which repeat at intervals of
2
π
{\displaystyle 2\pi }
radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.
a) Proof:
Consider the system ## \dot{x}=x+x^3-y^2 ## and ## \dot{y}=x^2-x^4+y^5 ##.
By theorem, Bendixson's negative criterion states that there are no closed paths in a simply connected region of the phase plane on which ## \frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y} ## is of...