**Abstract**

**The****The subject of the study was a chaotic particle motion initiated by a nonlinear periodic force in a mechanical system. The appearance of chaotic dynamics in simple mechanical systems, is a fundamental classical phenomena.**

*objective:*** Methods/Materials**

Non-linear oscillations were simulated through computer modeling. The computer model was realized as a sequence of matrix multiplication. Each matrix describes linear and non-linear oscillations of a mechanical body under external forces.

** Results**

The result of the matrix multiplication was a chaotic motion performed by a particle in the presence of linear and non-linear forces.

** Conclusions/Discussion**

This type of particle action is called the "beam-beam effect." It is one of the main reasons that limits the intensity of a particle collider.

The project is the study of particle motion inside a collider under external linear and nonlinear forces.

**Science Fair Project done By Konstantin Y. Batygin **