The article states an approach for developing a distributed application architecture for dynamic systems analysis in a hybrid cloud. The problem that was posed as an example is based on the calculation of the Lyapunov characteristic exponents for trajectories of dynamical systems in phase space. Parallelization of the computational process is achieved by supplying the list of running tasks for computing Lyapunov characteristic exponents, depending on the result of finished tasks during calculations. We have implemented a distributed application to make this experiment automated. The architecture of application elements is established on the principles of the enterprise desktop grid and cloud systems. A vital detail of the application architecture is the deployment process based on a cloud hybrid computing environment using the Jupyter server running on the DigitalOcean cloud platform and the Everest cloud platform simultaneously with the Maple package.