Dynamical Systems and Stability: From Basics to Applications

Stability and equilibria, phase space analysis, and real-world applications

What you’ll learn
✓ Understand the evolution of systems over time, state-space representation, and types of dynamical systems..
✓ Determine equilibrium points and assess their stability using linearization and qualitative methods.
✓ Describe the contributions of key figures and understand the development of nonlinear dynamics, chaos, and bifurcations.
✓ Visualize system behavior through phase portraits, flows, and trajectories, and predict qualitative system behavior without solving equations analytically.
✓ Apply dynamical system concepts to real-world problems in physics, engineering, biology, and ecology.

Requirements
● Basics of Differential Equations, Linear Algebra.

Description
This course introduces students to the fascinating world of dynamical systems and stability theory, bridging mathematical foundations with real-world applications. A dynamical system is any process that evolves over time according to a set of rules, and such systems are found in physics, biology, engineering, economics, and even social sciences. Students will begin by exploring the historical development of the subject, from Newton’s deterministic laws of motion to Poincaré’s qualitative insights, and gain an appreciation of how nonlinear dynamics emerged as a modern discipline.

The course emphasizes the concepts of equilibrium points, stability analysis, and state-space representations, equipping students with tools to interpret and predict system behavior without always relying on exact solutions. Phase portraits, flows, and trajectories will be used to develop an intuitive understanding of local and global dynamics. Special attention will be given to bifurcation theory, which explains how small changes in parameters can cause dramatic shifts in system behavior — such as the onset of oscillations, chaos, or sudden collapses.

Applications are highlighted throughout, including predator–prey interactions in ecology, epidemic models in biology, oscillations in engineering, and stability problems in physics. By the end of the course, students will be able to connect rigorous mathematical analysis with practical phenomena, preparing them for advanced study and interdisciplinary research.

Who this course is for
■ Graduate Students, Masters Students

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