Bifurcation Analysis in Dynamical Systems
Consider the one-dimensional autonomous differential equation \(\frac{dx}{dt} = rx – x^3\), where \(r\) is a real parameter. Analyze the system with respect to changes in \(r\).
Global Phase Portrait Analysis of a Dynamical System
Consider the autonomous differential equation \( \frac{dx}{dt} = x^3 – 4x^2 + 3x \). Analyze the system by addressing the following parts: (a) Find all equilibrium points. (b) Determine the stability (stable/unstable) of each equilibrium point. (c) Summarize the overall behavior of solutions and describe the global phase portrait.
Introduction to Phase Portraits
Define a phase portrait in the context of dynamical systems. In your answer, clearly describe the information it provides about the behavior of solutions and explain one key advantage of using phase portraits to analyze system stability.