Difference between Index Theory and Lyapunov Stability Analysis for Planar Systems

📚 Introduction to Index Theory and Lyapunov Stability

Index Theory and Lyapunov Stability Analysis are both powerful tools for understanding the behavior of dynamical systems, particularly planar systems (systems described by two variables). However, they address different aspects of stability and use distinct mathematical approaches. Let's explore their differences.

🔑 Core Concepts

• 🔍 Index Theory: Focuses on the topological properties of vector fields, particularly around isolated critical points (equilibrium points). It calculates the index of a critical point, which is an integer that describes how trajectories wind around that point.
• 🌱 Lyapunov Stability Analysis: Examines the stability of equilibrium points based on the existence of a Lyapunov function. A Lyapunov function is a scalar function that decreases along trajectories near the equilibrium point, indicating stability.

🧮 Mathematical Formulation

• 📐 Index Theory: The index $I$ of a critical point $x_0$ is defined as the winding number of the vector field around a small closed curve enclosing $x_0$. Formally, if $F(x, y) = (P(x, y), Q(x, y))$ is the vector field, then $I = \frac{1}{2\pi} \oint_C d(\arctan(\frac{Q}{P}))$, where $C$ is the closed curve.
• 🌡️ Lyapunov Stability Analysis: Requires finding a Lyapunov function $V(x, y)$ such that $V(x_0) = 0$, $V(x, y) > 0$ for $x \neq x_0$ in a neighborhood of $x_0$, and $\dot{V}(x, y) \leq 0$ along trajectories of the system. If $\dot{V}(x, y) < 0$, the equilibrium is asymptotically stable.

⚖️ Differences in Approach

• 🧭 Index Theory: Is a qualitative approach relying on the geometry of the vector field. It helps classify the nature of critical points (e.g., saddles, nodes, spirals) based on their index. It does *not* require explicit solutions of the system.
• 📈 Lyapunov Stability Analysis: Is a more quantitative approach. It requires constructing a suitable Lyapunov function, which can sometimes be challenging. It directly addresses the stability of the equilibrium point.

🎯 When to Use Which

• 📍 Index Theory: Useful when you want to understand the global behavior of the system, classify critical points, and determine the existence of periodic orbits (via the Poincaré-Hopf Index Theorem).
• ✅ Lyapunov Stability Analysis: Ideal when you want to rigorously prove the stability of an equilibrium point. It provides a direct way to assess stability based on the properties of the Lyapunov function.

📉 Limitations

• 🚧 Index Theory: Does not provide information about the rate of convergence to an equilibrium point or the size of the region of attraction.
• ⚠️ Lyapunov Stability Analysis: Finding a suitable Lyapunov function can be difficult or impossible for some systems. The method can be conservative, meaning it might fail to prove stability even when the system is stable.

💡 Example: Planar System

Consider the system $\dot{x} = -y - x^3$, $\dot{y} = x - y^3$.

• 🔎 Index Theory: The origin (0,0) is an isolated critical point. The index of the origin can be determined by analyzing the vector field around it. In this case, the index is +1, suggesting a stable or unstable node or spiral.
• 🧪 Lyapunov Stability Analysis: A Lyapunov function can be $V(x, y) = \frac{1}{2}(x^2 + y^2)$. Then $\dot{V}(x, y) = x\dot{x} + y\dot{y} = x(-y - x^3) + y(x - y^3) = -x^4 - y^4 \leq 0$. This demonstrates that the origin is Lyapunov stable. Since $\dot{V} < 0$ except at (0,0), the origin is asymptotically stable.

🧮 Summary Table