Common Mistakes When Calculating the Poincaré Index for Planar Systems

📚 Understanding Common Mistakes in Poincaré Index Calculation

The Poincaré index is a crucial concept in dynamical systems, providing insights into the behavior of trajectories around singular points in a vector field. However, its calculation can be tricky, leading to common errors. This guide will help you avoid these pitfalls.

🎯 Objectives

• 🧭 Learn to correctly identify singular points in a planar system.
• 🔄 Understand the importance of orientation when calculating the Poincaré index.
• 🔢 Avoid common arithmetic errors in index calculation.

🧰 Materials

• Graph paper
• Pencils/Pens
• Equations of planar systems

🚀 Warm-up (5 mins)

Review the definitions of singular points and the concept of a closed trajectory. Briefly discuss how trajectories behave around different types of singular points (e.g., nodes, saddles, spirals).

👨‍🏫 Main Instruction

Here's a breakdown of common mistakes and how to avoid them:

• 🧭 Incorrectly Identifying Singular Points: A singular point $(x_0, y_0)$ must satisfy $F(x_0, y_0) = 0$ and $G(x_0, y_0) = 0$ for the system $\frac{dx}{dt} = F(x, y)$, $\frac{dy}{dt} = G(x, y)$. Always double-check by substituting the point into both equations.
• 🔄 Ignoring Orientation: The orientation of the closed curve around the singular point is critical. The index is $+1$ if the trajectories move counter-clockwise and $-1$ if they move clockwise. Use the vector field to determine the direction.
• ➕ Miscounting Rotations: The Poincaré index is calculated by tracking the rotation of the vector field along a closed curve. Ensure you accurately count the number of full rotations (positive or negative). A partial rotation does not contribute to the index.
• 📐 Errors with Saddles: Saddle points always have an index of -1. Students often miscalculate this due to the complex trajectory behavior near the saddle.
• 🌀 Spirals vs. Centers: Differentiate between stable/unstable spirals and centers. Spirals have an index of +1, similar to nodes, while centers also have an index of +1, but their trajectory behavior is fundamentally different.
• 📊 Complex Systems: For systems with multiple singular points inside the closed curve, remember that the Poincaré index is the sum of the indices of all enclosed singular points.
• ✍️ Algebraic Errors: Double-check all algebraic manipulations when solving for singular points or analyzing the vector field's direction. Simple mistakes can lead to incorrect conclusions.

📝 Assessment

Here are some practice questions to test your understanding:

1. Determine the singular points of the system $\frac{dx}{dt} = x - y$, $\frac{dy}{dt} = x^2 - 4$. Calculate the Poincaré index around each singular point.
2. For the system $\frac{dx}{dt} = x^2 + y^2 - 1$, $\frac{dy}{dt} = y - x^3$, find all singular points and their corresponding Poincaré indices.
3. Consider the system $\frac{dx}{dt} = -y + x(x^2 + y^2)$, $\frac{dy}{dt} = x + y(x^2 + y^2)$. Determine the nature of the singular point at the origin and calculate its index.
4. A planar system has singular points at (0, 0) (a saddle), (1, 0) (a node), and (0, 1) (a spiral). What is the Poincaré index around a closed curve enclosing all three singular points?
5. Given the system $\frac{dx}{dt} = x(1 - x - y)$, $\frac{dy}{dt} = y(0.5 - y - x)$, find the singular points and their indices.
6. Analyze the system $\frac{dx}{dt} = y$, $\frac{dy}{dt} = -x - y$. Calculate the index around the origin.
7. For the system $\frac{dx}{dt} = x - x^2 + xy$, $\frac{dy}{dt} = -y + 2xy$, determine all singular points and calculate the Poincaré index around a curve enclosing all of them.