Complex Trajectory Explorer
This interactive visualization explores the behavior of complex polynomial trajectories. It shows how a polynomial’s zeros evolve as we trace a circle in the complex plane, and analyzes the dominant (major) term’s contribution.
Complex Trajectory Explorer
About this visualization
The explorer displays five interconnected plots:
- Plot 1 (Top-Left): Trajectory in the Complex Plane
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Shows the trajectory of the point \(z(t)\) where \(z = x_0 + Re^{it}\) (a circle) passes through the polynomial. - Black line: the full trajectory - Green line: vector from origin to current point - Red marker: current position on trajectory
- Plot 2 (Top-Right): Polynomial Response
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Shows how the polynomial \(p(z)\) transforms the input circle. - Blue line: full polynomial response \(p(z(t))\) - Red dashed line: contribution from the major (dominant) term - Red marker: current position
- Plot 3 (Bottom-Left): Argument (Angle) Evolution
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Tracks how the argument \(\arg(p(z(t)))\) changes as we move along the circle. - Blue line: argument of the full polynomial - Red line: argument of the major term - The slope indicates how quickly the angle changes (related to winding numbers)
- Plot 4 (Bottom-Center): Absolute Value Evolution
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Shows the magnitude \(|p(z(t))|\) and \(|a_n z^n|\) (major term). - Blue line: \(|p(z(t))|\) - Red line: \(|a_n z^n|\) - Useful for understanding where the polynomial is largest/smallest
- Plot 5 (Right): Heatmap of Log-Reciprocal
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A heatmap of \(|1/p(z)|\) in the region around the circle. - Black line: the circle \(z(t)\) - Shows where the polynomial is close to zero (bright colors) - Turbo colorscale for visualization
Parameters
- Polynomial Coefficients (a₀, a₁, a₂, a₃, a₄):
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Defines the polynomial \(p(z) = a_0 + a_1 z + a_2 z^2 + a_3 z^3 + a_4 z^4\) - Use the +/− buttons or drag the sliders to adjust - Range: −5 to +5 - The major term is the coefficient with the highest power (usually determines asymptotic behavior)
- R:
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Radius of the circle in the complex plane - Range: 0.1 to 30 - Larger R explores behavior at infinity; smaller R explores behavior near origin
- x₀, y₀:
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Center of the circle (offset from the origin) - Range: −5 to +5 - Allows you to explore different regions of the complex plane
- position:
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Time parameter slider (0 to 500) - Controls which point on the circle is highlighted - Range: 0 to 500 (representing 0 to 2π − ε) - Use +/− buttons or drag to animate through the circle
What to explore
Argument Winding: Watch the argument plots. The total winding around the origin is related to the number of zeros by the Argument Principle.
Major Term Dominance: Compare the blue and red lines. When they align closely, the major term dominates the polynomial’s behavior.
Zeros and Poles: Look at the heatmap to see where the polynomial approaches zero. Adjust R and the center (x₀, y₀) to explore different scales.
Phase Transitions: Change the coefficients to see how the topology of the trajectory changes.
Resonance Effects: Certain coefficient combinations can create interesting patterns in the angle and magnitude evolution.
Tips
Use the +/− buttons for fine adjustments (changes by one step)
Drag the sliders for continuous exploration
The text at the top shows the current polynomial and highlights the major term in color
The heatmap colorbar ranges from dark (far from zero) to bright (close to zero)