Source-linked AI summary
A new tool to study real dynamics: The Convergence Plane
Angel Alberto Magreñan
TL;DR
The paper addresses the difficulty of studying real dynamics across all starting points and parameter values in iterative-method families. It introduces The Convergence Plane, a colored two-dimensional representation of these pairs, and demonstrates it on Damped Newton’s method for a cubic polynomial. The example identifies convergence, cycles, and chaotic or divergent behavior across parameter zones, while showing how attraction basins vary with the parameter.
Problem
Studying real dynamics for every starting point and parameter value in an iterative-method family is difficult with existing point-focused tools.
Method
The Convergence Plane maps starting points and parameter values to a plane and classifies their dynamical behavior using a colored computational grid.
Results
For Damped Newton’s method on p−(x) = x^3 − x, the example distinguishes root convergence for λ ∈(0, 2), cycles of orders 2, 4, and 8, and chaotic behavior or cycles with periods different from 2^n.
Takeaways & Limitations
The tool provides a global view of convergence and basin changes, helping identify parameter values associated with larger convergence zones.
Takeaways & Limitations
The algorithm requires a prefixed study region, maximum iteration count, and tolerance, and the example assumes the orbit is not asymptotically periodic.
Abstract
from arXiv · showhide
In this paper, the author presents a new tool, called The Convergence Plane, that allows to study the real dynamics of iterative methods whose iterations depends on one parameter in an easy and compact way. This tool can be used, inter alia, to find the elements of a family that have good convergence properties and discard the bad ones or to see how the basins of attraction changes along the elements of the family. To show the applicability of the tool an example of the dynamics of the Damped Newton's method applied to a cubic polynomial is presented.
1 Introduction and Motivation
The paper introduces The Convergence Plane to make real dynamics easier to study across parameterized iterative-method families. It responds to the difficulty of analyzing real behavior point by point while retaining information from parameter-based tools.
- The Convergence Plane is proposed as a tool for studying the real dynamics of iterative-method families depending on a parameter.
- Real dynamics differs from complex dynamics because monotone convergence and asymptotes have real-line meanings absent from the complex-plane setting.
- Existing real-line tools such as Feigenbaum diagrams and Lyapunov exponents focus on concrete points, making separate analysis of every point difficult.
- The Convergence Plane uses a horizontal starting-point axis and a vertical parameter axis, so each point represents an initial estimate and a family member.
- The paper presents the algorithm, validates the tool with Damped Newton’s method on p−(x) = x^3 − x, and then gives conclusions.
2 Algorithm of The Convergence Plane
The algorithm constructs a colored grid over starting points and parameter values, classifying convergence and other dynamical behaviors. The resulting plane gives a global view of convergence regions and parameter effects.
- Each plane point represents the pair (x0, λ0), combining a starting point with a parameter value for the family study.
- The procedure computes fixed points, assigns each fixed point a color, and specifies the study region D, iteration limit M, and tolerance ε.
- A d × d grid of initial points and parameter values is selected within D for evaluating the family.
- Points that fail to converge to a fixed point after M iterations are colored black, while cycles, extraneous fixed points, and divergence can receive additional colors.
- The computed plane identifies starting-point–parameter pairs that converge to roots and helps select parameters with larger basins of attraction.
- Mathematica program: The example uses Mathematica plotting routines to generate the plane, with a 1024 × 1024 grid and explicit settings for region, iterations, and error handling.
- Mathematica program: The implementation defines roots and root identification, applies the iteration method, checks convergence or cycles, and maps outcomes to colors.
3 Example: Damped Newton’s method applied to the polynomial
The example applies Damped Newton’s method to p−(x) = x3 −x and compares Lyapunov exponents, Feigenbaum diagrams, and the Convergence Plane across damping parameters and starting points. The analysis identifies three dynamical zones, showing where iterations converge to roots, settle into cycles, or exhibit chaotic behavior and other-period cycles.
- Example setup: The Damped Newton iteration is applied to the cubic p−(x) = x3 −x to demonstrate the Convergence Plane.The study uses the method as a test case for the new tool.
- Diagnostic tools: Lyapunov exponents identify chaotic orbits, while Feigenbaum diagrams distinguish convergence to cycles, points, chaos, or infinity.The two tools provide information about the orbit being iterated.
- Dynamical zones: For λ ∈(0, 2), iterations converge to fixed points, equivalently the roots of p−(x).The corresponding Convergence Plane zone includes every point except poles and their preimages.
- Dynamical zones: The second zone contains cycles of orders 2, 4, and 8, whereas the third contains chaotic behavior or cycles with periods different from 2n.The analysis focuses on λ ∈(0, 2.6) to expose these distinct regimes.
- Convergence Plane: The Convergence Plane maps starting points horizontally and parameter values vertically, revealing how basins of attraction change with λ.For example, the basin of r2 = 0 decreases as λ approaches 2, while the other two root basins increase.
- Conclusions: The study concludes that Damped Newton’s method is a good root-finding algorithm for λ ∈(0, 2], while outside that interval iterations can cycle or diverge.The Julia set becomes more intricate as the damping factor increases toward 2.
4 Conclusions
The Convergence Plane studies convergence across every initial point and parameter value, helping identify favorable parameter choices and track basin changes. The paper also presents extensions to parameterized polynomial families and other iterative methods.
- The Convergence Plane studies convergence for every initial point and parameter value in a family of iterative methods.
- It can help select a parameter whose convergence zone is as large as possible.
- The tool shows how basins of attraction change as the parameter varies.
- The technique can be adapted to parameterized polynomial families, two-point methods, and methods for non-differentiable functions.
- The paper presents the tool as making real-dynamics analysis easier, deeper, and more compact.