[Paper Review] Box-counting dimension of solution curves for a class of two-dimensional nonautonomous linear differential systems

Masakazu Onitsuka, Satoshi Tanaka|arXiv (Cornell University)|Mar 6, 2017

Mathematical Control Systems and Analysis3 citations

TL;DR

This paper establishes the box-counting dimension of solution curves for a class of two-dimensional nonautonomous linear differential systems, specifically $ x' = y $, $ y' = -x - h(t)y $, under conditions ensuring asymptotic stability. It proves that the dimension is $ \frac{2}{1+\alpha} $ when the damping coefficient $ h(t) \sim \alpha t^{-1} $, and 1 when $ h(t) $ is bounded, providing a precise fractal characterization of spiral trajectories approaching the origin.

ABSTRACT

A class of two-dimensional linear differential systems is considered. The box-counting dimension of the graphs of solution curves is calculated. Criteria to obtain the box-counting dimension of spirals are also established.

Motivation & Objective

• To determine the box-counting dimension of solution curves for a class of two-dimensional nonautonomous linear differential systems.
• To analyze the fractal geometry of spiral trajectories arising from damped linear oscillators with time-varying damping.
• To establish criteria based on the asymptotic behavior of the damping coefficient $ h(t) $ that govern the dimension of solution curves.
• To extend existing results on rectifiability and attractivity to include fractal dimension analysis.

Proposed method

• Transform the system into polar coordinates $ x = r(t)\cos\theta(t) $, $ y = r(t)\sin\theta(t) $, analyzing the radial and angular dynamics.
• Use the asymptotic behavior of $ h(t) $, particularly $ h(t) \sim \alpha t^{-1} $, to derive estimates on $ r(t) $ and $ \theta(t) $.
• Define a new function $ f(\varphi) = r(\eta^{-1}(\varphi)) $, where $ \eta(t) = -\theta(t) $, to reparametrize the solution curve in terms of angle $ \varphi $.
• Apply known results on box-counting dimension for spiral curves, particularly Corollary 1.1 and Corollary 1.2, to compute the dimension based on the decay rate of $ f(\varphi) $.
• Establish bounds on $ |f'(\varphi)| $ and $ \varphi f(\varphi) $ to verify the conditions for the dimension formula.
• Use symmetry and invariance under sign change of $ y $ to relate the dimension of $ \Gamma_{(x,y;t_1)} $ to the transformed curve $ \Gamma $.

Experimental results

Research questions

• RQ1 What is the box-counting dimension of the solution curve $ \Gamma_{(x,y;t_1)} $ for nontrivial solutions of the system $ x' = y $, $ y' = -x - h(t)y $?
• RQ2 How does the box-counting dimension depend on the asymptotic behavior of the damping coefficient $ h(t) $?
• RQ3 Under what conditions is the solution curve a spiral with finite box-counting dimension greater than 1?
• RQ4 Can the dimension be exactly computed when $ h(t) \sim \alpha t^{-1} $ as $ t \to \infty $?
• RQ5 What is the dimension when $ h(t) $ is bounded but not decaying, and how does this differ from the power-law case?

Key findings

• When $ h(t) \sim \alpha t^{-1} $ with $ \alpha > 0 $, the box-counting dimension of the solution curve $ \Gamma_{(x,y;t_1)} $ is $ \frac{2}{1+\alpha} $.
• If $ h(t) $ is bounded and $ \int_{t_0}^\infty e^{-H(t)/2} dt = \infty $, the solution curve has box-counting dimension 1.
• The dimension $ \frac{2}{1+\alpha} $ is achieved when $ r(t) \sim C t^{-1} $ and $ |\theta(t)| \sim \alpha \log t $, leading to a slow spiral.
• The result holds for all nontrivial solutions under the assumptions of Theorem A, including attractivity and spiral rotation.
• The dimension is invariant under sign reversal of $ y $, so $ \dim_B \Gamma_{(x,y;t_1)} = \dim_B \Gamma_{(x,-y;t_1)} $.
• The analysis confirms that the fractal dimension depends critically on the decay rate of $ h(t) $, with slower decay leading to higher dimension.