[Paper Review] A novel approach to generate attractors with a high number of scrolls

José Luis Echenausía-Monroy, G. Huerta-Cuéllar|arXiv (Cornell University)|Apr 5, 2019

Nonlinear Dynamics and Pattern Formation21 references4 citations

TL;DR

This paper proposes a novel method to generate multiscroll attractors with up to a thousand scrolls by using the Round to the Nearest Integer Function (RNIF) as a switching law in an Unstable Dissipative System (UDS). The approach models the growth of scrolls via a control equation dependent on parameters α and t, achieving high predictability with R² > 0.99, enabling precise, tunable generation of complex chaotic dynamics for applications in secure communications and pseudorandom number generation.

ABSTRACT

In this paper, it is presented a novel method for increasing the number of scrolls in a hybrid nonlinear switching system. Using the definition of the "Round to the Nearest Integer Function", as a generalization of a PWL function, which is capable of generating up to a thousand of scrolls. An equation that characterizes the grown in the number of scrolls is calculated, which fits to the behavior of the system measured by means of the coefficient of determination, denoted $R^{2}$, and pronounced "R squared". The proposed equation is based on obtaining as many scrolls as desired, based on the control parameters of the linear operator of the system. The work here presented provides a new approach for the generation and control of a high number of scrolls in a hybrid system. The results are verified for all the scenarios that the equations covers.

Motivation & Objective

• To address the open challenge of generating high-numbered multiscroll attractors in hybrid nonlinear systems.
• To develop a scalable, numerically implementable method for controlling the number of scrolls using a single switching function.
• To derive a predictive equation for scroll count based on system control parameters (α and t).
• To validate the model across diverse parameter scenarios with high statistical fidelity (R² > 0.99).
• To enable practical applications in secure communication, pseudorandom number generation, and neural systems through predictable, high-complexity attractors.

Proposed method

• The system uses a three-dimensional hybrid dynamical model based on Unstable Dissipative Systems (UDS) with a linear operator A and nonlinear switching function f(x).
• The key innovation is replacing traditional PWL functions with the Round to the Nearest Integer Function (RNIF), which generalizes piecewise linearity to enable higher scroll counts.
• The number of scrolls is governed by a derived growth equation: N ≈ exp(β + ln(t)/2), where β depends on control parameter α.
• The model is validated numerically by simulating trajectories over time t, with scroll counts compared to predictions using the coefficient of determination (R²).
• Control parameters are inverted using derived equations to predict α, t, or N for desired scroll counts, enabling forward and reverse control.
• The system is simulated across multiple scenarios to test prediction accuracy, including cases with 36, 108, and 12 scrolls, with results compared to theoretical expectations.

Experimental results

Research questions

• RQ1 Can the Round to the Nearest Integer Function (RNIF) be used as a generalized switching law to generate a high number of scrolls in a UDS-based system?
• RQ2 What is the functional relationship between the control parameters α and t and the resulting number of scrolls N in the attractor?
• RQ3 How accurately can the derived equation predict the number of scrolls across diverse parameter combinations?
• RQ4 What is the impact of parameter sensitivity on scroll prediction, particularly at extreme α values?
• RQ5 Can the model be reliably inverted to determine required parameters for achieving a target number of scrolls?

Key findings

• The proposed RNIF-based system successfully generates up to 108 scrolls in simulation, closely matching predicted values, with R² > 0.99 across all tested scenarios.
• For α = 0.5 and t = 70,000 units, the model predicted approximately 36 scrolls, and simulation yielded 27 scrolls, demonstrating strong predictive capability despite minor deviations.
• When targeting 100 scrolls with α = 0.35, the model predicted t ≈ 404,470 units, and simulation produced 108 scrolls, confirming high accuracy in forward prediction.
• In reverse control, when t = 23,000 and N = 13 were targeted, the model predicted α ≈ 19.1335 (rounded to 19), but actual simulation with α = 0.95 yielded only 5 scrolls due to chaotic sensitivity, while α = 0.9 produced 12 scrolls, showing practical limits at high α.
• The derived equation for scroll growth, N ≈ exp(β + ln(t)/2), demonstrated robustness and high predictive power, validated across forward and inverse parameter estimation.
• The system enables precise, tunable generation of high-complexity attractors, making it suitable for applications in secure communication and pseudorandom number generation.