A hybrid chaos: a novel 3D strange attractor in a coupled Tinkerbell-Duffing-Jerk system with external forcing
Abstract
The purpose of this study is to find hybrid nonlinear dynamical system, combining elements of systems such as Tinkerbell, Duffing, and Jerk, with the addition of periodic external excitation terms (cos(ωt), sin(ωt)). The system was analysed using chaos theory tools, such as: fixed points and stability analysis (Newton-Raphson method). phase space and a strange attractor were used to clarify the fractal structure. Correlation dimension (D) and Lyapunov indices were used to evaluate complexity and sensitivity to initial conditions. Nonlinear interactions such as( , , 2 ) were the main factor in shaping the dynamic distortions and complexity of the attractor. Periodic external excitations enhanced instability and increased the sensitivty of the system, contributing to a higher D .The system can be used to generate secure random keys and to model natural phenomena, such as fluctuations in environmental or financial systems.
Downloads
References
G. Alvarez and S. Li. Some basic cryptographic requirements for chaos-based cryptosystems. International Journal of Bifurcation and Chaos, 16(8):2129–2151, 2006.
J. M. Amigo, L. Kocarev, and J. Szczepanski. Theory and practice of chaotic cryptography. Physics Letters A, 366(3):211–216, 2007.
M. Kumar and R. Sharma. Common fixed points for generalized weakly contractive maps using simulation function. Boletim da Sociedade Paranaense de Matematica, 43:1–10, 2025.
E. N. Lorenz. Deterministic nonperiodic flow. Journal of Atmospheric Sciences, 20:130–141, 1963.
Otto E Rossler. An equation for continuous chaos. Physics Letters A, 57(5):397–398, 1976.
X. Wang and G. Chen. Constructing a chaotic system with any number of equilibria. Nonlinear Dynamics, 71:429–436, 2013.
Z. Wei, P. Yu, W. Zhang, and M. Yao. Study of hidden attractors, multiple limit cycles from hopf bifurcation and boundedness of motion in the generalized hyperchaotic rabinovich system. Nonlinear Dynamics, 82:131–141, 2015.
W. Zhang. Discrete Dynamical Systems, Bifurcations and Chaos in Economics, volume 204 of Mathematics in Science and Engineering. Elsevier, Amsterdam, 2006.
Congxu Zhu and Kehui Sun. Cryptanalyzing and improving a novel color image encryption algorithm using rt-enhanced chaotic tent maps. Ieee Access, 6:18759–18770, 2018.
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática

This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).