Design of modified Duffing pendulum for trajectory generation

Abstract

Relevance. The article proposes a new approach to the construction and analysis of chaotic dynamical systems based on distance-based state variables. In contrast to traditional state-space models, where the state of the system is described directly by coordinate variables, the proposed method represents the system using distances between the current state and a set of predefined reference (base) points. These base points can remain fixed in space or move along given trajectories. Such a geometric representation of the system state offers additional flexibility for modeling nonlinear dynamics and allows the construction of new types of chaotic behavior. Purpose and objectives. In this framework, the evolution of the system is described in terms of distances to selected base points, rather than through initial coordinates. This transformation generalizes traditional state-space descriptions and allows the developer to influence the structure of the dynamics by choosing the number, configuration, and movement of the reference points. As a result, the proposed formulation provides a convenient way to introduce additional nonlinear interactions into the system while maintaining a clear geometric interpretation. Methods. To demonstrate the proposed approach, the Duffing pendulum is used as a representative example. The classical Duffing oscillator is a well-known nonlinear system that can exhibit periodic, quasi-periodic, and chaotic oscillations. In this work, the Duffing dynamics is reformulated using distance-based variables defined with respect to several reference points. Two scenarios are considered: systems with fixed reference points and systems where the reference points move in time. In the latter case, the motion of the reference points can be periodic or chaotic, which further enriches the behavior of the system as a whole. Both continuous and discrete formulations of the model are developed. The continuous representation allows for the analysis of theoretical properties of the system, while the discrete version is well suited for numerical simulation and implementation on digital platforms and embedded devices. Results. Numerical experiments show that the modified systems reproduce the main qualitative features of the classical Duffing oscillator, as well as create new types of attractors. In particular, the models can generate unipolar and more complex multidimensional chaotic attractors. Conclusions. These results indicate that distance-based observability equations can serve as an effective tool for the design of new chaotic systems. The proposed approach can be useful in applications such as secure communication, parallel generation of chaotic signals, and the study of complex nonlinear dynamics.