Interesting and complex behaviour of Duffing equations within the frame of Caputo fractional operator

İlhan, Esin

doi:10.1088/1402-4896/ac5ff5

Cited by 7 publications

(2 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Duffing equation, as a nonlinear second-order differential equation, describes a damped oscillator's motion via a more complex potential compared to the one concerning simple harmonic motion, which aids with regard to the description of an oscillator that displays some times complex, at other times chaotic behavior. The manuscript examines the coupled system illustrating the damped and driven oscillators, which is to say Duffing equations with a familiar and robust numerical method [7]. For this purpose, the authors employ reliable Caputo fractional operator to be able to capture the essential behaviors of the physical model used.…”

Section: Work In Progressmentioning

confidence: 99%

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Baleanu,

Karaca,

Vázquez

et al. 2023

Phys. Scr.

View full text Add to dashboard Cite

Most physical systems in nature display inherently nonlinear and dynamical properties; hence, it would be difficult for nonlinear equations to be solved merely by analytical methods, which has given rise to the emerging of engrossing phenomena such as bifurcation and chaos. Conjointly, due to nonlinear systems’ exhibiting more exotic behavior than harmonic distortion, it becomes compelling to test, classify and interpret the results in an accurate way. For this reason, avoiding preconceived ideas of the way the system is likely to respond is of pivotal importance since this facet would have effect on the type of testing run and processing techniques used in nonlinear systems. Paradigms of nonlinear science may suggest that it is ‘the study of every single phenomenon’ due to its interdisciplinary nature, which is another challenge encountered and needs to be addressed by generating and designing a systematic mathematical framework where the complexity of natural phenomena hints the requirement of identifying their commonalties and classifying their various manifestations in different nonlinear systems. Studying such common properties, concepts or paradigms can enable one to gain insight into nonlinear problems, their essence and consequences in a broad range of disciplines all forthwith. Fractional differential equations associated with non-local phenomena in physics have arisen as a powerful mathematical tool within a multidisciplinary research framework. Fractional differential equations, as one extension of the fractional calculus theory, can yield the evolution of various systems properly, which reinforces its position in mathematics and science while setting stage for the description of dynamic, complicated and nonlinear events. Through the reflection of the systems’ actual properties, fractional calculus manifests unforeseeable and hidden variations, and thus, enables integration and differentiation, with the solutions to be approximated by numerical methods along with modeling and predicting the dynamics of multiphysics, multiscale and physical systems. Neural Networks (NNs), consisting of hidden layers with nonlinear functions that have vector inputs and outputs, are also considerably employed owing to their versatile and efficient characteristics in classification problems as well as their sophisticated neural network architectures, which make them capable of tackling complicated governing partial differential equation problems. Furthermore, partial differential equations are used to provide comprehensive and accurate models for many scientific phenomena owing to the advancements of data gathering and machine learning techniques which have raised opportunities for data-driven identification of governing equations derived from experimentally observed data. Given these considerations, while many problems are solvable and have been solved, efforts are still needed to be able to respond to the remaining open questions in the fields that have a broad range of spectrum ranging from mathematics, physics, biology, virology, epidemiology, chemistry, engineering, social sciences to applied sciences. With a view of different aspects of such questions, our special issue provides a collection of recent research focusing on the advances in the foundational theory, methodology and topical applications of fractals, fractional calculus, fractional differential equations, differential equations (PDEs, ODEs, to name some), delay differential equations (DDEs), chaos, bifurcation, stability, sensitivity, machine learning, quantum machine learning, and so forth in order to expound on advanced fractional calculus, differential equations and neural networks with detailed analyses, models, simulations, data-driven approaches as well as numerical computations.

show abstract

Section: Work In Progressmentioning

confidence: 99%

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Baleanu,

Karaca,

Vázquez

et al. 2023

Phys. Scr.

View full text Add to dashboard Cite

show abstract

“…which is a typical equation in nonlinear vibration systems, and serves as a paradigm of nonlinear dynamics. Many mathematical models of practical engineering nonlinear vibration problems can be transformed into this equation for study [43][44][45]. Specifically, we investigate the Duffing equation with amplitude modulation and without initial phase, expressed as…”

Section: Introductionmentioning

confidence: 99%

Amplitude modulation leads to the disappearance of relaxation oscillations in the Duffing system

Song,

Jiang,

Han

et al. 2024

Phys. Scr.

View full text Add to dashboard Cite

Relaxation oscillations are pervasive in diverse areas of natural sciences and engineering, and exploring the dynamical mechanisms of relaxation oscillations is one of the most significant issues. Typical relaxation oscillations can be observed in the Duffing system. Recently, amplitude modulation has emerged as a novel control mechanism for investigating the behavior of fast-slow dynamics in systemic tension oscillations. It has demonstrated the ability to prolong the quasi-static slow process of the system and increase the number of bifurcation points. However, the exploration of the mechanistic aspects of amplitude modulation is still in its early stages, with many unreported dynamical mechanisms. Among these, investigating the modes of relaxation oscillations induced by amplitude modulation is one of the most important issues. Therefore, this manuscript focuses on studying the effect of amplitude modulation on relaxation oscillations, using the classical forced Duffing system as a representative model. Significantly, we report an intriguing finding for the first time, revealing a new amplitude-modulated mechanism by which the disappearance of relaxation oscillations can be induced. By employing the fast-slow analysis, we have examined the underlying dynamical mechanisms, revealing a strong correlation with the modulation index of amplitude modulation. Notably, when the system operates under low amplitude modulation, an extension of the quasi-static process is observed, manifesting as a prolonged slow process. Conversely, under high amplitude modulation, relaxation oscillations suddenly disappear. Our results serve to enrich the potential mechanisms of amplitude modulation, and our analysis provides a reference for investigating the dynamical behavior induced by amplitude modulation in other dynamical systems.

show abstract

Dynamic analysis of a class of fractional‐order dry friction oscillators

Si,

Xie,

Zhao

et al. 2024

Math Methods in App Sciences

View full text Add to dashboard Cite

This article investigates a class of Duffing nonlinear dynamic systems with fractional‐order dry friction and conducts in‐depth research on the stability, chaotic characteristics, and erosion of the safety basin of this system; the results are verified through numerical simulation. First, the average method is used to approximate the amplitude–frequency relationship of the system, and the accuracy of the analytical results is verified through numerical experiments. Second, the Melnikov method is used to obtain the conditions for the system to enter chaos in the Smale horseshoe sense, and the Melnikov curve is drawn for further verification. Then, bifurcation diagrams are drawn for the changes in various parameters in the system, with a focus on analyzing the influence of friction factors on chaotic bifurcation. By applying the definition and calculation principle of the maximum Lyapunov exponent, and drawing and utilizing the maximum Lyapunov exponent graph, the chaotic state that the system enters under different parameters is more clearly defined. Finally, the evolution law of the safety basin under various parameter changes, especially dry friction changes, is analyzed, and the erosion and bifurcation mechanism of the safety basin is studied. Comparing with the bifurcation diagram, it reveals that chaos primarily contributes to the erosion of the safety basin.

show abstract

Interesting and complex behaviour of Duffing equations within the frame of Caputo fractional operator

Cited by 7 publications

References 38 publications

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Amplitude modulation leads to the disappearance of relaxation oscillations in the Duffing system

Dynamic analysis of a class of fractional‐order dry friction oscillators

Contact Info

Product

Resources

About