Damping efficiency of the Duffing system with additional fractional terms

Rysak, Andrzej; Sedlmayr, Martyna

doi:10.1016/j.apm.2022.06.037

Cited by 9 publications

(4 citation statements)

References 24 publications

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“…The results showed the accuracy of significant frequency bandwidths. Fractional derivatives were used to modify a Duffing system [35]. The results show that the fractional elements clearly modify the system dynamics and significantly increase the system energy efficiency.…”

Section: Fractional Order Calculusmentioning

confidence: 99%

Fractional Order Sequential Minimal Optimization Classification Method

Zhao,

Dai,

Huang

2023

Fractal Fract

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Sequential minimal optimization (SMO) method is an algorithm for solving optimization problems arising from the training process of support vector machines (SVM). The SMO algorithm is mainly used to solve the optimization problem of the objective function of SVM, and it can have high accuracy. However, its optimization accuracy can be improved. Fractional order calculus is an extension of integer order calculus, which can more accurately describe the actual system and get more accurate results. In this paper, the fractional order sequential minimal optimization (FOSMO) method is proposed based on the SMO method and fractional order calculus for classification. Firstly, an objective function is expressed by a fractional order function using the FOSMO method. The representation and meaning of fractional order terms in the objective function are studied. Then the fractional derivative of Lagrange multipliers is obtained according to fractional order calculus. Lastly, the objective function is optimized based on fractional order Lagrange multipliers, and then some experiments are carried out on the linear and nonlinear classification cases. Some experiments are carried out on two-classification and multi-classification situations, and experimental results show that the FOSMO method can obtain better accuracy than the normal SMO method.

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Section: Fractional Order Calculusmentioning

confidence: 99%

Fractional Order Sequential Minimal Optimization Classification Method

Zhao,

Dai,

Huang

2023

Fractal Fract

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“…It is noteworthy that richer dynamic phenomena have been found in fractional-order nonlinear systems, such as limit cycle [17], bifurcation [18], resonance [19] and synchronization [20]. Interestingly, chaotic behavior has also been discovered in some fractional-order dynamical systems, such as the fractional-order Rösslor system [21], the fractional-order Duffing system [22], the fractional-order Lorenz system [23], the fractional-order Liu system [24] and so on [25][26][27][28]. It should be mentioned that these systems have many remarkable properties, including extreme sensitivity to initial conditions, unpredictability, internal randomness and ergodicity.…”

Section: Introductionmentioning

confidence: 99%

A novel entanglement functions-based 4D fractional-order chaotic system and its bifurcation analysis

Tang,

Li,

Huang

2024

Phys. Scr.

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A novel 4D fractional-order chaotic entanglement system based on sinusoidal functions is established in this paper. We aim to reveal the relationship between the dynamical behavior of the new system and its entanglement coefficients. It is found that the equilibrium point of the system varies regularly with the successive change of the entanglement coefficient. The supercritical pitchfork bifurcation phenomenon of the new system is discussed based on the fractional-order stability theory. Furthermore, sufficient conditions and threshold for supercritical Hopf bifurcation caused by the entanglement coefficient are provided. Finally, the route to chaos of the new system is explored utilizing multiple numerical indicators, such as spectral entropy complexity, bifurcation diagrams, Lyapunov exponential spectrum, phase portraits, and 0-1 test curves. The results indicate that in addition to various chaotic attractors, there are phenomena such as period-doubling bifurcations, period windows, and coexisting symmetric attractors (periodic or chaotic).

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“…Given that the direct method (Lyapunov first method) has been successfully used in the study of similar problems, this work will also use it in the stability analysis of fractional Duffing systems with a nonlinear time delay term [12]. The problem of motion stability is also important for the study of fractional Duffing system with nonlinear time delay term in this work [15,16].…”

Section: Introductionmentioning

confidence: 99%

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

Yang

Jiao

2023

Math Methods in App Sciences

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The dynamic characteristics of the fractional Duffing system with nonlinear time delay term are studied according to the average method and the harmonic balance method. First, the steady‐state solutions of the fractional Duffing system with a nonlinear time delay term are obtained by applying the average method and the harmonic balance method. The analytical expressions of the amplitude–frequency characteristic curves under the average method and the harmonic balance method are also established. Second, the stability criterion of the system is obtained according to the indirect method of studying the motion stability problem, and the concept of the stability condition parameter is proposed in this process. Third, the transient solution of the fractional Duffing system with a nonlinear time delay term is obtained utilizing the average method, and the approximate analytical solution of the fractional Duffing system with nonlinear time delay term is obtained. Finally, the amplitude–frequency characteristic curves of the different fractional differential terms, coefficients of different fractional differential terms, magnitude of different time delay quantities, and feedback strength of different delay are analyzed by numerical simulation. Furthermore, the amplitude–frequency characteristic curves obtained by the average method and the harmonic balance method under different order of fractional differential term are compared and analyzed by numerical simulation. The time sequence and phase diagrams of the system under different order of fractional differential term and different time delay quantities are analyzed by numerical simulation. In addition, the correctness of the analytical analysis is verified by comparing the analytical results with the numerical simulation results under the average method and the harmonic balance method, respectively, by numerical simulation.

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Damping efficiency of the Duffing system with additional fractional terms

Cited by 9 publications

References 24 publications

Fractional Order Sequential Minimal Optimization Classification Method

Fractional Order Sequential Minimal Optimization Classification Method

A novel entanglement functions-based 4D fractional-order chaotic system and its bifurcation analysis

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

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