scholarly journals Exact Solution of Impulse Response to a Class of Fractional Oscillators and Its Stability

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Ming Li ◽  
S. C. Lim ◽  
Shengyong Chen
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Impulse Response ◽  
Fractional Order ◽  
System Analysis ◽  
Stability Behavior ◽  
Single Degree ◽  
Fractional Oscillator ◽  
The Stability ◽  
Typical Model

Oscillator of single-degree-freedom is a typical model in system analysis. Oscillations resulted from differential equations with fractional order attract the interests of researchers since such a type of oscillations may appear dramatic behaviors in system responses. However, a solution to the impulse response of a class of fractional oscillators studied in this paper remains unknown in the field. In this paper, we propose the solution in the closed form to the impulse response of the class of fractional oscillators. Based on it, we reveal the stability behavior of this class of fractional oscillators as follows. A fractional oscillator in this class may be strictly stable, nonstable, or marginally stable, depending on the ranges of its fractional order.

scholarly journals Further Study on Dynamics for a Fractional-Order Competitor-Competitor-Mutualist Lotka–Volterra System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Bingnan Tang
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Natural World ◽  
Volterra Model ◽  
Sufficient Criterion ◽  
Volterra System ◽  
Stability Behavior ◽  
Predator Model ◽  
Set Up ◽  
The Stability

On the basis of the previous publications, a new fractional-order prey-predator model is set up. Firstly, we discuss the existence, uniqueness, and nonnegativity for the involved fractional-order prey-predator model. Secondly, by analyzing the characteristic equation of the considered fractional-order Lotka–Volterra model and regarding the delay as bifurcation variable, we set up a new sufficient criterion to guarantee the stability behavior and the appearance of Hopf bifurcation for the addressed fractional-order Lotka–Volterra system. Thirdly, we perform the computer simulations with Matlab software to substantiate the rationalisation of the analysis conclusions. The obtained results play an important role in maintaining the balance of population in natural world.

scholarly journals Bifurcation Study on Fractional-Order Cohen–Grossberg Neural Networks Involving Delays

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Bingnan Tang
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Fractional Order ◽  
Sufficient Criterion ◽  
Software Simulation ◽  
Stability Behavior ◽  
Delayed Neural Networks ◽  
Simulation Results ◽  
Set Up ◽  
The Stability

This work is chiefly concerned with the stability behavior and the appearance of Hopf bifurcation of fractional-order delayed Cohen–Grossberg neural networks. Firstly, we study the stability and the appearance of Hopf bifurcation of the involved neural networks with identical delay ϑ 1 = ϑ 2 = ϑ . Secondly, the sufficient criterion to guarantee the stability and the emergence of Hopf bifurcation for given neural networks with the delay ϑ 2 = 0 is set up. Thirdly, we derive the sufficient condition ensuring the stability and the appearance of Hopf bifurcation for given neural networks with the delay ϑ 1 = 0 . The investigation manifests that the delay plays a momentous role in stabilizing networks and controlling the Hopf bifurcation of the addressed fractional-order delayed neural networks. At last, software simulation results successfully verified the rationality of the analytical results. The theoretical findings of this work can be applied to design, control, and optimize neural networks.

GENERALIZED FRACTIONAL ORDER BLOCH EQUATION WITH EXTENDED DELAY

2012 ◽  
Vol 22 (04) ◽  
pp. 1250071 ◽  
Author(s):  
SACHIN BHALEKAR ◽  
VARSHA DAFTARDAR-GEJJI ◽  
DUMITRU BALEANU ◽  
RICHARD MAGIN
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Time Derivative ◽  
Bloch Equation ◽  
Extended Model ◽  
Order Ordinary Differential Equation ◽  
Stability Behavior ◽  
System Memory ◽  
The Stability ◽  
Fractional Time Derivative

The fundamental description of relaxation (T1 and T2) in nuclear magnetic resonance (NMR) is provided by the Bloch equation, an integer-order ordinary differential equation that interrelates precession of magnetization with time- and space-dependent relaxation. In this paper, we propose a fractional order Bloch equation that includes an extended model of time delays. The fractional time derivative embeds in the Bloch equation a fading power law form of system memory while the time delay averages the present value of magnetization with an earlier one. The analysis shows different patterns in the stability behavior for T1 and T2 relaxation. The T1 decay is stable for the range of delays tested (1 μsec to 200 μsec), while the T2 relaxation in this extended model exhibits a critical delay (typically 100 μsec to 200 μsec) above which the system is unstable. Delays arise in NMR in both the system model and in the signal excitation and detection processes. Therefore, by adding extended time delay to the fractional derivative model for the Bloch equation, we believe that we can develop a more appropriate model for NMR resonance and relaxation.

scholarly journals Drive-Response Synchronization of a Fractional-Order Hyperchaotic System and Its Circuit Implementation

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Darui Zhu ◽  
Chongxin Liu ◽  
Bingnan Yan
Keyword(s):  
Numerical Simulation ◽  
Fractional Order ◽  
Circuit Simulation ◽  
Hyperchaotic System ◽  
Oscillator Circuit ◽  
Response System ◽  
Fractional Oscillator ◽  
Simulation Results ◽  
The Stability ◽  
Synchronization Scheme

A novel fractional-order hyperchaotic system is proposed; the theoretical analysis and numerical simulation of this system are studied. Based on the stability theory of fractional calculus, we propose a novel drive-response synchronization scheme. In order to achieve this synchronization control, the Adams-Bashforth-Moulton algorithm is studied. And then, a drive-response synchronization controller is designed to realize the synchronization of the drive and response system, and the simulation results are given. At last, the fractional oscillator circuit of the new fractional-order hyperchaotic system is designed based on the EWB software, and it is verified that the simulation results of the fractional-order oscillator circuit are consistent with the numerical simulation results through circuit simulation.

On the Stability of Most General Exact Solution for Isentropic Superdense Stars

2005 ◽  
Vol 8 (1) ◽  
pp. 60-62
Author(s):  
Dhuha Majeed Saleh ◽  
◽  
Abdul Samee A. Al-Janabi ◽  
M. K. Jasim ◽  
◽  
...  
Keyword(s):  
Exact Solution ◽  
Superdense Stars ◽  
The Stability

scholarly journals Observer-based adaptive neural network backstepping sliding mode control for switched fractional order uncertain nonlinear systems with unmeasured states

2021 ◽  
pp. 002029402110211
Author(s):  
Tao Chen ◽  
Damin Cao ◽  
Jiaxin Yuan ◽  
Hui Yang
Keyword(s):  
Neural Network ◽  
Nonlinear Systems ◽  
Sliding Mode Control ◽  
Fractional Order ◽  
Sliding Mode ◽  
Tracking Error ◽  
Adaptive Neural Network ◽  
Dynamic Surface ◽  
Mode Control ◽  
The Stability

This paper proposes an observer-based adaptive neural network backstepping sliding mode controller to ensure the stability of switched fractional order strict-feedback nonlinear systems in the presence of arbitrary switchings and unmeasured states. To avoid “explosion of complexity” and obtain fractional derivatives for virtual control functions continuously, the fractional order dynamic surface control (DSC) technology is introduced into the controller. An observer is used for states estimation of the fractional order systems. The sliding mode control technology is introduced to enhance robustness. The unknown nonlinear functions and uncertain disturbances are approximated by the radial basis function neural networks (RBFNNs). The stability of system is ensured by the constructed Lyapunov functions. The fractional adaptive laws are proposed to update uncertain parameters. The proposed controller can ensure convergence of the tracking error and all the states remain bounded in the closed-loop systems. Lastly, the feasibility of the proposed control method is proved by giving two examples.

Global stability analysis of fractional-order gene regulatory networks with time delay

2019 ◽  
Vol 12 (06) ◽  
pp. 1950067 ◽  
Author(s):  
Zhaohua Wu ◽  
Zhiming Wang ◽  
Tiejun Zhou
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Gene Regulatory Networks ◽  
Existence And Uniqueness ◽  
Regulatory Networks ◽  
Lyapunov Method ◽  
Regulation Mechanism ◽  
Global Stability Analysis ◽  
Gene Regulatory ◽  
The Stability

Fractional-order gene regulatory networks with time delay (DFGRNs) have proven that they are more suitable to model gene regulation mechanism than integer-order. In this paper, a novel DFGRN is proposed. The existence and uniqueness of the equilibrium point for the DFGRN are proved under certain conditions. On this basis, the conditions on the global asymptotic stability are established by using the Lyapunov method and comparison theorem for the DFGRN, and the stability conditions are dependent on the fractional-order [Formula: see text]. Finally, numerical simulations show that the obtained results are reasonable.

scholarly journals Small world and stability analysis of industrial coupling symbiosis network of ecological industrial park of oil and gas resource cities

2021 ◽  
pp. 014459872098361
Author(s):  
Yanqiu Wang ◽  
Zhengxin Sun ◽  
Pengtai Li ◽  
Zhiwei Zhu
Keyword(s):  
System Analysis ◽  
Oil And Gas ◽  
Small World ◽  
Industrial Park ◽  
Sharp Change ◽  
Connected Subgraph ◽  
Network Efficiency ◽  
Complex Network Theory ◽  
The Stability ◽  
Agglomeration Coefficient

This paper analyzes the small cosmopolitan and stability of the industrial coupling symbiotic network of eco-industrial parks of oil and gas resource-based cities. Taking Daqing A Ecological Industrial Park as an example, we constructed the characteristic index system and calculated the topological parameters such as the agglomeration coefficient and the average shortest path length of the industrial coupling symbiotic network. Based on the complex network theory we analyzed the characteristics of the scaled world, constructed the adjacency matrix of material and information transfers between enterprises, drew the network topology diagram. We simulated the system analysis and analyzed the stability of the industrial coupling symbiotic network of the eco-industrial park using the network efficiency and node load and maximum connected subgraph. The analysis results are as follows: the small world degree δ of Daqing A Eco-industrial Park is 0.891, which indicates that the industrial coupled symbiotic network has strong small world characteristics; the average path is 1.268, and the agglomeration coefficient is 0.631. The probability of edge connection between two nodes in a symbiotic network is 63.1%, which has a relatively high degree of aggregation, indicating that energy and material exchanges are frequent among all enterprises in the network, the degree of network aggregation is high, and the dependence between nodes is high; when the tolerance parameter is 0 to 0.3, the network efficiency and the maximum connected subgraphs show a sharp change trend, indicating that the topology of the industrial coupling symbiotic network of the eco-industrial park changes drastically when the network is subjected to deliberate attacks. It is easy to cause the breakage of material flow and energy flow in the industrial park, which leads to the decline of the stability of the industrial coupling symbiotic network of the eco-industrial park.

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