Fractional-order Oscillator Research Articles

Nonlinear model and characteristic analysis of fractional-order high frequency oscillator

Recently, fractional calculus theory has been widely used in various fields, including the design of oscillators. However, there are still some issues that need to be studied in depth. Firstly, it is necessary to classify the fractional-order oscillators according to the properties of the frequency selective network components, and analyze the influence of the fractional order of components on the characteristics of different types of fractional-order oscillators. Secondly, for fractional-order oscillators with nonlinear devices such as transistors, the traditional linear models cannot describe them accurately. Thirdly, due to the complexity of fractional differential equation, it is difficult to obtain the general analytical solution of the fractional-order oscillator. For these reasons, this paper takes the fractional-order capacitive feedback LC oscillator as an example, and establishes a nonlinear model of this oscillator based on the transistor characteristic mechanism. With the help of the equivalent small parameter method, the influence of the fractional-order characteristics of the components on the output waveform is analyzed. It is revealed that for the fractional-order capacitive feedback LC oscillator, the order of capacitance C1 has the most significant effect on frequency and amplitude, followed by the order of inductance L2, and the order of the feedback capacitor C2 has very little effect on the oscillator. In addition, although the introduction of fractional order increases the oscillation frequency, it will also lead to an increase in THD in some cases. Simulation and experimental results verify the correctness of the proposed nonlinear model and characteristic analysis of fractional-order high frequency oscillator.

Dynamics of Fractional-Order Chaotic Rocard Relaxation Econometric System

The chaotic Rocard relaxation econometric system was proposed earlier than the Lorenz system. In this paper, the fractional-order Rocard system is investigated and its dynamical behaviors are analyzed by bifurcation diagram, Lyapunov exponent and spectral entropy (SE) complexity. The stability analysis is conducted and the conditions for the fractional-order system to maintain chaotic or periodic state are investigated. Compared with the integer-order counterpart, the fractional-order system generates more periodic parameter areas, and they play vital roles in economic cycles and economic chaos control. We also found that the fractional-order oscillator has a larger Lyapunov exponent than its integer-order counterpart. By applying the 0–1 test method, the minimum order for the system for chaotic oscillation is determined. The SE complexity evidence further confirms involving fractional calculus that enriches dynamical behaviors. These results indicate the promising applications in chaos control of the economic system.

CIM applications in fractional domain: Fractional-order universal filter & fractional-order oscillator

In this work, applications of current inverting metamutator (CIM) are explored in fractional domain for the very first time. The realization of a current-mode universal fractional-order filter (FOF) and a sinusoidal fractional-order oscillator (FOO) circuit utilizing one CIM block is presented. The functionality of the proposed structures are verified through SPICE simulations using 180nm CMOS technology parameters. Stability analysis has been presented for the proposed circuits. Robustness of the proposed circuits is verified through sensitivity, Monte-Carlo and mismatch analysis.

Design and implementation of OTA based fractional-order oscillator

Design and implementation of OTA based fractional-order oscillator

On the numerical study of fractional and non-fractional model of nonlinear Duffing oscillator: a comparison of integer and non-integer order approaches

ABSTRACT This manuscript analyzes nonlinear Duffing oscillators via integer and non-integer order differential operators known as the Caputo, Caputo–Fabrizio, and the Atangana–Baleanu taken in the Caputo sense. Application of the fixed-point theory shows that the fractional-order nonlinear Duffing oscillator does have a unique solution subject to appropriate initial conditions. Furthermore, the fractional-order parameters α,β and γ∈]0,1] for the Caputo, Caputo–Fabrizio, and the Atangana–Baleanu operators, respectively, when kept at variation, revealed exciting changes in the amplitude and the frequency responses for the fractional displacement function of the nonlinear oscillator. Lastly, numerical simulations have been carried out and presented in graphical illustrations to observe the effects of each parametric value of the nonlinear Duffing oscillator under the classical, the Caputo, the Caputo–Fabrizio, and the Atangana–Baleanu fractional differential operators wherein the Markovian nature of the oscillator is retained when α=β=γ=1. Based upon the comparative mathematical analysis of the oscillator, it has been shown that the Atangana–Baleanu fractional-order operator possessing non-locality and non-singular convolution kernel has the better capability to retain the memory effects of the system.

Oscillators Based on Fractional-Order Memory Elements

This paper deals with the new oscillator structures that contain new elements, so-called memory elements, known as memristor, meminductor, and memcapacitor. Such circuits can exhibit oscillations as well as chaotic behavior. New mathematical models of fractional-order elements and whole oscillator circuits are proposed as well. An illustrative example to demonstrate the oscillations and the chaotic behavior through the numerical solution of the fractional-order circuit model is provided.

Different Stochastic Resonances Induced by Multiplicative Polynomial Trichotomous Noise in a Fractional Order Oscillator with Time Delay and Fractional Gaussian Noise

A general investigation on the mechanism of stochastic resonance is reported in a time-delay fractional Langevin system, which endues a nonlinear form multiplicative colored noise and fractional Gaussian noise. In terms of theoretical analysis, both the expressions of output steady-state amplitude and that of the first moment of system response are obtained by utilizing stochastic averaging method, fractional Shapiro and Laplace methods. Due to the presence of trichotomous colored noise, the excitation frequency can induce fresh multimodal Bona fide stochastic resonance, exhibiting much more novel dynamical behaviors than the non-disturbance case. It is verified that multimodal pattern only appears with small noise switching rate and memory damping order. The explicit expressions of critical noise intensity corresponding to the generalized stochastic resonance are given for the first time, by which it is determined that nonlinear form colored noise induces much more of a comprehensive resonant phenomena than the linear form. In the case of slow transfer rate noise, a newfangled phenomenon of double hypersensitive response induced by a variation in noise intensity is discovered and verified for the first time, with the necessary range of parameters for this phenomenon given. In terms of numerical scheme, an efficient and feasible algorithm for generating trichotomous noise is proposed, by which an algorithm based on the Caputo fractional derivative are applied. The numerical results match well with the analytical ones.

Novel Fractional-Order Current-Mode VCOs/SRCOs with All-Grounded-Passive-Elements

Five new structures of linear fractional-order voltage-controlled oscillators (FO-VCOs) utilizing only Multiplication-Mode Current Conveyors (MMCCs) and canonic numbers of all grounded passive elements (AGPEs) are presented in this paper. The derivation procedure has been detailed along with the relevant nonideal analysis. The work also presents three fractional-order cases of the oscillators and brings out the effects of port nonidealities and port parasitics on the performance of presented FO-VCOs. The workability of all the five new FO-VCOs has been validated on PSPICE utilizing library files of AD844 and AD835 for realizing MMCC employed in the FO-VCOs. For all the three special fractional-order cases, the simulation results have been tabulated along with relevant MATLAB plots that facilitate comparison among oscillation parameters and the fractional order. The harmonic distortion figures have been found out by Fourier analysis. Experimental workability of a VCO, employing MMCC realized with AD835 and AD844 of our investigation is included to support hardware functionality.

Forecasting models for chaotic fractional-order oscillators using neural networks

Forecasting models for chaotic fractional-order oscillators using neural networks

Chaotic resonance in a fractional-order oscillator system with application to mechanical fault diagnosis

This paper is to generalize the research on chaotic resonance (CR) towards fractional-order chaotic systems and then develop a new technique for detecting weak signals embedded in strong background noise. For illustration, a fractional-order Duffing oscillator system is evaluated by means of bifurcation analysis, revealing the phenomenon of chaotic resonance, with the optimal driving amplitude falling within a chaotic interval. It is found that the weak signal can be amplified by the intrinsic fluctuations in the chaotic system instead of stochastic noise. Based on this investigation, a novel weak signal detection method is developed and successfully applied to mechanical fault diagnosis without the need of signal preprocessing. Extensive numerical results show that the signal-to-noise ratio of the incipient fault signal of machinery can be greatly improved.

On the performance of electronically tunable fractional-order oscillator using grounded resonator concept

Recent works target the design of fractional-order oscillators. However, some features of such circuits are not frequently considered in the design despite their importance in practice. This work provides an analysis of the fractional-order oscillator design procedure with a simple but still beneficial electronic tuning feature. The presented design allows us to keep a stable and non-standard phase shift between produced harmonic signals while tuning the oscillation frequency of the oscillator. Grounded fractional-order elements and modern commercially available active elements are implemented in the designed topology. Time domain results as well as spectral analysis are obtained from experimental measurements. Moreover, several values of non-standard phase shifts are tested. The experimental verification targets the low-frequency bandwidth from several hundreds Hz up to several kHz because of possible application areas in these bands (audio) and due to a very low-impedance character of the used RC constant phase elements as approximants of fractional-order capacitors.

Control of amplitude death by coupling range in a network of fractional-order oscillators

We investigate the oscillating dynamics in a ring of network of nonlocally delay-coupled fractional-order Stuart-Landau oscillators. It is concluded that with the increasing of coupling range, the structures of death islands go from richness to simplistic, nevertheless, the area of amplitude death (AD) state is expanded along coupling delay and coupling strength directions. The increased coupling range can prompt the coupled systems with low frequency to occur AD. When system size varies, the area of death islands changes periodically, and the linear function relationship between periodic length and coupling range can be deduced. Thus, one can modulate the oscillating dynamics by adjusting the relationship between coupling range and system size. Furthermore, the results of numerical simulations are consistent with theoretical analysis.

Fractional-Order Hidden Attractor Based on the Extended Liu System

In this paper, a new commensurate fractional-order chaotic oscillator is presented. The mathematical model with a weak feedback term, which is named hypogenetic flow, is proposed based on the Liu system. And with changing the parameters of the system, the hidden attractor can have no equilibrium points or line equilibrium. What is more interesting is that under the occasion that no equilibrium point can be obtained, the phase trajectory can converge to a minimal field under the lead of some initial conditions, similar to the fixed point. We call it the virtual equilibrium point. On the other hand, when the value of parameters can produce an infinite number of equilibrium points, the line equilibrium points are nonhyperbolic. Moreover than that, there are coexistence attractors, which can present hyperchaos, chaos, period, and virtual equilibrium point. The dynamic characteristics of the system are analyzed, and the parameter estimation is also studied. Then, an electronic circuit implementation of the system is built, which shows the feasibility of the system. At last, for the fractional system with hidden attractors, the finite-time synchronization control of the system is carried out based on the finite-time stability theory of the fractional system. And the effectiveness of the controller is verified by numerical simulation.

Synchronization Analysis on Two-Layer Networks of Fractional-Order Systems: IntraIayer and InterIayer Synchronization

The paper focuses on a two-layer network composed of fractional-order oscillators, and investigates the mesoscale collective dynamical behaviors of the network, i.e., intra-layer synchronization and inter-layer synchronization. Specifically, the necessary constraint conditions are firstly presented for achieving intra-layer synchronization and inter-layer synchronization. With the constraints, based on the Lyapunov stability theory and the fractional-order system theory, sufficient conditions for inter-layer synchronization and intra-layer synchronization are obtained respectively. Interestingly, the conditions cover the basic synchronization criteria bridging the relationship of interacting between the topologies, inner coupling functions and coupling strengthes of the intra-layer and the inter-layer. Finally, numerical examples are provided to further verify the theoretical results.

Realisation of some current-mode fractional-order VCOs/SRCOs using multiplication mode current conveyors

In this paper for the first time a catalogue of linear voltage-controlled fractional-order oscillators employing multiplication-mode current conveyor (MMCC) have been systematically derived using state variable approach. The work also includes detailed relevant analysis of the derived oscillators. Furthermore, three special cases are considered for each of the derived oscillators. Non-ideal analysis has been done for all the oscillators to show the impact of port non-idealities on the frequency. Port parasitics are also examined to justify the deviation between the theoretical and the simulated frequency values against the controlling voltage. Fourier analysis has been carried out to find out the total harmonic distortion figures. Functionality of all the oscillators have been verified on PSPICE utilizing the library files of AD844 and AD835 to realize MMCC. Simulation results are also provided for each special case including the integer-order case. Relevant MATLAB plots are provided for one of the oscillators to facilitate comparison between factional-order and oscillation parameters of the oscillator.

Bifurcation and stability analysis of commensurate fractional-order van der Pol oscillator with time-delayed feedback

The stability and existence conditions of Hopf bifurcation of a commensurate fractional-order van der Pol oscillator with time-delayed feedback are studied. Firstly, the necessary and sufficient conditions for the asymptotic stability of the equilibrium point of fractional-order van der Pol oscillator with linear displacement feedback are obtained, and it is found that the conditions are not only related to the feedback gain, but also to the fractional order. Secondly, regarding time delay as a bifurcation parameter, the stability of the commensurate fractional-order van der Pol system with time-delayed feedback is investigated based on the characteristic equation. Under some conditions, the critical value of time delay is calculated. The equilibrium point is stable when the parameter is less than the critical value and will be unstable if the parameter is greater than it. Moreover, the conditions for the occurrence of Hopf bifurcation are obtained. Finally, choosing four typical system parameters, some numerical simulations are carried out to verify the correctness of the obtained theoretical results.

Aging transition in mixed active and inactive fractional-order oscillators.

Recently, a large number of studies have concentrated on aging transition, but they have so far been restricted to coupled integer-order oscillators. Here, we report the first study of aging transition in mixed active and inactive fractional-order oscillators. It has been demonstrated that while the heterogeneity is caused by the distance parameter, both the coupling strength and the fractional-order derivative can modulate the critical ratio at which aging transition occurs. In addition, a small fractional-order derivative may ruin the ability of oscillation and, thus, reduce the critical ratio in globally coupled fractional-order Stuart-Landau oscillators. Remarkably, the larger the natural frequency is the more easily the aging transition occurs in coupled fractional-order oscillators. Further studies have shown that, being diverse from an integer-order Stuart-Landau oscillator, the natural frequency may induce a Hopf bifurcation in a fractional-order Stuart-Landau oscillator, accordingly, introducing a new heterogeneity in the coupled fractional-order Stuart-Landau oscillators. Therein, a counterintuitive phenomenon has been found that the critical ratio depends unmonotonously on the coupling strength, which implies that the coupled fractional-order Stuart-Landau oscillators possess the weakest robustness of oscillation at a certain level of coupling strength.

Antimonotonicity and multistability in a fractional order memristive chaotic oscillator

A memristor diode bridge chaotic circuit is proposed in this paper. The proposed oscillator has only one nonlinear element in the form of memristor. Dynamical properties of the proposed oscillator are investigated. The fractional order model of the oscillator is designed using Grünwald–Letnikov (GL) method. Bifurcation diagrams are plotted which shows that the proposed oscillator exhibits multistability. Finally, the antimonotonicity property of the fractional order oscillator is discussed in detail with two control parameters. Such property has not been explored for fractional order systems before.

Hopf bifurcation, antimonotonicity and amplitude controls in the chaotic Toda jerk oscillator: analysis, circuit realization and combination synchronization in its fractional-order form

ABSTRACTIn this paper, an autonomous Toda jerk oscillator is proposed and analysed. The autonomous Toda jerk oscillator is obtained by converting an autonomous two-dimensional Toda oscillator with an exponential nonlinear term to a jerk oscillator. The existence of Hopf bifurcation is established during the stability analysis of the unique equilibrium point. For a suitable choice of the parameters, the proposed autonomous Toda jerk oscillator can generate antimonotonicity, periodic oscillations, chaotic oscillations and bubbles. By introducing two additional parameters in the proposed autonomous Toda jerk oscillator, it is possible to control partially or totally the amplitude of its signals. In addition, electronic circuit realization of the proposed Toda jerk oscillator is carried out to confirm results found during numerical simulations. The commensurate fractional-order version of the proposed autonomous chaotic Toda jerk oscillator is studied using the stability theorem of fractional-order oscillators and numerical simulations. It is found that periodic oscillations and chaos exist in the fractional-order form of the proposed Toda jerk oscillator with order less than three. Finally, combination synchronization of two fractional-order proposed autonomous chaotic Toda jerk oscillators with another fractional-order proposed autonomous chaotic Toda jerk oscillator is analysed using the nonlinear feedback control method.

Amplitude death islands in globally delay-coupled fractional-order oscillators

In this paper, amplitude death (AD) is investigated theoretically and numerically in N globally delay-coupled fractional-order oscillators. Due to the presence of fractional-order derivative and coupling delay, Laplace transform method has been utilized to obtain the characteristic equations. Then, based on Lyapunov stability, we theoretically get the boundaries and number of death islands. It has been found that with the introduction of the fractional-order derivative, many more death islands emerge, and the oscillation quenching dynamics are facilitated. We find AD only occurs between two critical fractional-order derivatives $$\alpha _c^ - $$ (lower-bounded value) and $$\alpha _c^ + $$ (upper-bounded value) which are affected by natural frequency and system size. With the increment of system size, the oscillation quenching dynamics are weakened. The number of death islands is closely geared to the fractional-order derivative and the system size. Furthermore, the results from numerical simulations best confirm the theoretical analyses.