Double-well Duffing Oscillator Research Articles

Realization of logic gates in bi-directionally coupled nonlinear oscillators.

Implementation of logic gates has been investigated in nonlinear dynamical systems from various perspectives over the years. Specifically, logic gates have been implemented in both single nonlinear systems and coupled nonlinear oscillators. The majority of the works in the literature have been done on the evolution of single oscillators into OR/AND or NOR/NAND logic gates. In the present study, we demonstrate the design of logic gates in bi-directionally coupled double-well Duffing oscillators by applying two logic inputs to the drive system alone along with a fixed bias. The nonlinear system, comprising both bi-directional components, exhibits varied logic behaviors within an optimal range of coupling strength. Both attractive and repulsive couplings yield similar and complementary logic behaviors in the first and second oscillators. These couplings play a major role in exhibiting fundamental and universal logic gates in simple nonlinear systems. Under a positive bias, both the first and second oscillators demonstrate OR logic gate for the attractive coupling, while exhibiting OR and NOR logic gates, respectively, for the repulsive coupling. Conversely, under a negative bias, both the first and second oscillators display AND logic gate for the attractive coupling, and AND and NAND logical outputs for the repulsive coupling. Furthermore, we confirm the robustness of the bi-directional oscillators against moderate noise in maintaining the desired logical outputs.

Experimental investigation of the chaotification of a Duffing-like electronic oscillator under two-frequency excitation.

Two-frequency excitation has recently emerged as an efficient method to generate strong chaotification of Duffing and Duffing-like dynamical systems with both single- and double-well potentials. For the systems with a double-well potential, large continuous regions with robust chaos (chaotic attractor insensitive to changes in the system parameters) have been predicted to exist when the method is applied. Motivated by these theoretical results, in this work, we investigate experimentally the chaotification under two-frequency excitation of a simple electronic circuit analogous to the double-well Duffing oscillator. The experimental results confirm the theoretical expectations, and a strong chaotification is observed. Evidences are also presented that the chaotic attractor is robust. Therefore, the work establishes experimentally the two-frequency excitation as a simple and reliable method of chaotification. Furthermore, because of its ease of fabrication, the experimental results turn the particular circuit considered in this work into an interesting practical alternative as a reliable source of continuous-time chaotic signals.

Dynamics of coexisting rotating waves in unidirectional rings of bistable Duffing oscillators

We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies.

Coupled non-oscillatory Duffing oscillators: Multistability, multiscroll chaos generation and circuit realization

We consider the dynamics of two coupled double-well non-oscillatory cubic Duffing oscillators. The coupling is realized by perturbing each one’s amplitude with a signal proportional to the amplitude of the other one. We prove that the coupling induces in the resulting fourth-order self-driven nonlinear system up to nine equilibrium points and engendered extremely complex dynamic features amongst which the coexistence of several bifurcation branches, multiple Hopf bifurcations (when a single parameter is considered), multistable behaviors (e.g. a pair of coexisting double scroll chaos, four coexisting mono-scroll chaos), and four-scroll chaotic attractor. A detailed investigation of these latter features is carried out by resorting to both analytical (i.e. Hopf bifurcation theorem, Routh–Hurwitz criterion) and numerical tools (e.g. Lyapunov exponent diagrams, 1D bifurcation diagrams, frequency spectrum, phase space trajectory plots, time series and attraction basins). An analog circuit implementing the coupled Duffing oscillators is performed by using simple electronic components (i.e. capacitors, resistors, quadruple operational amplifiers and analog multipliers chips). Experimentally captured phase portraits support the conclusions of the theoretical study. One of the important conclusions of this work is that the coupling of non-oscillatory Duffing type oscillators can be considered as an alternative approach to the generation of multiscroll chaos.

Nonlinear system identification of a double-well Duffing oscillator with position-dependent friction

Nonlinear system identification of a double-well Duffing oscillator with position-dependent friction

Demonstrating Filtered Feedback Control Near a Boundary Crisis

In this paper we address the problem of controlling chaos by using a selective filter inserted in a negative feedback loop. This has been achieved in driven double-well Duffing oscillator in a parameter region exhibiting both a boundary crisis and generalized bistability. The optimization of the filter response allows us to improve the reduction of the control signal when the periodic solution competing with the chaotic attractor is approached. The advantage in using a selective frequency approach with respect to the Pyragas's method mainly resides in the tunability of the maximum of its amplitude response and zero phase condition. Numerical simulations performed in the temporal domain match with the experiment and make us glimpse the potential of an automatic adaptive strategy adopting filter parameter variations in order to reach an optimal feedback signal reduction.

Dynamics of a ring of three unidirectionally coupled Duffing oscillators with time-dependent damping

We study dynamics of a ring of three unidirectionally coupled double-well Duffing oscillators for three different values of the damping coefficient: fixed, proportional to time, and inversely proportional to time. The system dynamics in all cases are analyzed using time series, Fourier and Hilbert transforms, Poincaré sections, bifurcation diagrams, and Lyapunov exponents for various coupling strengths and damping coefficients. In the first case, we observe a well-known route from a stable steady state to hyperchaos through Hopf bifurcation and a series of torus bifurcations, as the coupling strength is increased. In the second case, the system is highly dissipative and converges into one of the stable equilibria. Finally, in the third case, transient toroidal hyperchaos takes place.

Strong chaotification and robust chaos in the Duffing oscillator induced by two-frequency excitation

In this work, we demonstrate numerically that two-frequency excitation is an effective method to produce chaotification over very large regions of the parameter space for the Duffing oscillator with single- and double-well potentials. It is also shown that chaos is robust in the last case. Robust chaos is characterized by the existence of a single chaotic attractor which is not altered by changes in the system parameters. It is generally required for practical applications of chaos to prevent the effects of fabrication tolerances, external influences, and aging that can destroy chaos. After showing that very large and continuous regions in the parameter space develop a chaotic dynamics under two-frequency excitation for the double-well Duffing oscillator, we demonstrate that chaos is robust over these regions. The proof is based upon the observation of the monotonic changes in the statistical properties of the chaotic attractor when the system parameters are varied and by its uniqueness, demonstrated by changing the initial conditions. The effects of a second frequency in the single-well Duffing oscillator is also investigated. While a quite significant chaotification is observed, chaos is generally not robust in this case.

Strange nonchaotic attractors for computation.

We investigate the response of quasiperiodically driven nonlinear systems exhibiting strange nonchaotic attractors (SNAs) to deterministic input signals. We show that if one uses two square waves in an aperiodic manner as input to a quasiperiodically driven double-well Duffing oscillator system, the response of the system can produce logical output controlled by such a forcing. Changing the threshold or biasing of the system changes the output to another logic operation and memory latch. The interplay of nonlinearity and quasiperiodic forcing yields logical behavior, and the emergent outcome of such a system is a logic gate. It is further shown that the logical behaviors persist even for an experimental noise floor. Thus the SNA turns out to be an efficient tool for computation.

Efficient potential well escape for bi-stable Duffing oscillators

The problem of escape from a potential well of bi-stable oscillators has attracted attention given the diversity of physical and engineering systems described by this mathematical model. Most previous studies have considered quasi-static dynamics leading to escape. In devising efficient escape strategies for structures, transient conditions have not yet received adequate consideration. In this study, the intra-well nonlinear resonant dynamics of bi-stable systems are studied and exploited, yielding a time-efficient strategy for triggering minimal amplitude escape by employing transient perturbations. The response characteristics of both, the symmetric and asymmetric double-well Duffing oscillators are explored analytically to identify the stable solution branches for any given forcing configuration. Based on the basins of attraction of the stable attractors, a novel actuation methodology employing controlled perturbations in the phase of the forcing for driving the system into a series of high-amplitude limit cycle oscillations and eventual escape to the desired stable solution is proposed. Additionally, accelerated settling to the desired configuration is achieved by implementing state feedback techniques. The proposed algorithm serves as a potential tool for implementing fast shape adaptation in bi-stable structural systems.

Characterizing Complex Dynamics in the Classical and Semi-Classical Duffing Oscillator Using Ordinal Patterns Analysis.

The driven double-well Duffing oscillator is a well-studied system that manifests a wide variety of dynamics, from periodic behavior to chaos, and describes a diverse array of physical systems. It has been shown to be relevant in understanding chaos in the classical to quantum transition. Here we explore the complexity of its dynamics in the classical and semi-classical regimes, using the technique of ordinal pattern analysis. This is of particular relevance to potential experiments in the semi-classical regime. We unveil different dynamical regimes within the chaotic range, which cannot be detected with more traditional statistical tools. These regimes are characterized by different hierarchies and probabilities of the ordinal patterns. Correlation between the Lyapunov exponent and the permutation entropy is revealed that leads us to interpret dips in the Lyapunov exponent as transitions in the dynamics of the system.

Power Series -Aftertreatment Technique for Nonlinear Cubic Duffing and Double-Well Duffing Oscillators

Modeling of large amplitude of structures such as slender, flexible cantilever beam and fluid-structure resting on nonlinear elastic foundations or subjected to stretching effects often lead to strongly nonlinear models of Duffing equations which are not amendable to exact analytical methods. In this work, explicit analytical solutions to the large amplitude nonlinear oscillation systems of cubic Duffing and double-well Duffing oscillators are provided using power series-aftertreatment technique. The developed analytical solutions are valid for both small and large amplitudes of oscillation. The accuracy and explicitness of the analytical solutions are carried out to establish the validity of the method. Good agreements are established between the solution of the new method and established exact analytical solution. The developed analytical solutions in this work can serve as a starting point for a better understanding of the relationship between the physical quantities of the problems as it provides continuous physical insights into the problems than pure numerical or computation methods.

Implementation of dynamic dual input multiple output logic gate via resonance in globally coupled Duffing oscillators.

We have used a system of globally coupled double-well Duffing oscillators under an enhanced resonance condition to design and implement Dual Input Multiple Output (DIMO) logic gates. In order to enhance the resonance, the first oscillator in the globally coupled system alone is excited by two forces out of which one acts as a driving force and the other will be either sub-harmonic or super-harmonic in nature. We report that for an appropriate coupling strength, the second force coherently drives and enhances not only the amplitude of the weak first force to all the coupled systems but also drives and propagates the digital signals if any given to the first system. We then numerically confirm the propagation of any digital signal or square wave without any attenuation under an enhanced resonance condition for an amplitude greater than a threshold value. Further, we extend this idea for computing various logical operations and succeed in designing theoretically DIMO logic gates such as AND/NAND, OR/NOR gates with globally coupled systems.

Predicting Non-Stationary and Stochastic Activation of Saddle-Node Bifurcation

Accurately predicting the onset of large behavioral deviations associated with saddle-node bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and microscale mass sensing. In many such practices, noise and non-stationarity are unavoidable and ever-present influences. As a result, it is critical to simultaneously account for these two factors toward the estimation of parameters that may induce sudden bifurcations. Here, a new analytical formulation is presented to accurately determine the probable time at which a system undergoes an escape event as governing parameters are swept toward a saddle-node bifurcation point in the presence of noise. The double-well Duffing oscillator serves as the archetype system of interest since it possesses a dynamic saddle-node bifurcation. The stochastic normal form of the saddle-node bifurcation is derived from the governing equation of this oscillator to formulate the probability distribution of escape events. Non-stationarity is accounted for using a time-dependent bifurcation parameter in the stochastic normal form. Then, the mean escape time is approximated from the probability density function (PDF) to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verifies that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.

Effects of asymmetric potentials on bursting oscillations in Duffing oscillator

In this paper, the Duffing oscillator with three different forms of asymmetrical double-well potentials is considered. We mainly focus our attention on how the three different forms of asymmetrical potentials can affect the dynamical behaviors of the bursting oscillations generated by double-well Duffing oscillator. Applying the dynamical mechanism of bursting oscillations, we carry out an analysis on the bifurcation diagram and dynamics of the system when the asymmetry parameter and amplitude of the external force vary. Our results show that the period, amplitude and numbers of spikes depend on the value of the control parameter.

Optimal Phase-Control Strategy for Damped-Driven Duffing Oscillators

Phase-control techniques of chaos aim to extract periodic behaviors from chaotic systems by applying weak harmonic perturbations with a suitably chosen phase. However, little is known about the best strategy for selecting adequate perturbations to reach desired states. Here we use experimental measures and numerical simulations to assess the benefits of controlling individually the three terms of a Duffing oscillator. Using a real-time analog indicator able to discriminate on-the-fly periodic behaviors from chaos, we reconstruct experimentally the phase versus perturbation strength stability areas when periodic perturbations are applied to different terms governing the oscillator. We verify the system to be more sensitive to perturbations applied to the quadratic term of the double-well Duffing oscillator and to the quartic term of the single-well Duffing oscillator.

Generation and Evolution of Chaos in Double-Well Duffing Oscillator under Parametrical Excitation

The generation and evolution of chaotic motion in double-well Duffing oscillator under harmonic parametrical excitation are investigated. Firstly, the complex dynamical behaviors are studied by applying multibifurcation diagram and Poincaré sections. Secondly, by means of Melnikov’s approach, the threshold value of parameterμfor generation of chaotic behavior in Smale horseshoe sense is calculated. By the numerical simulation, it is obvious that asμexceeds this threshold value, the behavior of Duffing oscillator is still steady-state periodic but the transient motion is chaotic; until the top Lyapunov exponent turns to positive, the motion of system turns to permanent chaos. Therefore, in order to gain an insight into the evolution of chaotic behavior afterμpassing the threshold value, the transient motion, basin of attraction, and basin boundary are also investigated.

Reconstructing the transient, dissipative dynamics of a bistable Duffing oscillator with an enhanced averaging method and Jacobian elliptic functions

Systems characterized by the governing equation of the bistable, double-well Duffing oscillator are ever-present throughout the fields of science and engineering. While the prediction of the transient dynamics of these strongly nonlinear oscillators has been a particular research interest, the sufficiently accurate reconstruction of the dissipative behaviors continues to be an unrealized goal. In this study, an enhanced averaging method using Jacobian elliptic functions is presented to faithfully predict the transient, dissipative dynamics of a bistable Duffing oscillator. The analytical approach is uniquely applied to reconstruct the intrawell and interwell dynamic regimes. By relaxing the requirement for small variation of the transient, averaged parameters in the proposed solution formulation, the resulting analytical predictions are in excellent agreement with exact trajectories of displacement and velocity determined via numerical integration of the governing equation. A wide range of system parameters and initial conditions are utilized to assess the accuracy and computational efficiency of the analytical method, and the consistent agreement between numerical and analytical results verifies the robustness of the proposed method. Although the analytical formulations are distinct for the two dynamic regimes, it is found that directly splicing the inter- and intrawell predictions facilitates good agreement with the exact dynamics of the full reconstructed, transient trajectory.

Periodic motions in a double-well Duffing oscillator under periodic excitation through discrete implicit mappings

In this paper, periodic motions of a periodically forced, damped, Duffing oscillator with double-well potential are analytically predicted through discrete implicit mappings. The discrete implicit maps are obtained from the differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions of the Duffing oscillator are predicted analytically, and the corresponding stability and bifurcation analysis of periodic motions are carried out through the eigenvalue analysis. Finally, from the analytical prediction, numerical results of periodic motions are performed by the numerical method of the differential equation to verify the semi-analytical prediction, and the corresponding harmonic amplitudes are computed through discrete Fourier series of the analytically predicted node points of periodic motions, and the complexity of periodic motions can be measured by the harmonic amplitude. The frame work presented in this paper can provide a semi-analytical method to find periodic motions and to determine routes of periodic motions to chaos rather than numerical simulation only in nonlinear dynamical systems, and the stable and unstable periodic motions and even chaos can be predicted analytically.

Energy Harvesting Enhancement by Vibrational Resonance

The idea to use environmental energy to power electronic portable devices is becoming very popular in recent years. In fact, the possibility of not relying only on batteries can provide devices longer operating periods in a fully sustainable way. Vibrational kinetic energy is a reliable and widespread environmental energy, that makes it a suitable energy source to exploit. In this paper, we study the electrical response of a bistable system, by using a double-well Duffing oscillator, connected to a circuit through piezoceramic elements and driven by both a low (LF) and a high frequency (HF) forcing, where the HF forcing is the environmental vibration, while the LF is controlled by us. The response amplitude at low-frequency increases, reaches a maximum and then decreases for a certain range of HF forcing. This phenomenon is called vibrational resonance. Finally, we demonstrate that by enhancing the oscillations we can harvest more electric energy. It is important to take into account that by doing so with a forcing induced by us, the amplification effect is highly controllable and easily reproducible.