Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos

This letter proposes a new 3D quadratic autonomous chaotic system which displays an extremely complicated dynamical behavior over a large range of parameters. The new chaotic system has five real equilibrium points. Interestingly, this system can generate one-wing, two-wing, three-wing and four-wing chaotic attractors and periodic motion with variation of only one parameter. Besides, this new system can generate two coexisting one-wing and two coexisting two-wing attractors with different initial conditions. Furthermore, the transient chaos phenomenon happens in the system. Some basic dynamical behaviors of the proposed chaotic system are studied. Furthermore, the bifurcation diagram, Lyapunov exponents and Poincaré mapping are investigated. Numerical simulations are carried out in order to demonstrate the obtained analytical results. The interesting findings clearly show that this is a special strange new chaotic system, which deserves further detailed investigation.     

Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems

In this paper, unstable dynamics is considered for the models of vibro-impact systems with linear differential equations coupled to an impact map. To provide a skeleton for the organization of chaotic attractors, we propose a method for detecting unstable periodic orbits embedded in chaotic attractors through a combination of unconstrained optimization technique and Poincaré map. Three numerical examples from different vibro-impact models demonstrate that the strategy can efficiently detect unstable periodic orbits in chaotic attractors. In order to explore the mechanism responsible for the creation of multi-dimensional tori attractors, we also present another method to detect unstable quasiperiodic orbits embedded multi-dimensional tori attractors by examining a specially transient time series. The upper bound and lower bound of the transient time series (in the Poincaré map) can be obtained by analyzing transient Lyapunov exponent and transient Lyapunov dimension. Some examples verify the effectiveness of the numerical algorithm.     

3-scroll and 4-scroll chaotic attractors generated from a new 3-D quadratic autonomous system

This article introduces a new chaotic system of 3-D quadratic autonomous ordinary differential equations, which can display 2-scroll chaotic attractors. Some basic dynamical behaviors of the new 3-D system are investigated. Of particular interest is that the chaotic system can generate complex 3-scroll and 4-scroll chaotic attractors. Finally, bifurcation analysis shows that the system can display extremely rich dynamics. The obtained results clearly show that this is a new chaotic system which deserves further detailed investigation.     

周期激励下广义蔡氏电路混沌运动中的概周期行为

由于广义蔡氏电路存在2个对称的稳定平衡点,周期激励可能导致系统出现相应于不同初值的2种共存的分岔模式. 概周期解由环面破裂进入混沌,混沌吸引子从相位不同步逐渐演化为同步,并进一步随着参数的变化,产生分裂现象. 分裂后的2个相互对称的混沌吸引子仍存在相位同步效应,这2个混沌吸引子再次相互作用后形成扩大了的混沌吸引子,并交替围绕2个子混沌结构来回振荡. 同时,在混沌过程中,其轨迹在相当长的一段时间内严格按照概周期行为振荡,即混沌结构中存在局部概周期行为,这种局部概周期行为随参数的变化会逐步减弱,直至消失.      

Nonlinear oscillations and chaotic motions in a road vehicle system with driver steering control

The nonlinear dynamics of a differential system describing the motion of a vehicle driven by a pilot is examined. In a first step, the stability of the system near the critical speed is analyzed by the bifurcation method in order to characterize its behavior after a loss of stability. It is shown that a Hopf bifurcation takes place, the stability of limit cycles depending mainly on the vehicle and pilot model parameters. In a second step, the front wheels of the vehicle are assumed to be subjected to a periodic disturbance. Chaotic and hyperchaotic motions are found to occur for some range of the speed parameter. Numerical simulations, such as bifurcation diagrams, Poincaré maps, Fourier spectrums, projection of trajectories, and Lyapunov exponents are used to establish the existence of chaotic attractors. Multiple attractors may coexist for some values of the speed, and basins of attraction for such attractors are shown to have fractal geometries.     

Estimation of Chaotic and Regular (Stick–Slip and Slip–Slip) Oscillations Exhibited by Coupled Oscillators with Dry Friction

In this paper, we present a novel approach to quantify regular or chaotic dynamics of either smooth or non-smooth dynamical systems. The introduced method is applied to trace regular and chaotic stick–slip and slip–slip dynamics. Stick–slip and slip–slip periodic and chaotic trajectories are analyzed (for the investigated parameters, a stick–slip dynamics dominates). Advantages of the proposed numerical technique are given.     

The effects of fractional order on a 3-D quadratic autonomous system with four-wing attractor

In this paper, a fractional 3-dimensional (3-D) 4-wing quadratic autonomous system (Qi system) is analyzed. Time domain approximation method (Grunwald–Letnikov method) and frequency domain approximation method are used together to analyze the behavior of this fractional order chaotic system. It is found that the decreasing of the system order has great effect on the dynamics of this nonlinear system. The fractional Qi system can exhibit chaos when the total order less than 3, although the regular one always shows periodic orbits in the same range of parameters. In some fractional order, the 4 wings are decayed to a scroll using the frequency domain approximation method which is different from the result using time domain approximation method. A surprising finding is that the phase diagrams display a character of local self-similar in the 4-wing attractors of this fractional Qi system using the frequency approximation method even though the number and the characteristics of equilibria are not changed. The frequency spectrums show that there is some shrinking tendency of the bandwidth with the falling of the system states order. However, the change of fractional order has little effect on the bandwidth of frequency spectrum using the time domain approximation method. According to the bifurcation analysis, the fractional order Qi system attractors start from sink, then period bifurcation to some simple periodic orbits, and chaotic attractors, finally escape from chaotic attractor to periodic orbits with the increasing of fractional order α in the interval [0.8,1]. The simulation results revealed that the time domain approximation method is more accurate and reliable than the frequency domain approximation method.     

Definition and Empirical Characterization of Creative Processes

Creative processes exhibit a new, thus far unrecognized, form of dynamical behavior distinct from the known classes of mechanical and chaotic dynamics. We present quantitative methods of time series analysis that distinguish creative processes from random and chaotic systems. Creative processes exhibit diversification, indicating an expanding phase space volume, which contrasts with processes that converge to equilibrium, or to periodic or chaotic attractors. Creative processes exhibit novelty, that is, they produce less recurrence than obtains from random series. Creative processes exhibit arrangement, a measure of patterned recurrences that indicates nonrandom complexity. These three measures, diversification, novelty, and arrangement, reliably identify creative dynamics and distinguish creativity from chaos and from randomness.     

A simple inductor-free memristive circuit with three line equilibria

In this work, a novel inductor-free fourth-order two-memristor-based chaotic circuit is proposed. This new circuit is developed from a current feedback op amp-based sinusoidal oscillator through replacing a linear resistor with a memristor and adding another different parallel memristor to the cascaded memristor–capacitor net. The proposed circuit can perform chaotic, fixed point, and period behaviors. The most striking feature is that this system has three line equilibria and exhibits the extreme multistability phenomenon of the coexisting infinitely many attractors. Specially, amplitude death behavior and transient transition behavior can also be found in the proposed system. By using standard nonlinear analysis tools including system dissipation, equilibrium point stability, phase portrait, Lyapunov exponent spectrum, and bifurcation diagram, the fundamental dynamical characteristics of the circuit are investigated in detail. Moreover, a MULTISIM circuit is designed to verify the numerical simulations.     

Introduction to Nonlinear Dynamics of Electronic Systems: Tutorial

The study of chaos has generated enormous interest in exploring the complexity of the behavior in nature and in technology. Many of the important features of chaotic dynamical systems can be seen using experimental and computational methods in simple nonlinear mechanical systems or electronic circuits. Starting with the study of a chaotic nonlinear mechanical system (driven damped pendulum) or a nonlinear electronic system (circuit Chua) we introduce the reader into the concepts of chaos order in Sharkovsky's sense, and topological invariants (topological entropy and topological frequencies). The Kirchhoff's circuit laws are a pair of laws that deal with the conservation of charge and energy in electric circuits, and the algebraic theory of graphs characterizes these linear systems in terms of cycles and cocycles (or cuts). Here we discuss methods (topological semiconjugacy to piecewise linear maps and Markov graphs) to find a similar situation for the nonlinear dynamics, to understanding chaotic dynamics. Thus to chaotic dynamics we associate a Markov graph, where the dynamical and topological invariants will be seen as graph theoretical quantities.     

Dissipative chaos, Shilnikov chaos and bursting oscillations in a three-dimensional autonomous system: theory and electronic implementation

A three-dimensional autonomous chaotic system is presented and physically implemented. Some basic dynamical properties and behaviors of this system are described in terms of symmetry, dissipative system, equilibria, eigenvalue structures, bifurcations, and phase portraits. By tuning the parameters, the system displays chaotic attractors of different shapes. For specific parameters, the system exhibits periodic and chaotic bursting oscillations which resemble the conventional heart sound signals. The existence of Shilnikov type of heteroclinic orbit in the three-dimensional system is proven using the undetermined coefficients method. As a result, Shilnikov criterion guarantees that the three-dimensional system has the horseshoe chaos. The corresponding electronic circuit is designed and implemented, exhibiting experimental chaotic attractors in accord with numerical simulations.     

Implementation and study of the nonlinear dynamics of a memristor-based Duffing oscillator

An electronic model of Duffing oscillator with a characteristic memristive nonlinear element is proposed instead of the classical cubic nonlinearity. The memristive Duffing oscillator circuit system is mathematically modeled, and the stability analysis presents the evolution of the proposed system. The dynamical behavior of this circuit is investigated through numerical simulations, statistical analysis, and real-time hardware experiments, which have been carried out under the external periodic force. The chaotic dynamics of the circuit is studied by means of phase diagram. It is found that the proposed circuit system shows complex behaviors, like bifurcations and chaos, three tori, transient chaos, and intermittency for a certain range of circuit parameters. The observed phenomena and scenario are illustrated in detail through experimental and numerical studies of memristive Duffing oscillator circuit. The existence of regular and chaotic behaviors is also verified by using 0–1 test measurements. In addition, the robustness of the signal strength is confirmed through signal-to-noise ratio. The numerically observed results are confirmed from the laboratory experiment.     

Chaotic dynamics in a neural network under electromagnetic radiation

In this paper, a small Hopfield neural network with three neurons is studied, in which one of the three neurons is considered to be exposed to electromagnetic radiation. The effect of electromagnetic radiation is modeled and considered as magnetic flux across membrane of the neuron, which contributes to the formation of membrane potential, and a feedback with a memristive type is used to describe coupling between magnetic flux and membrane potential. With the electromagnetic radiation being considered, the previous steady neural network can present abundant chaotic dynamics. It is found that hidden attractors can be observed in the neural network under different conditions. Moreover, periodic motion and chaotic motion appear intermittently with variations in some system parameters. Particularly, coexistence of periodic attractor, quasiperiodic attractor, and chaotic strange attractor, coexistence of bifurcation modes and transient chaos can be observed. In addition, an electric circuit of the neural network is implemented in Pspice, and the experimental results agree well with the numerical ones.     

Single amplifier biquad based inductor-free Chua’s circuit

The present paper reports an inductor-free realization of Chua??s circuit, which is designed by suitably cascading a single amplifier biquad based active band pass filter with a Chua??s diode. The system has been mathematically modeled with three-coupled first-order autonomous nonlinear differential equations. It has been shown through numerical simulations of the mathematical model and hardware experiments that the circuit emulates the behaviors of a classical Chua??s circuit, e.g., fixed point behavior, limit cycle oscillation, period doubling cascade, chaotic spiral attractors, chaotic double scrolls and boundary crisis. The occurrence of chaotic oscillation has been established through experimental power spectrum, and quantified with the dynamical measure like Lyapunov exponents.     

Dynamics of a linear system with time-dependent mass and a coupled light mass with non-smooth potential

Multi-scale dynamics of two coupled oscillators, the linear one with varying mass and a non-smooth light system is studied. The light system, namely nonlinear energy sink (NES) is implemented for passively controlling the linear system against external impulses and/or forces. Obtained invariant manifold of the system at fast time scale let us explain the system behavior during its transient and steady-state regime while predicted dynamics of the system at slow time scale let us detect the positions of fixed and fold singularities by explanation of its behavior.     

Dynamics of heteroclinic tangles with an unbroken saddle connection

This paper studies a second-order differential equation with two heteroclinic solutions to two saddle fixed points. When an equation is periodically perturbed, one heteroclinic solution generates tangle while the other remains unbroken. We illustrate chaotic dynamics in the sense of Smale horseshoes and Hénon-like attractors with SRB measures. More explicitly, we obtain three different dynamical phenomena, namely the transient heteroclinic tangles containing no physical measures, heteroclinic tangles dominated by sinks representing stable dynamical behavior, and heteroclinic tangles with Hénon-like attractors admitting SRB measures representing chaos. We also demonstrate that three types of phenomena repeat periodically as the forcing magnitude goes to zero.     

Orbital decomposition for ill-behaved event sequences: transients and superordinate structures

Time series analysis is often challenged by the presence of transient functions. We examined some types of transients found in time series of events that lend themselves to symbolic dynamics analysis through the method of orbital decomposition, which is based on the principle that chaotic series arise from coupled oscillators. Synthetic data sets were constructed to study the impact of intrusive events, intrusive series, merged functions, non-coupled oscillators, and driving oscillations on the patterns of final statistics obtained from orbital decomposition analysis. Two real-world data sets - a logbook of the ritual behaviors of a patient with obsessive compulsive disorder and a time series of kill dates from an infamous serial murderer - were examined for non-ergodic properties similar to those found in the synthetic data.     

Analysis of a new three-dimensional system with multiple chaotic attractors

In this paper, a new three-dimensional autonomous system with complex dynamical behaviors is reported. This new system has three quadratic nonlinear terms and one constant term. One remarkable feature of the system is that it can generate multiple chaotic and multiple periodic attractors in a wide range of system parameters. The presence of coexisting chaotic and periodic attractors in the system is investigated. Moreover, it is easily found that the new system also can generate four-scroll chaotic attractor. Some basic dynamical behaviors of the system are investigated through theoretical analysis and numerical simulation.     

Nonlinear control in an electromechanical transducer with chaotic behaviour

The electromechanical transducer considered in this work is composed of a mechanical oscillator linked to an electronic circuit. Simulations results have determined that for some combination of parameters the electromechanical system is subject to chaotic motion with resonant transient behavior, and after the resonant transient the mechanical system (MS) synchronizes with the electrical system (ES). In order to improve the transient response, avoiding both the chaotic and resonant behaviors, a nonlinear control system is designed, a feedback control strategy is used to drive the system into the desired periodic orbit, and a nonlinear feedforward strategy is used to keep the system into the periodic orbit, obtained by the Fourier series. Two control techniques are used and compared, namely: the state dependent Ricatti equation and the optimal linear feedback control. Numerical simulations results are shown in order to compare the results, considering parametric uncertainties. Additionally, the energy transfer “pumping” between the ES and the MS is also analysed.