Hyperchaotic memristive system with hidden attractors and its adaptive control scheme

In this research work a novel 4-D memristive system is presented. The proposed system belongs to the category of dynamical systems with hidden attractors as it displays a line of equilibrium points. Also, it has an hyperchaotic dynamical behavior in a particular range of its parameters space. System’s behavior is investigated through numerical simulations, by using well-known tools of nonlinear theory, such as phase portrait, bifurcation diagram, Lyapunov exponents and Poincaré map. Next, the case of chaos control of the system with unknown parameters using adaptive control method is investigated. Finally, an electronic circuit realization of the novel hyperchaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model.     

Bifurcation, response scenarios and dynamic integrity in a single-mode model of noncontact atomic force microscopy

The nonlinear dynamical behavior of a single-mode model of noncontact AFM is analyzed in terms of attractors robustness and basins integrity. The model considered for the analyses, proposed in (Hornstein and Gottlieb in Nonlinear Dyn. 54:93–122, 2008), consistently includes the nonlinear atomic interaction and is studied under scan excitation (which appears as parametric excitation) and vertical excitation (which is prevalently external). Local bifurcation analyses are carried out to identify the overall stability boundary in the excitation parameter space as the envelope of system local escapes, to be compared with the one obtained via numerical simulations. The dynamical integrity of periodic bounded solutions is studied, and basin erosion is evaluated by means of two different integrity measures. The obtained erosion profiles allow us to dwell on the possible lack of homogeneous safety of the stability boundary in terms of robustness of the attractors, and to identify practical escape thresholds ensuring an a priori design safety target.     

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.     

Chaos and Hopf bifurcation of a finance system

The complex dynamical behavior of a finance system is investigated in this paper. The Ruelle–Takens route to chaos and strange nonchaotic attractors (SNA) are found through numerical simulations. Then the system with time-delayed feedback is considered and the stability and Hopf bifurcation of the controlled system are investigated. This research has important theoretical and practical meanings.     

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.     

Partially unstable attractors in networks of forced integrate-and-fire oscillators

The asymptotic attractors of a nonlinear dynamical system play a key role in the long-term physically observable behaviors of the system. The study of attractors and the search for distinct types of attractor have been a central task in nonlinear dynamics. In smooth dynamical systems, an attractor is often enclosed completely in its basin of attraction with a finite distance from the basin boundary. Recent works have uncovered that, in neuronal networks, unstable attractors with a remote basin can arise, where almost every point on the attractor is locally transversely repelling. Herewith we report our discovery of a class of attractors: partially unstable attractors, in pulse-coupled integrate-and-fire networks subject to a periodic forcing. The defining feature of such an attractor is that it can simultaneously possess locally stable and unstable sets, both of positive measure. Exploiting the structure of the key dynamical events in the network, we develop a symbolic analysis that can fully explain the emergence of the partially unstable attractors. To our knowledge, such exotic attractors have not been reported previously, and we expect them to arise commonly in biological networks whose dynamics are governed by pulse (or spike) generation.     

Absolute term introduced to rebuild the chaotic attractor with constant Lyapunov exponent spectrum

When positive or negative feedback of absolute terms are introduced in dynamic equations of improved chaotic system with constant Lyapunov exponent spectrum, diverse structures of chaotic attractors can be rebuilt, numbers of novel attractors found and subsequently the dynamical behavior property analyzed. Drawing on the concept of global phase reversal and its implementation methods, three main features are discussed and a systematic conclusion is made, that is, the unique class of chaotic system which utilizes merely absolute terms to realize nonlinear function possesses the following three properties: adjustable amplitude, adjustable phase reversal and constant Lyapunov exponent spectrum.     

非光滑动力系统胞映射计算方法

针对非光滑动力学系统特点,在胞映射思想基础上,引入拉回积分等分析手段,得到了非光滑系统吸引子和吸引域的胞映射计算方法.并以一类碰振系统为例,给出了其吸引子和具有复杂分形边界的吸引域,并验证了该方法的有效性.     

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.     

粘弹性圆薄板的动力学行为

基于线性粘弹性力学的Boltzmann叠加原理,给出粘弹性圆薄板动力学分析的初边值问题。通过一定的简化后得到描述薄板力学行为的四维非线性非自治动力系统。综合使用非线性动力学中的数值分析方法,研究了参数对粘弹性圆薄板动力学行为的影响。同时计算了吸引子的Lyapunov维、相关维和点形维。     

Synchronization and circuit simulation of a new double-wing chaos

A new three-dimensional double-wing chaotic system with three quadratic terms was proposed. And the parameters which can induce the system are analyzed. The system with five equilibrium points has sophisticated dynamical behaviors and it is further investigated in details, including phase trajectory, Lyapunov exponent spectrum, Poincaré map, spectrogram map and dissipativity analysis. The circuit simulation results of the chaotic attractors are in agreement with numerical simulations. Furthermore, numerical simulations indicate that mismatch synchronization can be achieved and circuit simulations of the system synchronization are also presented.     

Synthesizing attractors of Hindmarsh–Rose neuronal systems

In this paper a periodic parameter-switching scheme is applied to the Hindmarsh–Rose neuronal system to synthesize certain attractors. Results show numerically, via computer graphic simulations, that the obtained synthesized attractor belongs to the class of all admissible attractors for the Hindmarsh–Rose neuronal system and matches the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This feature allows us to imagine that living beings are able to maintain vital behavior while the control parameter switches so that their dynamical behavior is suitable for the given environment.     

Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system

In this paper, we introduce a new chaotic complex nonlinear system and study its dynamical properties including invariance, dissipativity, equilibria and their stability, Lyapunov exponents, chaotic behavior, chaotic attractors, as well as necessary conditions for this system to generate chaos. Our system displays 2 and 4-scroll chaotic attractors for certain values of its parameters. Chaos synchronization of these attractors is studied via active control and explicit expressions are derived for the control functions which are used to achieve chaos synchronization. These expressions are tested numerically and excellent agreement is found. A Lyapunov function is derived to prove that the error system is asymptotically stable.     

Expected escape times from attractor basins due to low intensity noise

In this paper, a numerical approach is described to estimate escape times from attractor basins when a dynamical system is subjected to noise or stochastic perturbations. Noise can affect nonlinear system response by driving solution trajectories to different attractors. The changes in physical behavior can be observed as amplitude and phase change of periodic oscillations, initiation or annihilation of chaotic motion, phase synchronization, and so on. Estimating probability of transitions from one attractor to another, and predicting escape times are essential for quantifying the effects of noise on the system response. In this paper, a numerical approach is outlined where probability transition maps are generated between grids. Then, these maps are iterated to find the probability distribution after long durations, wherein, a constant escape rate can be observed between basins. The constant escape rate is then used to estimate the average escape times. The approach is applicable to systems subjected to low-intensity stochastic disturbances and with long escape times, where Monte Carlo simulations are impractical. Escape times up to \(10^{13}\) periods are estimated without relying on computationally expensive computations.

     

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.     

Finite-time stabilization of uncertain non-autonomous chaotic gyroscopes with nonlinear inputs

Gyroscopes are one of the most interesting and everlasting nonlinear nonautonomous dynamical systems that exhibit very complex dynamical behavior such as chaos.In this paper,the problem of robust stabi...     

Hopf bifurcation control for stochastic dynamical system with nonlinear random feedback method

Hopf bifurcation control in nonlinear stochastic dynamical system with nonlinear random feedback method is studied in this paper. Firstly, orthogonal polynomial approximation is applied to reduce the controlled stochastic nonlinear dynamical system with nonlinear random controller to the deterministic equivalent system, solvable by suitable numerical methods. Then, Hopf bifurcation control with nonlinear random feedback controller is discussed in detail. Numerical simulations show that the method provided in this paper is not only available to control the stochastic Hopf bifurcation in nonlinear stochastic dynamical system, but is also superior to the deterministic nonlinear feedback controller.     

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.     

Lyapunov type stability and Lyapunov exponent for exemplary multiplicative dynamical systems

This paper presents analysis of Lyapunov type stability for multiplicative dynamical systems. It has been formally defined and numerical simulations were performed to explore nonlinear dynamics. Chaotic behavior manifested for exemplary multiplicative dynamical systems has been confirmed by calculated Lyapunov exponent values.