Showing papers in "Nonlinear Dynamics in 2012"

Stability analysis of Caputo fractional-order nonlinear systems revisited

Abstract: In this paper stability analysis of fractional-order nonlinear systems is studied. An extension of Lyapunov direct method for fractional-order systems using Bihari’s and Bellman–Gronwall’s inequality and a proof of comparison theorem for fractional-order systems are proposed.

On intentional introduction of stiffness nonlinearities for energy harvesting under white Gaussian excitations

Abstract: A significant body of the open literature on vibratory energy harvesting is currently focused on the concept of purposeful inclusion of stiffness nonlinearities for broadband transduction. When compared to their linear resonant counterparts, nonlinear energy harvesters have a wider steady-state frequency bandwidth, leading to the idea that they can be utilized to improve performance especially in random and non-stationary vibratory environments. To further investigate this common belief, this paper studies the response of vibratory energy harvesters to white Gaussian excitations. Both mono- and bi-stable piezoelectric Duffing-type harvesters are considered. The Fokker–Plank–Kolmogorov equation governing the evolution of the system’s transition probability density function is formulated and used to generate the moment differential equations governing the response statistics. The moment equations are then closed using a fourth-order cumulant-neglect closure scheme and the relevant steady-state response statistics are obtained. It is demonstrated that the energy harvester’s time constant ratio, i.e., the ratio between the nominal period of the mechanical subsystem and the time constant of the harvesting circuit, plays a critical role in characterizing the performance of nonlinear harvesters in a random environment. When the time constant ratio is large, stiffness-type nonlinearities have very little influence on the voltage response. In such a case, no matter how the potential function of the harvester is altered, it does not affect the average output power of the device. When the time constant ratio is small, the influence of the nonlinearity on the voltage output becomes more prevalent. In this case, a Duffing-type mono-stable harvester can never outperform its linear counterpart. A bi-stable harvester, on the other hand, can outperform a linear harvester only when it is designed with the proper potential energy function based on the known noise intensity of the excitation. Such conclusions hold for harvesters with nonlinearities appearing in the restoring force.

A new image encryption method: parallel sub-image encryption with hyper chaos

Abstract: A new image encryption scheme, based on a total shuffling and parallel encryption algorithm is proposed in this paper. Two chaotic systems have been used in the encryption algorithm to confuse the re- lationship between the plain-image and the cipher- image. To make the encryption procedure more con- fusing and complex, the plain-image is first divided into 4 sub-images and then the position of each sub- image is changed pseudo-randomly according to a lo- gistic map. Next, a total shuffling matrix is used to shuffle the position of pixels in the whole image and then sub-images are encrypted simultaneously in a parallel manner. The experimental results on USC data base demonstrate that the proposed encryption algo- rithm has a low time complexity and has the advan- tages of large key space and high security. Moreover, the robustness of this locally encryption method is much more in contrast with other encryption schemes and the distribution of gray values has a random-like behavior in the encrypted image.

Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique

Abstract: In this paper, a novel fractional-order terminal sliding mode control approach is introduced to control/synchronize chaos of fractional-order nonautonomous chaotic/hyperchaotic systems in a given finite time. The effects of model uncertainties and external disturbances are fully taken into account. First, a novel fractional nonsingular terminal sliding surface is proposed and its finite-time convergence to zero is analytically proved. Then an appropriate robust fractional sliding mode control law is proposed to ensure the occurrence of the sliding motion in a given finite time. The fractional version of the Lyapunov stability is used to prove the finite-time existence of the sliding motion. The proposed control scheme is applied to control/synchronize chaos of autonomous/nonautonomous fractional-order chaotic/hyperchaotic systems in the presence of both model uncertainties and external disturbances. Two illustrative examples are presented to show the efficiency and applicability of the proposed finite-time control strategy. It is worth to notice that the proposed fractional nonsingular terminal sliding mode control approach can be applied to control a broad range of nonlinear autonomous/nonautonomous fractional-order dynamical systems in finite time.

Image encryption algorithm using chaotic Chebyshev generator

Abstract: In this paper, we present a chaotic image encryption algorithm in which the key stream is generated by nonlinear Chebyshev function. The novel method of designing pseudorandom chaotic sequence is carried out with the created secret keys depending on with each other. We then make multiple permutation of pixels to decrease the strong correlation between adjacent pixels in original plain image. Further, a two-dimensional Chebyshev function is considered to avoid known-plaintext and chosen-plaintext attacks in diffusion process, i.e., even with a one-bit change in original plain image, the encrypted image would become different greatly. Simulation results are given to show that the proposed method can offer us an efficient way of encrypting image.

Adaptive sliding mode control of dynamic system using RBF neural network

Abstract: This paper presents a robust adaptive sliding mode control strategy using radial basis function (RBF) neural network (NN) for a class of time varying system in the presence of model uncertainties and external disturbance. Adaptive RBF neural network controller that can learn the unknown upper bound of model uncertainties and external disturbances is incorporated into the adaptive sliding mode control system in the same Lyapunov framework. The proposed adaptive sliding mode controller can on line update the estimates of system dynamics. The asymptotical stability of the closed-loop system, the convergence of the neural network weight-updating process, and the boundedness of the neural network weight estimation errors can be strictly guaranteed. Numerical simulation for a MEMS triaxial angular velocity sensor is investigated to verify the effectiveness of the proposed adaptive RBF sliding mode control scheme.

Nonlinear damping in a micromechanical oscillator

Abstract: Nonlinear elastic effects play an important role in the dynamics of microelectromechanical systems (MEMS). A Duffing oscillator is widely used as an archetypical model of mechanical resonators with nonlinear elastic behavior. In contrast, nonlinear dissipation effects in micromechanical oscillators are often overlooked. In this work, we consider a doubly clamped micromechanical beam oscillator, which exhibits nonlinearity in both elastic and dissipative properties. The dynamics of the oscillator is measured in both frequency and time domains and compared to theoretical predictions based on a Duffing-like model with nonlinear dissipation. We especially focus on the behavior of the system near bifurcation points. The results show that nonlinear dissipation can have a significant impact on the dynamics of micromechanical systems. To account for the results, we have developed a continuous model of a geometrically nonlinear beam-string with a linear Voigt–Kelvin viscoelastic constitutive law, which shows a relation between linear and nonlinear damping. However, the experimental results suggest that this model alone cannot fully account for all the experimentally observed nonlinear dissipation, and that additional nonlinear dissipative processes exist in our devices.

Modeling and analysis of piezoaeroelastic energy harvesters

Abstract: This work investigates the influence of structural and aerodynamic nonlinearities on the dynamic behavior of a piezoaeroelastic system. The system is composed of a rigid airfoil supported by nonlinear torsional and flexural springs in the pitch and plunge motions, respectively, with a piezoelectric coupling attached to the plunge degree of freedom. The analysis shows that the effect of the electrical load resistance on the flutter speed is negligible in comparison to the effects of the linear spring coefficients. The effects of aerodynamic nonlinearities and nonlinear plunge and pitch spring coefficients on the system’s stability near the bifurcation are determined from the nonlinear normal form. This is useful to characterize the effects of different parameters on the system’s output and ensure that subcritical or “catastrophic” bifurcation does not take place. Numerical solutions of the coupled equations for two different configurations are then performed to determine the effects of varying the load resistance and the nonlinear spring coefficients on the limit-cycle oscillations (LCO) in the pitch and plunge motions, the voltage output and the harvested power.

Control of a class of fractional-order chaotic systems via sliding mode

Abstract: This paper investigates the chaos control of a class of fractional-order chaotic systems via sliding mode. First, the sliding mode control law is derived to make the states of the fractional-order chaotic systems asymptotically stable. Second, the designed control scheme guarantees asymptotical stability of the uncertain fractional-order chaotic systems in the presence of an external disturbance. Finally, simulation results are given to demonstrate the effectiveness of the proposed sliding mode control method.

A new car-following model accounting for varying road condition

Abstract: In this paper, we develop a new car-following model with consideration of varying road condition based on the empirical data. Firstly, we explore the effects of road condition on uniform flow from analytical and numerical perspectives. The results indicate that road condition has great influences on uniform flow, i.e., good road condition can enhance the velocity and flow and their increments will increase when road condition becomes better; bad road conditions will reduce the velocity and flow and their reductions will increase when road condition turns worse. Secondly, we study the effects of road conditions on the starting and braking processes. The numerical results show that good road condition will speed up the two processes and that bad road condition will slow down the two processes. Finally, we study the effects of road condition on small perturbation. The numerical results indicate that the stop-and-go phenomena resulted by small perturbation will become more serious when the road condition becomes better.

Flight dynamics and control of flapping-wing MAVs: a review

Abstract: This paper provides a thorough review of the significant work done so far in the area of flight dynamics and control of flapping-wing micro-air-vehicles (MAVs). It provides the background necessary to do research in that area. Furthermore, it raises questions that need to be addressed in the future. The three main blocks constituting the flight dynamic framework of flapping MAVs are reviewed. These blocks are the flapping kinematics, the aerodynamic modeling, and the body dynamics. The design and parametrization of the flapping kinematics necessary to produce high-control authority over the MAV, as well as design of kinematics suitable for different flight conditions, are reviewed. Aerodynamic models used for analysis of flapping flight are discussed. Particular attention is given to the physical aspects captured by these models. The issues and consequences of averaging the dynamics and neglecting the wing inertia are discussed. The dynamic stability analysis of flapping MAVs is usually performed by either averaging, linearization and subsequent analysis or using Floquet theory. Both approaches are discussed. The linear and nonlinear control design techniques for flapping MAVs are also reviewed and discussed.

An efficient chaotic image encryption algorithm based on a generalized Arnold map

Abstract: An efficient image encryption algorithm using the generalized Arnold map is proposed. The algorithm is composed of two stages, i.e., permutation and diffusion. First, a total circular function, rather than the traditional periodic position permutation, is used in the permutation stage. It can substantially reduce the correlation between adjacent pixels. Then, in the stage of diffusion, double diffusion functions, i.e., positive and opposite module, are utilized with a novel generation of the keystream. As the keystream depends on the processed image, the proposed method can resist known- and chosen-plaintext attacks. Experimental results and theoretical analysis indicate the effectiveness of our method. An extension of the proposed algorithm to other chaotic systems is also discussed.

Dynamic analysis and control of a new hyperchaotic finance system

Abstract: In this paper, a new hyperchaotic finance system which is constructed based on a chaotic finance system by adding an additional state variable is presented. The basic dynamical behaviors of this hyperchaotic finance system are investigated, such as the equilibrium, stability, hyperchaotic attractor, Lyapunov exponents, and bifurcation analysis. Furthermore, effective speed feedback controllers and linear feedback controllers are designed for stabilizing hyperchaos to unstable equilibrium points. Numerical simulations are given to illustrate and verify the results.

Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters

Abstract: A global nonlinear distributed-parameter model for a piezoelectric energy harvester under para- metric excitation is developed. The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geomet- ric, inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler- Lagrange principle and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient to evaluate the per- formance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence on the harvester's behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that, for small values of the quadratic damping, there is an overhang associated with a sub- critical pitchfork bifurcation.

A driver’s memory lattice model of traffic flow and its numerical simulation

Abstract: A new lattice model of traffic flow based on Nagatani’s model is proposed by taking the effect of driver’s memory into account. The linear stability condition of the extended model is obtained by using the linear stability theory. The analytical results show that the stabile area of the new model is larger than that of the original lattice hydrodynamic model by adjusting the driver’s memory intensity parameter p of the past information in the system. The modified KdV equation near the critical point is derived to describe the traffic jam by nonlinear analysis, and the phase space could be divided into three regions: the stability region, the metastable region, and the unstable region, respectively. Numerical simulation also shows that our model can stabilize the traffic flow by considering the information of driver’s memory.

A novel technique for the construction of strong S-boxes based on chaotic Lorenz systems

Abstract: In cryptographic systems, the encryption process relies on the nonlinear mapping of original data or plaintext to the secure data. The mapping of data is facilitated by the application of the substitution process embedded in the cipher. It is desirable to have resistance against differential cryptanalysis, which assists in providing clues about the composition of keys, and linear secret system, where a simple approximation is created to emulate the original cipher characteristics. In this work, we propose the use of nonlinear functional chaos-based substitution process which employs a continuous time Lorenz system. The proposed substitution system eliminates the need of independent round keys in a substitution-permutation network. The performance of the new substitution box is evaluated by nonlinearity analysis, strict avalanche criterion, bit independence criterion, linear approximation probability, and differential approximation probability.

A hyperchaotic system without equilibrium

Abstract: This article introduces a new chaotic system of 4-D autonomous ordinary differential equations, which has no equilibrium. This system shows a hyper-chaotic attractor. There is no sink in this system as there is no equilibrium. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, and Poincare maps. There is little difference between this chaotic system and other chaotic systems with one or several equilibria shown by phase portraits, Lyapunov exponents and time series methods, but the Poincare maps show this system is a chaotic system with more complicated dynamics. Moreover, the circuit realization is also presented.

An anti-disturbance PD control scheme for attitude control and stabilization of flexible spacecrafts

Abstract: This paper studies the attitude control problem of spacecrafts with flexible appendages. It is well known that the unwanted vibration modes, model uncertainty and space environmental disturbances may cause degradation of the performance of attitude control systems for a flexible spacecraft. In this paper, the vibration from flexible appendages is modeled as a derivative-bounded disturbance to the attitude control system of the rigid hub. A disturbance-observer-based control (DOBC) is formulated for feedforward compensation of the elastic vibration. The model uncertainty and space environmental disturbances as well as other noises are merged into an “equivalent” disturbance. We design a composite controller with a hierarchical architecture by combining DOBC and PD control, where DOBC is used to reject the vibration effect from the flexible appendages. Numerical simulations are performed to demonstrate that by using the composite hierarchical control law, disturbances can be effectively attenuated and the robust dynamic performances be enhanced.

Power harvesting from transverse galloping of square cylinder

Abstract: The concept of exploiting galloping of square cylinders to harvest energy is investigated. The energy is harvested by attaching a piezoelectric transducer to the transverse degree of freedom. A representative model that accounts for the coupled cylinder displacement and harvested voltage is used to determine the levels of the harvested power. The focus is on the effect of the Reynolds number on the aerodynamic force, the onset of galloping, and the level of the harvested power. The quasi steady approximation is used to model the aerodynamic loads. A linear analysis is performed to determine the effects of the electrical load resistance and the Reynolds number on the onset of galloping, which is due to a Hopf bifurcation. We derive the normal form of the dynamic system near the onset of galloping to characterize the type of the instability and to determine the effects of the system parameters on its outputs near the bifurcation. The results show that the electrical load resistance and the Reynolds number play an important role in determining the level of the harvested power and the onset of galloping. The results also show that the maximum levels of harvested power are accompanied with minimum transverse displacements for both low- and high-Reynolds number configurations.

Pattern formation of an epidemic model with diffusion

Abstract: One subject of spatial epidemiology is spatial variation in disease risk or incidence. The spread of epidemics can result in strong spatial patterns of such risk or incidence: for example, pathogen dispersal might be highly localized, vectors or reservoirs for pathogens might be spatially restricted, or susceptible hosts might be clumped. Here, spatial pattern of an epidemic model with nonlinear incidence rates is investigated. The conditions for Hopf bifurcation and Turing bifurcation are gained and, in particular, exact Turing domain is found in the two parameters space. Furthermore, numerical results show that force of infection, namely β, plays an important role in the spatial pattern. More specifically, different patterns emerge as β increases. The mathematical analysis and numerical results well extend the finding of pattern formation in the epidemic models and may well explain the field observed in some areas.

An SIR model with infection delay and propagation vector in complex networks

Abstract: Based on the SIR (Susceptible-Infected-Removed) model, we propose a novel epidemic model to investigate the impact of infection delay and propagation vector on the spreading behaviors in complex networks. Mean-field approximations and extensive numerical simulations indicate that the infection delay and propagation vector can largely reduce the critical threshold and promote the outbreak of epidemics, and even lead to the case that the infectious diseases transform from the disease-free state to endemic one. The current results are greatly instructive for us to further understand the epidemic spreading and design some effective prevention and containment strategies to fight the epidemics.

Modified projective synchronization of fractional-order chaotic systems via active sliding mode control

Abstract: This paper is devoted to study the problem of modified projective synchronization of fractional-order chaotic system. Base on the stability theorems of fractional-order linear system, active sliding mode controller is proposed to synchronize two different fractional-order systems. Moreover, the controller is robust to the bounded noise. Numerical simulations are provided to show the effectiveness of the analytical results.

On the simplest fractional-order memristor-based chaotic system

Abstract: In 1695, G. Leibniz laid the foundations of fractional calculus, but mathematicians revived it only 300 years later. In 1971, L.O. Chua postulated the existence of a fourth circuit element, called memristor, but Williams’s group of HP Labs realized it only 37 years later. By looking at these interdisciplinary and promising research areas, in this paper, a novel fractional-order system including a memristor is introduced. In particular, chaotic behaviors in the simplest fractional-order memristor-based system are shown. Numerical integrations (via a predictor–corrector method) and stability analysis of the system equilibria are carried out, with the aim to show that chaos can be found when the order of the derivative is 0.965. Finally, the presence of chaos is confirmed by the application of the recently introduced 0-1 test.

pth moment exponential synchronization for stochastic delayed Cohen–Grossberg neural networks with Markovian switching

Abstract: This paper is a contribution to the analysis of the pth moment exponential synchronization problem for a class of stochastic delayed Cohen–Grossberg neural networks with Markovian switching. The jumping parameters are determined by a continuous-time, discrete-state Markov chain, and the delays are time-varying delays. By using the Lyapunov–Krasovskii functional, stochastic analysis theory, a generalized Halanay-type inequality as well as output coupling with delay feedback control technique, some novel sufficient conditions are derived to achieve complete pth moment exponential synchronization of the addressed neural networks. In particular, the traditional assumptions on the differentiability of the time varying delay and the boundedness of its derivative are removed in this paper. The results obtained in this paper generalize and improve many known results. Moreover, a numerical example and its simulation are also provided to demonstrate the effectiveness and applicability of the theoretical results.

Finite-time convergent guidance law with impact angle constraint based on sliding-mode control

Abstract: In this paper, a finite-time convergent sliding-mode guidance law with terminal impact angle constraint is presented. The guidance law insures that the line-of-sight angular rate will converge to zero before the final time of the guidance process. Meanwhile the flight-path angle will meet the terminal impact angle requirement. Based on the finite-time convergence stability theory and the variable structure control theory, the finite convergence time is determined. Finally, the simulation results show that the guidance law is effective.

Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink

Abstract: We explore the conservative and dissipative dynamics of a two-degree-of-freedom (2-DoF) system consisting of a linear oscillator and a lightweight nonlinear rotator inertially coupled to it. When the total energy of the system is large enough, the motion of the rotator is, generically, chaotic. Moreover, we show that if the damping of the rotator is sufficiently small and the damping of the linear oscillator is even smaller, then the system passes through a cascade of resonance captures (transient internal resonances) as the total energy gradually decreases. Rather unexpectedly, all these captures have the same principal frequency but correspond to different nonlinear normal modes (NNMs). In each NNM, the rotator is phase-locked into periodic motion with two frequencies. The NNMs differ by the ratio of these frequencies, which is approximately an integer for each NNM. Essentially non-integer ratios lead to incommensurate periods of ‘slow’ and ‘fast’ motions of the rotator and, thus, to its chaotic behavior between successive resonance captures. Furthermore, we show that these cascades of resonance captures lead to targeted energy transfer (TET) from the linear oscillator to the rotator, with the latter serving, in essence, as a nonlinear energy sink (NES). Since the inertially-coupled NES that we consider has no linearized natural frequency, it is capable of engaging in resonance with the linear oscillator over broad frequency and energy ranges. The results presented herein indicate that the proposed rotational NES appears to be a promising design for broadband shock mitigation and vibration energy harvesting.

Darboux transformation for an integrable generalization of the nonlinear Schrödinger equation

Abstract: A Darboux transformation for an integrable generalization of the coupled nonlinear Schrodinger equation is derived with the help of the gauge transformation between the Lax pair. As a reduction, a Darboux transformation for an integrable generalization of the nonlinear Schrodinger equation is obtained, from which some new solutions for the integrable generalization of the nonlinear Schrodinger equation are given.

Cryptanalysis and improvement of an extended chaotic maps-based key agreement protocol

Abstract: Very recently, Lee et al. (C. Lee, C. Chen, C. Wu, S. Huang, An extended chaotic maps-based key agreement protocol with user anonymity, Nonlinear Dynamics, doi: 10.1007/s11071-011-0247-4 ) proposed a chaotic maps-based key agreement protocol with user anonymity and claimed their protocol could resist various attacks. In this paper, we will point out that Lee et al.’s protocol suffers from three weaknesses: (1) inability of resisting the privileged insider attack; (2) inability of resisting the denial-of-service attack; and (3) inability of providing anonymity. To overcome the weaknesses, we also proposed an improved protocol. The analysis shows our protocol is more suitable for practical applications.

Phenomena and modeling of piezoelectric energy harvesting from freely oscillating cylinders

Abstract: We investigate energy harvesting from vortex-induced vibrations of a freely moving rigid circular cylinder with a piezoelectric transducer attached to its transverse degree of freedom. The power levels that can be generated from these vibrations and variations of these levels with the freestream velocity are determined. A mathematical model that accounts for the coupled lift force, cylinder motion, and harvested voltage is presented. Linear analysis is performed to determine the effect of the electrical load resistance of the transducer on the natural frequency of the cylinder and the onset of synchronization (the shedding frequency is equal to the cylinder oscillating frequency) region. The impact of the nonlinearities on the cylinder response and harvested energy is investigated. The results show that the load resistance shifts the onset of synchronization to higher freestream velocities. For two different system parameters, the results show that the nonlinearities result in a hardening behavior for some values of the load resistance.

Nonlinear dynamic response of nanotube-reinforced composite plates resting on elastic foundations in thermal environments

Abstract: This paper presents an investigation on the nonlinear dynamic response of carbon nanotube-reinforced composite (CNTRC) plates resting on elastic foundations in thermal environments. Two configurations, i.e., single-layer CNTRC plate and three-layer plate that is composed of a homogeneous core layer and two CNTRC surface sheets, are considered. The single-walled carbon nanotube (SWCNT) reinforcement is either uniformly distributed (UD) or functionally graded (FG) in the thickness direction. The material properties of FG-CNTRC plates are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The motion equations are based on a higher-order shear deformation theory with a von Karman-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. The equations of motion that includes plate-foundation interaction are solved by a two-step perturbation technique. Two cases of the in-plane boundary conditions are considered. Initial stresses caused by thermal loads or in-plane edge loads are introduced. The effects of material property gradient, the volume fraction distribution, the foundation stiffness, the temperature change, the initial stress, and the core-to-face sheet thickness ratio on the dynamic response of CNTRC plates are discussed in detail through a parametric study.