Showing papers in "Nonlinear Dynamics in 2016"

Model of electrical activity in a neuron under magnetic flow effect

Abstract: The electric activities of neurons are dependent on the complex electrophysiological condition in neuronal system, and it indicates that the complex distribution of electromagnetic field could be detected in the neuronal system. According to the Maxwell electromagnetic induction theorem, the dynamical behavior in electric activity in each neuron can be changed due to the effect of internal bioelectricity of nervous system (e.g., fluctuation of ion concentration inside and outside of cell). As a result, internal fluctuation of electromagnetic field is established and the effect of magnetic flux across the membrane should be considered during the emergence of collective electrical activities and signals propagation among a large set of neurons. In this paper, the variable for magnetic flow is proposed to improve the previous Hindmarsh–Rose neuron model; thus, a four-variable neuron model is designed to describe the effect of electromagnetic induction on neuronal activities. Within the new neuron model, the effect of magnetic flow on membrane potential is described by imposing additive memristive current on the membrane variable, and the memristive current is dependent on the variation of magnetic flow. The dynamics of this modified model is discussed, and multiple modes of electric activities can be observed by changing the initial state, which indicates memory effect of neuronal system. Furthermore, a practical circuit is designed for this improved neuron model, and this model could be suitable for further investigation of electromagnetic radiation on biological neuronal system.

Review and comparison of dry friction force models

Abstract: Friction force models play a fundamental role for simulation of mechanical systems. Their choice affects the matching of numerical results with physically observed behavior. Friction is a complex phenomenon depending on many physical parameters and working conditions, and none of the available models can claim general validity. This paper focuses the attention on well-known friction models and offers a review and comparison based on numerical efficiency. However, it should be acknowledged that each model has its own distinctive pros and cons. Suitability of the model depends on physical and operating conditions. Features such as the capability to replicate stiction, Stribeck effect, and pre-sliding displacement are taken into account when selecting a friction formulation. For mechanical systems, the computational efficiency of the algorithm is a critical issue when a fast and responsive dynamic computation is required. This paper reports and compares eight widespread engineering friction force models. These are divided into two main categories: those based on the Coulomb approach and those established on the bristle analogy. The numerical performances and differences of each model have been monitored and compared. Three test cases are discussed: the Rabinowicz test and other two test problems casted for this occurrence.

Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation

Abstract: With symbolic computation, two classes of lump solutions to the dimensionally reduced equations in (2+1)-dimensions are derived, respectively, by searching for positive quadratic function solutions to the associated bilinear equations. To guarantee analyticity and rational localization of the lumps, two sets of sufficient and necessary conditions are presented on the parameters involved in the solutions. Localized characteristics and energy distribution of the lump solutions are also analyzed and illustrated.

A novel chaos-based image encryption using DNA sequence operation and Secure Hash Algorithm SHA-2

Abstract: In this paper, we propose a novel image encryption algorithm based on a hybrid model of deoxyribonucleic acid (DNA) masking, a Secure Hash Algorithm SHA-2 and the Lorenz system. Our study uses DNA sequences and operations and the chaotic Lorenz system to strengthen the cryptosystem. The significant advantages of this approach are improving the information entropy which is the most important feature of randomness, resisting against various typical attacks and getting good experimental results. The theoretical analysis and experimental results show that the algorithm improves the encoding efficiency, enhances the security of the ciphertext and has a large key space and a high key sensitivity, and it is able to resist against the statistical and exhaustive attacks.

A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems

Abstract: This study is aimed at examining and comparing several friction force models dealing with different friction phenomena in the context of multibody system dynamics. For this purpose, a comprehensive review of present literature in this field of investigation is first presented. In this process, the main aspects related to friction are discussed, with particular emphasis on the pure dry sliding friction, stick–slip effect, viscous friction and Stribeck effect. In a simple and general way, the friction force models can be classified into two main groups, namely the static friction approaches and the dynamic friction models. The former group mainly describes the steady-state behavior of friction force, while the latter allows capturing more properties by using extra state variables. In the present study, a total of 21 different friction force models are described and their fundamental physical and computational characteristics are discussed and compared in details. The application of those friction models in multibody system dynamic modeling and simulation is then investigated. Two multibody mechanical systems are utilized as demonstrative application examples with the purpose of illustrating the influence of the various frictional approaches on the dynamic response of the systems. From the results obtained, it can be stated that both the choice of the friction force model and friction parameters involved can significantly affect the simulated/modeled dynamic response of mechanical systems with friction.

Lump solutions to dimensionally reduced $$\varvec{p}$$ p -gKP and $$\varvec{p}$$ p -gBKP equations

Abstract: Based on generalized bilinear forms, lump solutions, rationally localized in all directions in the space, to dimensionally reduced p-gKP and p-gBKP equations in (2+1)-dimensions are computed through symbolic computation with Maple. The sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions are presented. The resulting lump solutions contain six parameters, two of which are totally free, due to the translation invariance, and the other four of which only need to satisfy the presented sufficient and necessary conditions. Their three-dimensional plots with particular choices of the involved parameters are made to show energy distribution.

Mathematical modeling of population dynamics with Allee effect

Abstract: Allee effect that refers to a positive relationship between individual fitness and population density provides an important conceptual framework in conservation biology. While declining Allee effect causes reduction in extinction risk in low-density population, it provides a benefit in limiting establishment success or spread of invading species. Population models that incorporated Allee effect confer the fundamental role which plays for shaping the population dynamics. In particular, non-spatial predator–prey and invasion models have shown the influence of Allee effects on population dynamics, and spatial models have illustrated its critical roles for pattern formation and the manifestation of traveling wave fronts. We highlight all such no-spatial and spatial population models and their contributions in deeper understanding of population dynamics. In addition, we briefly outline the trends for future research on Allee effect which we think are interesting and widely open.

Dynamical analysis of a simple autonomous jerk system with multiple attractors

Abstract: In recent years, tremendous research efforts have been devoted to simple chaotic oscillators based on jerk equation that involves a third-time derivative of a single variable In the present paper, we perform a systematic analysis of a simple autonomous jerk system with cubic nonlinearity The system is a linear transformation of Model MO5 first introduced in Sprott (Elegant chaos: algebraically simple flow World Scientific Publishing, Singapore, 2010) prior to the more detailed study by Louodop et al (Nonlinear Dyn 78:597–607, 2014) The basic dynamical properties of the model are investigated including equilibria and stability, phase portraits, frequency spectra, bifurcation diagrams, and Lyapunov exponent plots It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry-restoring crisis scenarios One of the key contributions of this work is the finding of a window in the parameter space in which the jerk system experiences the unusual and striking feature of multiple attractors (eg coexistence of four disconnected periodic and chaotic attractors) Basins of attraction of various coexisting attractors are computed showing complex basin boundaries Among the very few cases of lower-dimensional systems (eg Newton–Leipnik system) capable of displaying such type of behaviour reported to date, the jerk system with cubic nonlinearity considered in this work represents the simplest and the most ‘elegant’ prototype An appropriate electronic circuit describing the jerk system is designed and used for the investigations Results of theoretical analyses are perfectly traced by laboratory experimental measurements

Constructing lump solutions to a generalized Kadomtsev–Petviashvili–Boussinesq equation

Abstract: Associated with the prime number $$p=3$$ , a combined model of generalized bilinear Kadomtsev–Petviashvili and Boussinesq equation (gbKPB for short) in terms of the function f is proposed, which involves four arbitrary coefficients. To guarantee the existence of lump solutions, a constraint among these four coefficients is presented firstly, and then, the lump solutions are constructed and classified via searching for positive quadratic function solutions to the gbKPB equation. Different conditions posed on lump parameters are investigated to keep the analyticity and rational localization of the resulting solutions. Finally, 3-dimensional plots, density plots and 2-dimensional curves with particular choices of the involved parameters are given to show the profile characteristics of the presented lump solutions for the potential function $$u=2(\mathrm{{ln}}f)_x$$ .

Coexisting infinitely many attractors in active band-pass filter-based memristive circuit

Abstract: This paper presents an inductor-free memristive circuit, which is implemented by linearly coupling an active band-pass filter (BPF) with a parallel memristor and capacitor filter. Mathematical model is established, and numerical simulations are performed. The results verified by hardware experiments show that the active BPF-based memristive circuit exhibits the dynamical behaviors of point, period, chaos, and period-doubling bifurcation route. Most important of all, the newly proposed memristive circuit has a line equilibrium and its stability closely relies on memristor initial condition, which results in the emergence of extreme multistability. Stability distribution related to memristor initial condition is numerically estimated and the coexistence of infinitely many attractors is intuitively captured by numerical simulations and PSIM circuit simulations.

Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves

Abstract: In this paper, an analytical method based on the generalized Taylors series formula together with residual error function, namely residual power series method (RPSM), is proposed for finding the numerical solution of the coupled system of time–fractional nonlinear Boussinesq–Burger’s equations. The Boussinesq–Burger’s equations arise in studying the fluid flow in a dynamic system and describe the propagation of the shallow water waves. Subsequently, the approximate solutions of time-fractional nonlinear coupled Boussinesq–Burger’s equations obtained by RPSM are compared with the exact solutions as well as the solutions obtained by modified homotopy analysis transform method. Then, we provide a rigorous convergence analysis and error estimate of RPSM. Numerical simulations of the results are depicted through different graphical representations and tables showing that present scheme is reliable and powerful in finding the numerical solutions of coupled system of fractional nonlinear differential equations like Boussinesq–Burger’s equations.

A novel nonlinear resilient control for a quadrotor UAV via backstepping control and nonlinear disturbance observer

Abstract: This study proposes a novel nonlinear resilient trajectory control for a quadrotor unmanned aerial vehicle (UAV) using backstepping control and nonlinear disturbance observer. First, a nonlinear dynamic model for the quadrotor UAV that considers external disturbances from wind model uncertainties is developed. A nonlinear disturbance observer is then constructed separately from the controller to estimate the external disturbances and compensate for the negative effects of the disturbances. Based on the estimates from the given observer, a nominal nonlinear backstepping trajectory-tracking position controller is designed to stabilize the subsystems step by step until the ultimate control law is obtained. An extra term is added to the nominal controller to address the problem of actuator effectiveness loss and to ensure system resilience. The stability of the resilient controller is analyzed using Lyapunov stability theory. Simulation results are presented to demonstrate the effectiveness and robustness of the proposed nonlinear resilient controller.

Trial solution technique to chiral nonlinear Schrodinger’s equation in (1+2)-dimensions

Abstract: This paper applied the trial solution technique to chiral nonlinear Schrodinger’s equation in (1 $$+$$ 2)-dimensions. This led to solitons and other solutions to the model. Besides soliton and singular soliton solutions, this integration scheme also gave way to singular periodic solutions.

Trajectory tracking sliding mode control of underactuated AUVs

Abstract: This paper deals with the control of underactuated autonomous underwater vehicles (AUVs). AUVs are needed in many applications such as the exploration of oceans, scientific and military missions, etc. There are many challenges in the control of AUVs due to the complexity of the AUV model, the unmodelled dynamics, the uncertainties and the environmental disturbances. A trajectory tracking control scheme is proposed in this paper; this control scheme is designed using the sliding mode control technique in order to be robust against bounded disturbances. The control performance of an example AUV, using the proposed method, is evaluated through computer simulations. These simulation studies, which consider different reference trajectories, show that the proposed control scheme is robust under bounded disturbances.

New acoustic wave behaviors to the Davey–Stewartson equation with power-law nonlinearity arising in fluid dynamics

Abstract: This manuscript investigates the new acoustic wave behaviors of the Davey–Stewartson equation with power-law nonlinearity with the help of sine-Gordon expansion method. This technique yields many new acoustic wave behaviors such as hyperbolic, exponential and complex function structures to the problem considered. Wolfram Mathematica 9 has been used throughout the paper for mathematical calculations along with obtaining two- and three-dimensional surfaces of results.

Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction

Abstract: We propose a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach eliminates conflicting definitions, covers both autonomous and time-dependent systems and provides exact mathematical existence, uniqueness and robustness results. In this setting, a nonlinear normal mode (NNM) is a set filled with small-amplitude recurrent motions: a fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a NNM, serving as the smoothest nonlinear continuation of a spectral subspace of the linearized system along the NNM. The existence and uniqueness of SSMs turns out to depend on a spectral quotient computed from the real part of the spectrum of the linearized system. This quotient may well be large even for small dissipation; thus, the inclusion of damping is essential for firm conclusions about NNMs, SSMs and the reduced-order models they yield.

Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator

Abstract: Bursting, an important communication activity in biological neurons and endocrine cells, has been widely found in fast-slow dynamical systems In this paper, a modified second-order generalized memristor, memristive diode bridge cascaded with LC network, is presented and its fingerprints of the pinched hysteresis loops are analyzed By replacing the parallel resistor with the modified generalized memristor, a novel memristive Wien-bridge oscillator is constructed and its mathematical model is established, from which the dynamical behaviors of symmetric chaotic and periodic bursting oscillations are observed and the corresponding bifurcation mechanisms are explained Based on a hardware realization circuit, experimental observations are performed, which verify the numerical simulations

Spatiotemporal deformation of lump solution to (2+1)-dimensional KdV equation

Abstract: In this paper, we extend the recent work of Liu and his collaborators for the (2+1)-dimensional KdV equation. The lump solution under the small perturbation of parameter which decays to the background plane wave in all directions in the plane is obtained. This solution is analogous to the lump solution of the KP equation, but there are some novel different features. The deformation between and among bright, bright-dark and dark lump solution is investigated and exhibited mathematically and graphically. We also discuss that the deflection of lump solution not only depends on the perturbation parameter $$u_0$$ , but also has a relationship with the other parameters. These interesting nonlinear phenomena might provide us with useful information on the dynamics of the relevant fields in nonlinear science.

A novel image encryption algorithm based on genetic recombination and hyper-chaotic systems

Abstract: In this paper, a novel image encryption algorithm based on genetic recombination and hyper-chaotic system is proposed. The basic rules of genetic recombination are employed to scramble images because of its effectiveness. Specifically, the plain image is expanded into two compound images composed of selected four bit-planes and diffuse them at bit-plane level, the compound bit-planes and key streams are reconstructed based on the principles of genetic recombination, then perform traditional diffusion and obtain cipher images. The hyper-chaotic Lorenz system in this algorithm generates pseudorandom sequences in each phase. The experiment results and analysis have proved that the novel image encryption algorithm is effective for image encryption.

Optical solitons with complex Ginzburg–Landau equation

Abstract: The paper revisits in a systematic way the complex Ginzburg–Landau equation with Kerr and power law nonlinearities. Several integration techniques are applied to retrieve various soliton solutions to the model for both forms of nonlinearity. Bright, dark as well as singular soliton solutions are obtained. Several other solutions such as periodic singular solutions and plane waves emerge as a by-product of integration algorithms. Constraint conditions hold all of these solutions in place. The numerical simulations for bright soliton solutions are given for Kerr and power law.

Vibration control of a nonlinear beam with a nonlinear energy sink

Abstract: In this paper, targeted energy transfer from a nonlinear continuous system to a nonlinear energy sink (NES) is investigated. For this purpose, the equation of a nonlinear beam with simply supported ends, on which the NES is attached, is derived using Rayleigh–Ritz method and the Lagrange equation. Then, parameters of the NES are optimized by both sensitivity analysis and particle swarm optimization (PSO) method. Analysis of the energy transfer between the nonlinear beam and the NES is presented, using the complexification-averaging method, too. Attaching an NES to a nonlinear continuous system and using the PSO method to obtain the optimized parameters of the NES is a new development, presented in this work. Also, here, more than one mode of the beam has been considered for analysis of energy transfer between the NES and different modes of the primary system.

Adaptive backstepping output feedback control for a class of nonlinear fractional order systems

Abstract: This article presents a novel fractional order adaptive backstepping output feedback control scheme for nonlinear fractional order systems. The needed feedback information is constructed via a state estimation filter. To improve the control performance, tracking differentiators, nonlinear elements and fractional order update laws were introduced and applied to the control systems. With the aids of this frequency distribute model and the indirect Lyapunov method, the stability and the tracking convergence of the resulting closed-loop system were established. A careful simulation study was provided to illustrate the effectiveness of this novel scheme.

Optical solitons with Biswas–Milovic equation by extended trial equation method

Abstract: This paper addresses the Biswas–Milovic equation as a generalized model for soliton propagation through optical wave guides. The extended trail equation method reveals several forms of soliton solution such as bright, dark and singular solitons. Other wave solutions fall out as by-product of this integration algorithm.

Influence of isolation degree of spatial patterns on persistence of populations

Abstract: Spatial patterns are ubiquitous in nature, which have been identified as important factors in dynamics of ecosystems. However, how pattern structures have influence on persistence of populations is far from well being understood. Particularly, whether some characters of spatial pattern can be indicators for ecosystems collapse is not well studied. As a result, we presented a predator–prey system with spatial motion and found that isolation degree (average distance between patterns with high density) of spatial patterns plays an important role in the persistence of populations: If isolation degree is much smaller, then the population will persist; if isolation degree is much larger, then the population density will decrease with increasing space size and run a high risk of extinction as space size is large enough. Our results highlight the relationship between pattern structures and ecosystems collapse.

Chaos-based engineering applications with a 3D chaotic system without equilibrium points

Abstract: There has recently been an increase in the number of new chaotic system designs and chaos-based engineering applications. In this study, since homoclinic and heteroclinic orbits did not exist and analyses like Shilnikov method could not be used, a 3D chaotic system without equilibrium points was included and thus different engineering applications especially for encryption studies were realized. The 3D chaotic system without equilibrium points represents a new different phenomenon and an almost unexplored field of research. First of all, chaotic system without equilibrium points was examined as the basis and electronic circuit application of the chaotic system was realized and oscilloscope outputs of phase portraits were obtained. Later, chaotic system without equilibrium points was modelled on Labview Field Programmable Gate Array (FPGA) and then FPGA chip statistics, phase portraits and oscilloscope outputs were derived. With another study, VHDL and RK-4 algorithm were used and a new FPGA-based chaotic oscillators design was achieved. Results of Labview-based design on FPGA- and VHDL-based design were compared. Results of chaotic oscillator units designed here were gained via Xilinx ISE Simulator. Finally, a new chaos-based RNG design was achieved and internationally accepted FIPS-140-1 and NIST-800-22 randomness tests were run. Furthermore, video encryption application and security analyses were carried out with the RNG designed here.

Simple chaotic 3D flows with surfaces of equilibria

Abstract: Using a systematic computer search, twelve simple three-dimensional chaotic flows were found that have surfaces of equilibria. Although there are some four-dimensional systems with surfaces of equilibria, there is no such system in three-dimensional state space reported in the literature. Such systems are not difficult to design, but they can have some practical benefits. Study of chaotic flows with surfaces of equilibria provides a good reference for building systems with attractors that are protected from external influences, which can increase the safety of engineering.

Global stability of Clifford-valued recurrent neural networks with time delays

Abstract: In this paper, we study an issue of stability analysis for Clifford-valued recurrent neural networks (RNNs) with time delays. As an extension of real-valued neural network, the Clifford-valued neural network, which includes familiar complex-valued neural network and quaternion-valued neural network as special cases, has been an active research field recently. To the best of our knowledge, the stability problem for Clifford-valued systems with time delays has still not been solved. We first explore the existence and uniqueness for the equilibrium of delayed Clifford-valued RNNs, based on which some sufficient conditions ensuring the global asymptotic and exponential stability of such systems are obtained in terms of a linear matrix inequality (LMI). The simulation result of a numerical example is also provided to substantiate the effectiveness of the proposed results.

Spatiotemporal localizations in $$(3+1)$$ ( 3 + 1 ) -dimensional $${{\mathcal {PT}}}$$ PT -symmetric and strongly nonlocal nonlinear media

Abstract: A $$(3+1)$$ -dimensional coupled nonlocal nonlinear Schrodinger equation in the inhomogeneous $${\mathcal {PT}}$$ -symmetric and strongly nonlocal nonlinear media is studied, and analytical vector spatiotemporal localized solutions are obtained. From these solutions, Gaussian soliton clusters, multipole soliton clusters, and nested soliton clusters can be constructed. The expansion behavior and periodic expansion and compression of spatiotemporal localizations are also investigated in the diffraction decreasing system and periodic modulation system, respectively.

Spatiotemporal Hermite–Gaussian solitons of a (3 + 1)-dimensional partially nonlocal nonlinear Schrödinger equation

Abstract: A (3 + 1)-dimensional partially nonlocal nonlinear Schrodinger equation is considered, and approximate spatiotemporal Hermite–Gaussian soliton solutions are obtained using the Hirota method. Based on these results, some basic characteristics of spatiotemporal Hermite–Gaussian solitons are studied.

A pseudorandom number generator based on piecewise logistic map

Abstract: In order to overcome the disadvantages of logistic map in designing chaos-based cipher, the piecewise logistic map (PLM) is presented. Some properties related to cryptography of the PLM, such as ergodicity, Lyapunov exponent, and bifurcation, are analyzed and compared with the logistic map. From the view of cryptography, the PLM owns better properties than the logistic map. Then, a novel pseudorandom number generator (PRNG) based on the PLM is proposed. Since the cryptographic properties of the PLM are enhanced, the presented PRNG achieves a trade-off between efficiency and security. Both performance analysis and simulation test confirm that our scheme is simple, secure, and efficient, with high potential to be adopted as a stream cipher for secure communication.