Showing papers in "Nonlinear Dynamics in 2015"

Adaptive synchronization of fractional-order memristor-based neural networks with time delay

Abstract: This paper is concerned with the adaptive synchronization problem of fractional-order memristor-based neural networks with time delay. By combining the adaptive control, linear delay feedback control, and a fractional-order inequality, sufficient conditions are derived which ensure the drive–response systems to achieve synchronization. Finally, two numerical examples are given to demonstrate the effectiveness of the obtained results.

Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation

Abstract: The cable equation plays a central role in many areas of electrophysiology and in modeling neuronal dynamics. This paper reports an accurate spectral collocation method for solving one- and two-dimensional variable-order fractional nonlinear cable equations. The proposed method is based on shifted Jacobi collocation procedure in conjunction with the shifted Jacobi operational matrix for variable-order fractional derivatives, described in the sense of Caputo. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations. In addition, the method reduces the variable-order fractional nonlinear cable equation to a simpler problem that consists of solving a system of algebraic equations. The validity and effectiveness of the method are demonstrated by solving three numerical examples. The convergence of the method is graphically analyzed. The results demonstrate that the proposed method is a powerful algorithm with high accuracy for solving the variable-order nonlinear partial differential equations.

Parameter estimation and controller design for dynamic systems from the step responses based on the Newton iteration

Abstract: In this paper, a new Newton iterative identification method is presented for estimating the parameters of a second-order dynamic system utilizing the obtained data from the step response. In order to obtain the desired dynamic performance, a controller design method based on the root locus is presented to meet the requirement of the dynamic performance of the overshoot. The simulation results indicate that the proposed Newton iterative identification method is effective and the system response can meet the requirement of system dynamic performances.

Robust adaptive neural control of flexible hypersonic flight vehicle with dead-zone input nonlinearity

Abstract: This paper presents adaptive dynamic surface control for the flexible model of hypersonic flight vehicle in the presence of unknown dynamics and input nonlinearity. By modeling the flexible coupling as disturbance of rigid body, based on the functional decomposition, the dynamics is divided into attitude subsystem and velocity subsystem. Flight path angle, pitch angle, and pitching rate are involved in the attitude subsystem. To eliminate the inherent problem of “explosion of complexity” in back-stepping, the dynamic surface control is investigated to construct the controller. Furthermore, direct neural control with robust design is proposed without estimating the control gain function and in this way the singularity problem could be avoided. In the last step of dynamic surface design, through the use of Nussbaum-type function, stable adaptive control is presented for the unknown dynamics with time- varying control gain function. The uniform ultimate boundedness stability of the closed-loop system is guaranteed. Simulation result shows the feasibility of the proposed method.

Exponential input-to-state stability of stochastic Cohen–Grossberg neural networks with mixed delays

Abstract: In this paper, we study an issue of input-to-state stability analysis for a class of impulsive stochastic Cohen–Grossberg neural networks with mixed delays. The mixed delays consist of varying delays and continuously distributed delays. To the best of our knowledge, the input-to-state stability problem for this class of stochastic system has still not been solved, despite its practical importance. The main aim of this paper is to fill the gap. By constricting several novel Lyapunov–Krasovskii functionals and using some techniques such as the It $$\hat{o}$$ formula, Dynkin formula, impulse theory, stochastic analysis theory, and the mathematical induction, we obtain some new sufficient conditions to ensure that the considered system with/without impulse control is mean-square exponentially input-to-state stable. Moreover, the obtained results are illustrated well with two numerical examples and their simulations.

Optical solitons in nonlinear directional couplers by sine–cosine function method and Bernoulli’s equation approach

Abstract: This paper obtains soliton solutions to optical couplers by two methods. These are sine–cosine function method and Bernoulli’s equation approach. There are four laws that are touched upon in this paper. These are Kerr law, power law, parabolic law and dual-power law. The first integration scheme is applicable to Kerr and power laws only where bright soliton solutions are retrievable. The second tool is applicable to parabolic and dual-power laws only that leads to dark and singular solitons for these two nonlinear media.

In-plane and out-of-plane nonlinear size-dependent dynamics of microplates

Abstract: The aim of the current study is to examine the in-plane and out-of-plane nonlinear size-dependent dynamics of a microplate resting on an elastic foundation, constrained by distributed rotational springs at boundaries. Employing the von Karman plate theory as well as Kirchhoff’s hypotheses, the equations of motion for the in-plane and out-of-plane directions are derived by means of the Lagrange equations, based on the modified couple stress theory. The potential energies stored in a Winkler-type elastic foundation and the rotational springs at the edges of the microplate are taken into account. The set of second-order nonlinear ordinary differential equations, obtained via the Lagrange scheme, is recast into a double-dimensional set of first-order nonlinear ordinary differential equations with coupled terms by means of a change of variables. The linear natural frequencies of the system are obtained through use of an eigenvalue analysis upon the linear terms of the equations of motion. The nonlinear response, on the other hand, is obtained by means of the pseudo-arclength continuation method. The dynamical characteristics of the system are examined via plotting the frequency–response and force–response curves. The effect of the stiffness of the rotational and translational springs on the nonlinear size-dependent behaviour is also examined. Finally, the effect of employing the modified couple stress theory, rather than the classical theory, on the response is discussed.

A review of operational matrices and spectral techniques for fractional calculus

Abstract: Recently, operational matrices were adapted for solving several kinds of fractional differ- ential equations (FDEs). The use of numerical tech- niques in conjunction with operational matrices of someorthogonalpolynomials,forthesolutionofFDEs on finite and infinite intervals, produced highly accu- rate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, wepresenttheoperationalmatricesoffractionalderiva- tivesandintegrals,forseveralpolynomialsonbounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with differ- ent spectral techniques for solving the aforementioned equations on bounded domains. The operational matri- ces of fractional derivatives and integrals are also pre- sented for orthogonal Laguerre and modified general-

Dynamics of self-excited attractors and hidden attractors in generalized memristor-based Chua’s circuit

Abstract: A novel memristive chaotic circuit is presented, which is derived from the classical Chua’s circuit by substituting Chua’s diode with a first-order memristive diode bridge. The dynamical characteristics with the variations of circuit parameters are investigated both theoretically and numerically. The research results indicate that this circuit has three determined equilibrium points and displays complex nonlinear phenomena including coexisting bifurcation modes and coexisting attractors. Specifically, with another parameter setting, the memristive Chua’s circuit can generate hidden attractors and coexisting hidden attractors in a narrow parameter region. The phenomena of self-excited attractors and hidden attractors are experimentally captured from a physical circuit, which verify the numerical simulations.

Soliton solutions to resonant nonlinear Schrödinger’s equation with time-dependent coefficients by trial solution approach

Abstract: In this paper, the resonant nonlinear Schrodinger’s equation is studied with four forms of nonlinearity and time-dependent coefficients. The trial solution method is employed to solve the governing equations. Solitons and singular periodic solutions are obtained. The constraint conditions naturally emerge from the solution structure that are needed for its existence.

Unified nonlinear electroelastic dynamics of a bimorph piezoelectric cantilever for energy harvesting, sensing, and actuation

Abstract: Inherent nonlinearities of piezoelectric materials are pronounced in various engineering applications such as sensing, actuation, combined applications for vibration control, and energy harvesting from dynamical systems The existing literature focusing on the dynamics of electroelastic structures made of piezoelectric materials has explored such nonlinearities separately for the problems of mechanical and electrical excitation Similar manifestations of softening nonlinearities have been attributed to purely elastic nonlinear terms, coupling nonlinearities, hysteresis alone, or a combination of these effects by various authors In order to develop a unified nonlinear nonconservative framework with two-way coupling, the present work investigates the nonlinear dynamic behavior of a bimorph piezoelectric cantilever under low to moderately high mechanical and electrical excitation levels in energy harvesting, sensing, and actuation The highest voltage levels, for near resonance investigation, are well below the coercive field A distributed parameter electroelastic model is developed by accounting for softening and dissipative nonlinearities to analyze the primary resonance of a soft (eg, PZT-5A, PZT-5H) piezoelectric cantilever for the fundamental bending mode using the method of harmonic balance Excellent agreement between the model and experimental investigation is found, providing evidence that quadratic stiffness, damping, and electromechanical coupling effects accurately model predominantly observed nonlinear effects in geometrically linear vibration of piezoelectric cantilever beams The backbone curves of both energy harvesting and actuation frequency responses for a PZT-5A cantilever are experimentally found to be dominantly of first order and specifically governed by ferroelastic quadratic softening for a broad range of mechanical and electrical excitation levels Cubic and higher-order nonlinearities become effective only near the physical limits of the brittle and stiff (geometrically linear) bimorph cantilever when excited near resonance

Conservation laws for time-fractional subdiffusion and diffusion-wave equations

Abstract: A new technique for constructing conservation laws for fractional differential equations not having a Lagrangian is proposed. The technique is based on the methods of Lie group analysis and employs the concept of nonlinear self-adjointness which is enhanced to the certain class of fractional evolution equations. The proposed approach is demonstrated on subdiffusion and diffusion-wave equations with the Riemann–Liouville and Caputo time-fractional derivatives. It is shown that these equations are nonlinearly self-adjoint, and therefore, desired conservation laws can be calculated using the appropriate formal Lagrangians. The explicit forms of fractional generalizations of the Noether operators are also proposed for the equations with the Riemann–Liouville and Caputo time-fractional derivatives of order $$\alpha \in (0,2)$$ . Using these operators and formal Lagrangians, new conservation laws are constructed for the linear and nonlinear time-fractional subdiffusion and diffusion-wave equations by their Lie point symmetries.

Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria

Abstract: This paper studies a four-dimensional (4D) memristive system modified from the 3D chaotic system proposed by Lu and Chen. The new system keeps the symmetry and dissipativity of the original system and has an uncountable infinite number of stable and unstable equilibria. By varying the strength of the memristor, we find rich complex dynamics, such as limit cycles, torus, chaos, and hyperchaos, which can peacefully coexist with the stable equilibria. To explain such coexistence, we compute the unstable manifolds of the equilibria, find that the manifolds create a safe zone for the hyperchaotic attractor, and also find many heteroclinic orbits. To verify the existence of hyperchaos in the 4D memristive circuit, we carry out a computer-assisted proof via a topological horseshoe with two-directional expansions, as well as a circuit experiment on oscilloscope views.

Adaptive fractional-order switching-type control method design for 3D fractional-order nonlinear systems

Abstract: In this paper, an adaptive sliding mode technique based on a fractional-order (FO) switching-type control law is designed to guarantee robust stability for uncertain 3D FO nonlinear systems. A novel FO switching-type control law is proposed to ensure the existence of the sliding motion in finite time. Appropriate adaptive laws are shown to tackle the uncertainty and external disturbance. The calculation formula of the reaching time is analyzed and computed. The reachability analysis is visualized to show how to obtain a shorter reaching time. A stability criterion of the FO sliding mode dynamics is derived based on indirect approach to Lyapunov stability. Advantages of the proposed control scheme are illustrated through numerical simulations.

New class of chaotic systems with circular equilibrium

Abstract: This paper brings a new mathematical model of the third-order autonomous deterministic dynamical system with associated chaotic motion. Its unique property lies in the existence of circular equilibrium which was not, by referring to the best knowledge of the authors, so far reported. Both mathematical analysis and circuitry implementation of the corresponding differential equations are presented. It is shown that discovered system provides a structurally stable strange attractor which fulfills fractal dimensionality and geometrical density and is bounded into a finite state space volume.

Dynamic analysis and modeling of a novel fractional-order hydro-turbine-generator unit

Abstract: In order to study the stability of a hydro-turbine-generator unit in further depth, we establish a novel nonlinear fractional-order mathematical model considering a fractional-order damping force, a fractional-order oil-film force, an asymmetric magnetic pull and a hydraulic-asymmetric force. Furthermore, the nonlinear dynamics of the above fractional-order hydro-turbine-generator unit system with six typical fractional orders are studied in detail. Based on these, we analyze the effect of the fractional-order $$\alpha $$ on bifurcation points, the orbit of centroid of the rotor, the power and the frequency of the rotor. Fortunately, some variable laws are found from numerical simulation results. Finally, all of these results have enriched the dynamical behaviors of a hydro-turbine-generator system.

Coherently coupled solitons, breathers and rogue waves for polarized optical waves in an isotropic medium

Abstract: Under investigation in this paper is a coherently coupled nonlinear Schrodinger system which describes the propagation of polarized optical waves in an isotropic medium. By virtue of the Darboux transformation, some new solutions have been generated on the vanishing and non-vanishing backgrounds, including multi-solitons, bound solitons, one-breathers, bound breathers, two-breathers, first-order and higher-order rogue waves. Dynamic behaviors of those solitons, breathers and rogue waves have been discussed through graphic simulation.

Fractional-order delayed predator–prey systems with Holling type-II functional response

Abstract: In this paper, a fractional dynamical system of predator–prey with Holling type-II functional response and time delay is studied. Local and global stability of existence steady states and Hopf bifurcation with respect to the delay is investigated, with fractional-order $$0< \alpha \le 1$$ . It is found that Hopf bifurcation occurs when the delay passes through a sequence of critical values. Unconditionally, stable implicit scheme for the numerical simulations of the fractional-order delay differential model is introduced. The numerical simulations show the effectiveness of the numerical method and confirm the theoretical results. The presence of fractional order in the delayed differential model improves the stability of the solutions and enrich the dynamics of the model.

A novel color image encryption algorithm based on spatial permutation and quantum chaotic map

Abstract: In recent years, several algorithms of secure image encryption were studied and developed through chaotic processes. Most of the previous algorithms encrypt color components independently. In this paper, a novel image encryption algorithm based on quantum chaotic map and diffusion–permutation architecture is proposed. First, the new algorithm employs the quantum logistic map to diffuse the relationship of pixels in color components. Next, the keystreams generated by the two-dimensional logistic map are exploited to not only modify the value of diffused pixels, but also spatially permute the pixels of color components at the same time and make the three components affect one another. Finally, the random circular shift operation is applied to the result of the modified and permuted pixels to rearrange bits of each encrypted pixel. In order to achieve the high complexity and the high randomness between these generated keystreams, the two-dimensional logistic map and the quantum chaotic map are independently coupled with nearest-neighboring coupled-map lattices. The results of several experimental analyses about randomness, sensitivity and correlation of the cipher-images show that the proposed algorithm has high security level, high sensitivity and high speed which can be adopted for network security and secure communications.

Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system

Abstract: Based on Rabinovich system, a 4D Rabinovich system is generalized to study hidden attractors, multiple limit cycles and boundedness of motion. In the sense of coexisting attractors, the remarkable finding is that the proposed system has hidden hyperchaotic attractors around a unique stable equilibrium. To understand the complex dynamics of the system, some basic properties, such as Lyapunov exponents, and the way of producing hidden hyperchaos are analyzed with numerical simulation. Moreover, it is proved that there exist four small-amplitude limit cycles bifurcating from the unique equilibrium via Hopf bifurcation. Finally, boundedness of motion of the hyperchaotic attractors is rigorously proved.

Discrete chaos in fractional delayed logistic maps

Abstract: Recently the discrete fractional calculus (DFC) started to gain much importance due to its applications to the mathematical modeling of real world phenomena with memory effect. In this paper, the delayed logistic equation is discretized by utilizing the DFC approach and the related discrete chaos is reported. The Lyapunov exponent together with the discrete attractors and the bifurcation diagrams are given.

Model predictive control for improving operational efficiency of overhead cranes

Abstract: Model predictive control (MPC) has been successfully applied to many transportation systems. For the control of overhead cranes, existing MPC approaches mainly focus on improving the regulation performance, such as tracking error or steady-state error. In this paper, energy efficiency as well as safety is newly considered in our proposed MPC approach. Based on the system model designed, the MPC approach is applied to minimize an objective function that is formulated as the integration of energy consumption and swing angle. In our approach, promising results in terms of low energy consumption and small swing angle can be found, while the solutions obtained can satisfy all practical constraints. Our test results indicate that the MPC approach can ensure stability and robustness of improving energy efficiency and safety.

Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model

Abstract: Within the context of the Madelung fluid description, investigation has been carried out on the connection between the envelope soliton-like solutions of a wide family of nonlinear Schrodinger equations and the soliton-like solutions of a wide family of Korteweg–de Vries or Korteweg–de Vries-type equations. Under suitable hypothesis for the current velocity, the Gerdjikov–Ivanov envelope solitons are derived and discussed in this paper. For a motion with the stationary profile current velocity, the fluid density satisfies a generalized stationary Gardner equation, which possesses bright- and dark-type (including gray and black) solitary waves due to associated parametric constraints, and finally envelope solitons are found correspondingly for the Gerdjikov–Ivanov model. Moreover, this approach may be useful for studying other nonlinear Schrodinger-type equations.

Unknown input observer design for one-sided Lipschitz nonlinear systems

Abstract: This paper considers the observer design problem for one-sided Lipschitz nonlinear systems with unknown inputs. The systems under consideration are a larger class of nonlinearities than the well-studied Lipschitz systems and have inherent advantages with respect to conservativeness. For such systems, we first propose a full-order nonlinear unknown input observer (UIO) by using the linear matrix inequality (LMI) approach. Following a similar design procedure and using state transformation, the reduced-order nonlinear UIO is also constructed. Sufficient conditions to guarantee existence of full-order and reduced-order UIOs are established by carefully considering the one-sided Lipschitz condition together with the quadratic inner-bounded condition. Based on the matrix generalized inverse technique, the UIO conditions are formulated in terms of LMIs. Moreover, the proposed observers are applied to a single-link flexible joint robotic system with unknown inputs. Simulation results are finally given to illustrate the effectiveness of the proposed design.

Controllable combined Peregrine soliton and Kuznetsov–Ma soliton in {\varvec{\mathcal {PT}}}-symmetric nonlinear couplers with gain and loss

Abstract: We investigate a (2+1)-dimensional-coupled variable coefficient nonlinear Schrodinger equation in parity time symmetric nonlinear couplers with gain and loss and analytically obtain a combined structure solution via the Darboux transformation method. When the imaginary part of the eigenvalue $$n$$ is smaller or bigger than 1, we can obtain the combined Peregrine soliton and Akhmediev breather, or Kuznetsov–Ma soliton, respectively. Moreover, we study the controllable behaviors of this combined Peregrine soliton and Kuznetsov–Ma soliton structure in a diffraction decreasing system with exponential profile. In this system, the effective propagation distance $$Z$$ exists a maximal value $$Z_m$$ and the maximum amplitude of the KM soliton appears in the periodic locations $$Z_{i}$$ . By modulating the relation between values of $$Z_m$$ and $$Z_i$$ , we realize the control for the excitation of the combined Peregrine soliton and Kuznetsov–Ma soliton, such as the restraint, maintenance, and postpone.

1-Soliton solution of KdV6 equation

Abstract: This paper applies three forms of integration tools to integrate KdV6 equation that represents a nonholonomic deformation of the well-known KdV equation, which models shallow-water dynamics. The three integration algorithms applied are Kudryashov’s method, extended tanh scheme as well as $$G^{\prime }/G$$ -expansion mechanism. These tools lead to solitary waves, shock waves as well as singular periodic solutions to the equation. The corresponding integrability criteria, also known as constraint conditions, naturally emerge from the analysis of the problem.

Complex transient dynamics in periodically forced memristive Chua’s circuit

Abstract: When a sinusoidal voltage stimulus is applied, memristive Chua’s circuit becomes a non-autonomous periodically forced nonlinear circuit. By utilizing theoretical formulations, simulations and experimental verifications, the complex transient dynamics of the periodically forced memristive Chua’s circuit is investigated in this paper. It can be found that the equilibrium point of the circuit switches between a line equilibrium and no equilibrium with the time evolutions, and the circuit exhibits period, chaos and also hyperchaos in a parameter range of the stimulus frequency. Moreover, some abundant interesting nonlinear phenomena including transient chaos, transient hyperchaos and chaotic beats are revealed numerically and verified experimentally.

Nonautonomous solitons in modified inhomogeneous Hirota equation: soliton control and soliton interaction

Abstract: We have investigated the femtosecond soliton propagation in inhomogeneous fiber, which is described by the modified inhomogeneous Hirota equation with variable coefficient (MIH-vc). With the aid of AKNS method, corresponding Lax pair is constructed. By virtue of the Darboux transformation method and symbolic computation, the analytic one- and two-soliton solutions are explicitly obtained. Using obtained solutions, we graphically discuss the features of femtosecond solitons in modified inhomogeneous Hirota system by changing the profile of variable coefficients. We analyze various form of group velocity dispersion, third order dispersion and nonlinearity parameter for periodic amplification system, exponentially distributed system, parabolic solitons, periodic exponentially modulated system, which will be observable in the future experiments. These results are potentially useful in future experiments and soliton control for long-distance optical communication. Finally, the soliton solutions of the MIH-vc equation in double Wronskian form is constructed and further verified using the Wronskian technique by substitute in bilinear equations.

A novel image encryption scheme based on an improper fractional-order chaotic system

Abstract: Based on the features of digital image encryption and high-dimensional chaotic sequences, the paper proposes a symmetric digital image encryption algorithm by a new improper fractional-order chaotic system. The initial conditions, parameters and fractional orders of chaos are influenced by gray value of all pixels and used as secret key. Therefore, the total key length is large enough to resist any brute-force attacks. The original image is divided into four parts and encrypted by different encryption formulas. Theoretical analysis results show that the proposed encryption scheme has effective encryption and efficiencies.

Effect of multi-phase optimal velocity function on jamming transition in a lattice hydrodynamic model with passing

Abstract: In this paper, we study the effect of multi-phase optimal velocity function on a density difference lattice model with passing. The effect of reaction coefficient is examined through linear stability analysis and shown that it can significantly enlarge the stability region on the phase diagram for any rate of passing. Using nonlinear stability analysis, the critical value of passing constant is obtained and found independent of reaction coefficient. Below this critical value for which kink soliton solution of mKdV equation exists. By varying the density, multiple phase transitions are analyzed, which highly depend on the sensitivity, reaction coefficient and passing constant. It is observed that the number of stages in multi-phase transitions closely related to the number of the turning points in the optimal velocity function. The theoretical findings are verified using numerical simulation, which confirm that phase diagrams of multi-phase traffic in the case of passing highly depend on the choice of optimal velocity function as well as on other parameters such as sensitivity, reaction coefficient and rate of passing.