•Smooth solutions of the DNLSI in determinant form with a degenerate eigenvalue.•Smooth positon of the DNLSI, which has different property from soliton.•Several new patterns of the higher-order rogue wave of the DNLSI. In this paper, we provide a simple method to generate higher order position solutions and rogue wave solutions for the derivative nonlinear Schrödinger equation. The formulae of these higher order solutions are given in terms of determinants. The dynamics and structures of solutions generated by this method are studied.

•Kink dynamics in the ϕ4 field with PT-symmetric gain/loss prefactor is studied numerically.•Resonance between the kink’s internal mode and the spatial periodicity of the gain/loss prefactor was observed.•The kink’s translational and vibrational modes are coupled.•A two degree of freedom collective variable model captures the main effects qualitatively. The resonant interaction of the ϕ4 kink with a PT-symmetric perturbation is observed in the numerical study performed in the frame of the continuum model and with the help of a two degree of freedom collective variable model derived in PRA 89, 010102(R). The perturbation is in the form of first partial derivative in time term with a spatially periodic gain/loss coefficient. When the kink interacts with the perturbation, the kink’s internal mode is excited with the amplitude varying in time quasiperiodically. The maximal value of the amplitude was found to grow when the kink velocity is such that it travels one period of the gain/loss prefactor in nearly one period of the kink’s internal mode. It is also found that the kink’s translational and vibrational modes are coupled in a way that an increase in the kink’s internal mode amplitude results in a decrease in kink velocity. The results obtained with the collective variable method are in a good qualitative agreement with the numerical simulations for the continuum model. The results of the present study suggest that kink dynamics in open systems with balanced gain and loss can have new features in comparison with the case of conservative systems.

Nonlinear dynamic characteristics of rub-impact rotor system with fractional order damping are investigated. The model of rub-impact comprises a radial elastic force and a tangential Coulomb friction force. The fractional order damped rotor system with rubbing malfunction is established. The four order Runge–Kutta method and ten order CFE-Euler method are introduced to simulate the fractional order rub-impact rotor system equations. The effects of the rotating speed ratio, derivative order of damping and mass eccentricity on the system dynamics are investigated using rotor trajectory diagrams, bifurcation diagrams and Poincare map. Various complicated dynamic behaviors and types of routes to chaos are found, including period doubling bifurcation, sudden transition and quasi-periodic from periodic motion to chaos. The analysis results show that the fractional order rub-impact rotor system exhibits rich dynamic behaviors, and that the significant effect of fractional order will contribute to comprehensive understanding of nonlinear dynamics of rub-impact rotor.

•We presented a delayed-feedback control method for lattice hydrodynamic model.•We find the stability condition using Hurwitz criteria and condition for transfer function.•The control parameters are computed through Bode-plot of transfer function.•The effect of delayed-feedback control is examined theoretically as well as numerically. The delayed-feedback control (DFC) method for lattice hydrodynamic traffic flow model is investigated on a unidirectional road. By using the Hurwitz criteria and the condition for transfer function in term of H∞-norm, we designed the feedback gain and delay time to stabilize the traffic flow and suppress the traffic jam. The Bode-plot of transfer function have been plotted and discussed that the stability region enhances with delayed-feedback control. It is shown that the delayed-feedback control method stabilizes the traffic flow and suppresses the traffic jam efficiently. The simulation results are in good agreement with the theoretical analysis.

► Induced magnetic field effect on peristalsis is studied. ► Heat transfer effects are also considered. ► Long wavelength and low Reynolds number assumptions are utilized in the whole analysis. ► Trapping is analyzed. ► Physics for various parameters is pointed out. The effect of an induced magnetic field on peristaltic flow of an incompressible Carreau fluid in an asymmetric channel is analyzed. Perturbation solution to equations under long wavelength approximation is derived in terms of small Weissenberg number. Expressions have been constructed for the stream function, the axial induced magnetic field, the magnetic force function, the current density distribution and the temperature. Trapping phenomenon is examined with respect to emerging parameters of interest.

•Fourier spectral method is proposed as an alternative to both the predictor-corrector and finite difference schemes.•Computer simulations show that pattern formation in fractional reaction-diffusion equations is possible.•Problems that are of practical interest, arising in the application areas of biology, physics and engineering are formulated.•Different spatiotemporal dynamics are observed and displayed in 1D, 2D and 3D. In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L], x=x(x,y,z) and t ∈ [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α, for 0 < α < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < α < 1) and super-diffusive (1 < α < 2) scenarios. It is observed that computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed.

•The nonlocal symmetry for the Gardner equation is derived by the truncated Painlevé analysis or the Möbious (conformal) invariant form.•To solve the initial value problem related with the nonlocal symmetry, the original Gardner equation is prolonged the enlarged systems.•Many explicit interaction solutions among different types of solutions are given by using the symmetry reduction method to the enlarged systems. Based on the truncated Painlevé method or the Möbious (conformal) invariant form, the nonlocal symmetry for the (1+1)–dimensional Gardner equation is derived. The nonlocal symmetry can be localized to the Lie point symmetry by introducing one new dependent variable. Thanks to the localization procedure, the finite symmetry transformations are obtained by solving the initial value problem of the prolonged systems. Furthermore, by using the symmetry reduction method to the enlarged systems, many explicit interaction solutions among different types of solutions such as solitary waves, rational solutions, Painlevé II solutions are given. Especially, some special concrete soliton-cnoidal interaction solutions are analyzed both in analytical and graphical ways.

► An adaptive sliding mode backstepping control for mobile manipulator is proposed. ► A modelling of two-wheeled mobile manipulator is built in kinematic and dynamic. ► Estimation of the general nondeterminacy is adapted online. ► The adaptive law is designed according to the characteristics of the mobile manipulator. ► The controller does not require detailed parameters and accurate models. To solve disturbances, nonlinearity, nonholonomic constraints and dynamic coupling between the platform and its mounted robot manipulator, an adaptive sliding mode controller based on the backstepping method applied to the robust trajectory tracking of the wheeled mobile manipulator is described in this article. The control algorithm rests on adopting the backstepping method to improve the global ultimate asymptotic stability and applying the sliding mode control to obtain high response and invariability to uncertainties. According to the Lyapunov stability criterion, the wheeled mobile manipulator is divided into several stabilizing subsystems, and an adaptive law is designed to estimate the general nondeterminacy, which make the controller be capable to drive the trajectory tracking error of the mobile manipulator to converge to zero even in the presence of perturbations and mathematical model errors. We compare our controller with the robust neural network based algorithm in nonholonomic constraints and uncertainties, and simulation results prove the effectivity and feasibility of the proposed method in the trajectory tracking of the wheeled mobile manipulator.

A study of nite Larmor radius (FLR) eects on E B test particle chaotic transport in non- monotonic zonal ows with drift waves in magnetized plasmas is presented. Due to the non- monotonicity of the zonal ow, the Hamiltonian does not satisfy the twist condition. The electro- static potential is modeled as a linear superposition of a zonal ow and regular neutral modes of the Hasegawa-Mima equation. FLR eects are incorporated by gyro-averaging the EB Hamiltonian. It is shown that there is a critical value the Larmor radius for which the zonal ow transitions from a prole with one maximum to a prole with two maxima and a minimum. This bifurcation leads to the creation of additional shearless curves and resonances. The gyroaveraged nontwist Hamiltonian exhibits complex patterns of separatrix reconnection. A change in the Larmor ra- dius can lead to heteroclinic-homoclinic bifurcations and dipole formation. For Larmor radii for which the zonal ow has bifurcated, double heteroclinic-heteroclinic, homoclinic-homoclinic and heteroclinic-homoclinic topologies are observed. It is also shown that chaotic transport is typically reduced as the Larmor radius increases. Poincare sections shows that, for large enough Larmor radius, chaos can be practically suppressed. In particular, small changes on the Larmor radius can restore the shearless curve.

•The strongly coupled nonlinear space fractional Schrödinger equation is studied numerically.•An implicit difference scheme with the discrete conservative properties is constructed.•The solvability, boundedness and convergence in the maximum norm are proved.•A three-level linear difference scheme with two identities is also presented.•The performance of both schemes are verified numerically. This paper focuses on numerically solving the strongly coupled nonlinear space fractional Schrödinger equations. First, the laws of conservation of mass and energy are given. Then, an implicit difference scheme is proposed, under the assumption that the analytical solution decays to zero when the space variable x tends to infinity. We show that the scheme conserves the mass and energy and is unconditionally stable with respect to the initial values. Moreover, the solvability, boundedness and convergence in the maximum norm are established. To avoid solving nonlinear systems, a linear difference scheme with two identities is proposed. Several numerical experiments are provided to confirm the theoretical results.

•Non-linear size dependent Sheremetev-Pelekh beams governing PDEs are derived.•Reliability and validity of the governing PDEs is studied regarding statics and dynamics.•Symmetric and non-symmetric frequency spectra distributions are detected and discussed.•The Pomeau-Mannevile route to chaos is illustrated in our infinite dimensional system. In this work, a size-dependent model of a Sheremetev-Pelekh-Reddy-Levinson micro-beam is proposed and validated using the couple stress theory, taking into account large deformations. The applied Hamilton's principle yields the governing PDEs and boundary conditions. A comparison of statics and dynamics of beams with and without size-dependent components is carried out. It is shown that the proposed model results in significant, both qualitative and quantitative, changes in the nature of beam deformations, in comparison to the so far employed standard models. A novel scenario of transition from regular to chaotic vibrations of the size-dependent Sheremetev-Pelekh model, following the Pomeau-Manneville route to chaos, is also detected and illustrated, among others.

A novel messages spreading model is suggested in this paper. The model is a natural generalization of the SIS (susceptible-infective-susceptible) model, in which two relevant messages with same probability of acceptance may spread among nodes. One of the messages has a higher priority to be adopted than the other only in the sense that both messages act on the same node simultaneously. Node in the model is termed as supporter when it adopts either of messages. The transition probability allows that two kinds of supports may transform into each other with a certain rate, and it varies inversely with the associated levels which are discretely distributed in the symmetrical interval around original point. Results of numerical simulations show that individuals tend to believe the messages with a better consistency. If messages are conflicting with each other, the one with higher priority would be spread more and another would be ignored. Otherwise, the number of both supports remains at a uniformly higher level. Besides, in a network with lower connected degree, over a half of the individuals would keep neutral, and the message with lower priority becomes harder to diffuse than the prerogative one. This paper explores the propagation of multi-messages by considering their correlation degree, contributing to the understanding and predicting of the potential propagation trends.

In this paper, a sliding mode control design for fractional order systems with input and state time-delay is proposed. First, we consider a fractional order system without delay for which a sliding surface is proposed based on fractional integration of the state. Then, a stabilizing switching controller is derived. Second, a fractional system with state delay is considered. Third, a strategy including a fractional state predictor input delay compensation is developed. The existence of the sliding mode and the stability of the proposed control design are discussed. Numerical examples are given to illustrate the theoretical developments.

•A time-fractional Cahn-Hilliard model equipped with the Caputo derivative is studied.•A fresh lattice Boltzmann method for the fractional Cahn-Hilliard equation is proposed.•The proposed model accurately describes interface dynamics with the fractional effect.•The predicted system energy conforms to the energy dissipation law. Fractional phase field models have been reported to suitably describe the anomalous two-phase transport in heterogeneous porous media, evolution of structural damage, and image inpainting process. It is commonly different to derive their analytical solutions, and the numerical solution to these fractional models is an attractive work. As one of the popular fractional phase-field models, in this paper we propose a fresh lattice Boltzmann (LB) method for the fractional Cahn-Hilliard equation. To this end, we first transform the fractional Cahn-Hilliard equation into the standard one based on the Caputo derivative. Then the modified equilibrium distribution function and proper source term are incorporated into the LB method in order to recover the targeting equation. Several numerical experiments, including the circular disk, quadrate interface, droplet coalescence and spinodal decomposition, are carried out to validate the present LB method. It is shown that the numerical results at different fractional orders agree well with the analytical solution or some available results. Besides, it is found that increasing the fractional order promotes a faster evolution of phase interface in accordance with its physical definition, and also the system energy predicted by the present LB method conforms to the energy dissipation law.

•A periodic Chikungunya model with temperature and rainfall effects is studied.•Time-dependent maturation delay and extrinsic incubation period are incorporated.•Infected humans are divided into symptomatic and asymptomatic compartments.•Neglecting rainfall, seasonality and asymptomatic compartment, infection may be overestimated.•Numerical simulations are done with data of the largest outbreak of Brazil in 2017. In this paper, a periodic Chikungunya model with temperature and rainfall effects is proposed and studied, which incorporates time-dependent extrinsic incubation period, time-dependent maturation delay, asymptomatic and symptomatic infectious humans. Two threshold parameters for the extinction and persistence of mosquitos and the virus are derived, respectively: the mosquito reproduction number Rm and the basic reproduction number R0. Then the analytic results are verified by numerical simulation with the temperature and rainfall data of the state of Ceará, where the largest outbreak in Brazil’s history occurred in 2017. And the effects of rainfall, seasonality, and asymptomatic infection in humans on mosquito population and the Chikungunya transmission are explored. Our simulations show that if these factors are not taken into account, the number of mosquito population and people infected may be overestimated. Finally, the relationships between Rm and R0 and some parameters are established.

•We use fractional calculus (FC) to model a pn semiconductor diode.•We use electric impedance spectroscopy and describing function to measure impedance.•The measured data is approximated by means of FC models.•The FC models highlight dynamic characteristics overlooked by classical models. This paper adopts the fractional calculus to model a pn semiconductor diode under sinusoidal operation. The electrical impedance spectroscopy and the describing function techniques are used for measuring the device impedance. The experimental data is approximated by means of fractional-order models. The results demonstrate that the proposed approach describes the diode impedance using a limited number of parameters, while highlighting relevant dynamic characteristics that are overlooked by classical models.

•We study the FPUT system in the weakly-nonlinear regime of discrete wave turbulence.•M-wave resonances lead to a system of Diophantine nonlinear equations for M variables.•We solve these Diophantine equations using cyclotomic polynomials.•5-wave exact discrete resonances exist when the number of particles is divisible by 3.•Resonant modes form connected webs extending across scales, leading to equipartition. In systems of N coupled anharmonic oscillators, exact resonant interactions play an important role in the exchange of energy between normal modes. In the weakly nonlinear regime, those interactions may be responsible for the equipartition of energy in Fourier space. Here we consider analytically resonant wave-wave interactions for the celebrated Fermi-Pasta-Ulam-Tsingou (FPUT) system. Using a number-theoretical approach based on cyclotomic polynomials, we show that the problem of finding exact resonances for a system of N particles is equivalent to a Diophantine equation whose solutions depend sensitively on the number of particles N, in particular on the set of divisors of N. We provide an algorithm to construct all possible resonances for N particles, based on two basic methods: pairing-off and cyclotomic, which we introduce and use to build up explicit solutions to the 4-wave, 5-wave and 6-wave resonant conditions. Our results shed some light in the understanding of the long-standing FPUT paradox, regarding the sensitivity of the resonant manifolds with respect to the number of particles N and the corresponding time scale of the interactions leading to an eventual thermalisation. In this light we demonstrate that 6-wave resonances always exist for any N, while 5-wave resonances exist if N is divisible by 3 and N ⩾ 9. It is known that in the discrete case 4-wave resonances do not produce energy mixing across the spectrum, so we investigate whether 5-wave resonances can produce energy mixing across a significant region of the Fourier spectrum by looking at the structure of the interconnected network of Fourier modes that can interact nonlinearly via resonances. We obtain that the answer depends on the set of odd divisors of N which are not divisible by 3: the size of this set determines the number of dynamically independent components, corresponding to independent constants of motion (energies). We show that 6-wave resonances connect all these independent components, providing in principle a restoring mechanism for full-scale thermalisation.