Showing papers in "Nonlinear Dynamics in 2019"

Coexisting multi-stable patterns in memristor synapse-coupled Hopfield neural network with two neurons

Abstract: When possessing a potential difference between two neurons, an electromagnetic induction current appears in the Hopfield neural network (HNN), which can be emulated by a flux-controlled memristor synapse. Thus, a three-order two-neuron-based autonomous memristive HNN is presented in this paper, which is the lowest order and has not been reported in the previous studies. With the mathematical model, the detailed stability analyses for the line equilibrium are executed, so that the fold and Hopf bifurcation sets and stability region distributions in the parameter plane are obtained. Furthermore, numerical results of coexisting bifurcation patterns are investigated, which are confirmed effectively by local basins of attraction and phase plane plots. The numerical results demonstrate coexisting multi-stable patterns of the spiral chaotic patterns with different dynamic amplitudes, periodic patterns with different periodicities, and stable resting patterns with different positions in the memristive HNN. Besides, the circuit synthesis and breadboard experiments are performed to well validate the numerical simulations.

Electric load forecasting by complete ensemble empirical mode decomposition adaptive noise and support vector regression with quantum-based dragonfly algorithm

Abstract: Accurate electric load forecasting can provide critical support to makers of energy policy and managers of power systems. The support vector regression (SVR) model can be hybridized with novel meta-heuristic algorithms not only to identify fluctuations and the nonlinear tendencies of electric loads, but also to generate satisfactory forecasts. However, many such algorithms have numerous drawbacks, such as a low population diversity and trapping at local optima, which are problems of premature convergence. Accordingly, approaches to increase the accuracy of forecasting must be developed. In this investigation, quantum computing mechanism is used to quantamize dragonfly behaviors to enhance the searching effectiveness of the dragonfly algorithm, namely QDA. In addition, conducting the data preprocessing by the complete ensemble empirical mode decomposition adaptive noise (CEEMDAN) is useful to improve the forecasting accuracy. Thus, a new electric load forecasting model, the CEEMDAN-SVRQDA model, that combines the CEEMDAN and hybridizes the QDA with an SVR model, is proposed to provide more accurate forecasts. Two numerical examples from the Tokyo Electric Power Company (Japan) and the National Grid (UK) demonstrate that the proposed model outperforms other models.

Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation

Abstract: A new method named bilinear neural network is introduced in this paper, and the corresponding tensor formula is proposed to obtain the exact analytical solutions of nonlinear partial differential equations (PDEs). This is the first time that the neural network model is used to find the exact analytical solution, and this method covers almost all methods of constructing a function after bilinearization to solve nonlinear PDEs. Furthermore, this method is most likely a universal method to obtain the exact analytical solutions of nonlinear PDEs. Abundant arbitrary functions solutions of the reduced p-gBKP equation are obtained by using this method. Various beautiful plots of the presented solutions, which show diversity of exact solutions to PDEs, are made. By choosing appropriate values and functions, the fractal solitons waves are obtained and the self-similar characteristics of these waves are observed by reducing the observation range and magnifying local images. Via various three-dimensional plots, the evolution characteristics of these waves are exhibited.

One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada–Kotera equation

Abstract: The main concern of the present article is to study the fifth-order variable-coefficient Sawada–Kotera (VcSK) equation which describes the motion of long waves in shallow water under the gravity. A single- and double-soliton rational solutions for this model are formally retrieved through the generalized unified method. For a single-soliton wave, the velocity, the amplitude and the shape of the wave cannot be affected by variable coefficients. There is an inelastic collision (the collision that makes change in amplitude of the soliton waves and shifts in their trajectories) between the double-soliton waves due to the time-varying field in a graded-index waveguide. It hoped that the established solutions can be used to enrich the dynamic behaviors of the VcSK equation.

ELECTRE II method to deal with probabilistic linguistic term sets and its application to edge computing

Abstract: The edge node selection problem in edge computing is a typical multi-criteria group decision-making problem. In this paper, we put forward an ELECTRE II method with the probabilistic linguistic information to handle the edge node selection problem. First, a novel distance measure is developed for probabilistic linguistic term sets (PLTSs) and an entropy measure is devised to measure the uncertainty degree of PLTSs. Based on the score value and entropy, a novel method is put forward to compare two PLTSs. Next, a weight-determining method for criteria based on multiple correlation coefficient and a weight-determining method for experts based on entropy theory are proposed. After that, a novel probabilistic linguistic ELECTRE II method is put forward to deal with the edge node selection problem. Comparison with previous methods is provided to verify the superiority of our method.

Phase-shift controlling of three solitons in dispersion-decreasing fibers

Abstract: Phase-shift controlling can attenuate the interactions between solitons and gives practical advantage in optical communication systems. For the variable-coefficient nonlinear Schrodinger equation, which can be imitated the transmission of solitons in the dispersion-decreasing fiber, analytic three solitons solutions are derived via the Hirota method. Based on the obtained solutions, influences of the second-order dispersion parameters and other related parameters in different function types on the soliton transmission are discussed. Results declare that phase-shift controlling of solitons in dispersion-decreasing fiber can be achieved when the dispersion function is Gaussian one. In addition, by adjusting the constraint value, propagation distance of solitons can be further extended. This may be useful in the optical logic devices and ultra-short pulse lasers.

Dromion-like soliton interactions for nonlinear Schrödinger equation with variable coefficients in inhomogeneous optical fibers

Abstract: In this paper, a generalized nonlinear Schrodinger equation with variable coefficients is investigated. According to the analytic soliton solutions, the dromion-like structures between soliton interactions are revealed, and interaction properties are discussed. By changing the values of gain/loss, group velocity dispersion and self-phase modulation parameters, influences of them on interaction intensity and phase of solitons are presented. Besides, methods of controlling the path and spacing of solitons are suggested. Results of this paper may be potential valuable to the study of soliton interactions in inhomogeneous optical fibers and have applications in all-optical switches.

Interaction properties of solitonics in inhomogeneous optical fibers

Abstract: In this paper, a generalized nonlinear Schrodinger equation with variable dispersion and nonlinear coefficients, which can be used to describe the pulse transmission in inhomogeneous optical fibers, is investigated analytically. By virtue of the Hirota method, analytic multiple soliton solutions are obtained. Interactions between solitonics are presented through choosing specific nonlinearity functions, and interaction properties of them are analyzed. Results obtained may potentially be useful in the area of optical communications.

Periodic attenuating oscillation between soliton interactions for higher-order variable coefficient nonlinear Schrödinger equation

Abstract: According to the change in the amplitude of the oscillation, it can be divided into equal-amplitude oscillation, amplitude-reduced oscillation (attenuating oscillation) and amplitude-increasing oscillation (divergence oscillation). In this paper, the periodic attenuating oscillation of solitons for a higher-order variable coefficient nonlinear Schrodinger equation is investigated. Analytic one- and two-soliton solutions of this equation are obtained by the Hirota bilinear method. By analyzing the soliton propagation properties, we study how to choose the corresponding parameters to control the soliton propagation and periodic attenuation oscillation phenomena. Results might be of significance for the study of optical communications including soliton control, amplification, compression and interactions.

Generation and control of multiple solitons under the influence of parameters

Abstract: In this paper, the analytic three-soliton solution for a high-order nonlinear Schrodinger equation is obtained by the Hirota’s bilinear method. The transmission characteristics of three solitons are discussed. By selecting relevant parameters, soliton interactions are presented, and the method of generating new solitons is suggested. The influences of corresponding parameters on soliton transmission and interactions are analyzed. Results of this paper are helpful for enriching the soliton theory and studying the signal routing system.

Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation

Abstract: In this work, a non-isospectral and variable-coefficient Kadomtsev–Petviashvili equation is considered using Hirota’s bilinear form and a direct assumption with arbitrary functions. Analytical rational solutions in light of positive quadratic functions and lump solutions of the variable-coefficient Kadomtsev–Petviashvili equation are obtained. These lump solutions describe two types of characters by some three-dimensional graphs and contour plots, which contain bright lump wave and bright–dark lump wave. Meanwhile, periodic structure of the lump wave is also shown.

Control of imperfect dynamical systems

Abstract: Imperfections are unavoidable in production processes of real devices. Despite this, and despite the fact that real devices usually operate in regimes far from ideality, they still work. This is related to the fact that imperfections give rise to hidden dynamics, which, opportunely excited, have an overall positive effect on the device. In this paper, we focus on a complex and imperfect electromechanical structure which can be considered as a paradigm for imperfect systems. The electrical and mechanical interactions within the structure generate complex patterns of vibration which may prevent the system to reach the correct working conditions. A control strategy to ensure the optimal working conditions based on the excitation of the hidden dynamics induced by imperfections is discussed, characterizing its effect with respect to the control signal properties and to the power provided to the structure.

Elastic and viscoelastic foundations: a review on linear and nonlinear vibration modeling and applications

Abstract: This paper presents a comprehensive review on different theoretical elastic and viscoelastic foundation models in oscillatory systems. The review covers the simplest foundation models to the most complicated one and fully tracks the recent theories on the topic of mechanical foundations. It is fully discussed why each theory has been developed, what limitations each one contains, and which approaches have been applied to remove these limitations. Moreover, corresponding theories about structures supported by such foundations are briefly reviewed. Subsequently, an introduction to popular solution methods is presented. Finally, several important practical applications related to the linear and nonlinear foundations are reviewed. This paper provides a detailed theoretical background and also physical understanding from different types of foundations with applications in structural mechanics, nanosystems, bio-devices, composite structures, and aerospace-based mechanical systems. The presented information of this review article can be used by researchers to select an appropriate kind of foundation/structure for their dynamical systems. The paper ends with a new idea of intelligent foundations based on nanogenerators, which can be exploited in future smart cities for both energy harvesting and self-powered sensing applications.

Model electrical activity of neuron under electric field

Abstract: Continuous pump and transmission of charges such as calcium, potassium, sodium in the cell can induce time-varying electromagnetic field, and the induced electric field can further modulate the propagation of ions in the cell. Based on the physical laws of static electric field, the effect of electric field in isolate neuron is investigated by introducing additive variable E on the model. Each neuron is considered as a charged body with complex distribution of charges, and electric field is triggered to receive and give response to external electric field and electric stimulus. That is, the electric field is considered as a new variable to describe the polarization modulation of media resulting from external electric field and intrinsic change of density distribution in charges or ions. The dynamical behaviors in electrical activities are analyzed and discussed in the new neuron model, and it confirms that electric field can cause distinct mode transition in electrical activities of neuron exposed to different kinds of electric field. It could provide new insights to understand signal encoding and propagation in nervous system. Finally, it also suggests that new model can be used for signal propagation between neurons when synapse coupling is suppressed.

Hidden extreme multistability and dimensionality reduction analysis for an improved non-autonomous memristive FitzHugh–Nagumo circuit

Abstract: Due to the introduction of ideal memristors, extreme multistability has been found in many autonomous memristive circuits. However, such extreme multistability has not yet been reported in a non-autonomous memristive circuit. To this end, this paper presents an improved non-autonomous memristive FitzHugh–Nagumo circuit that possesses a smooth hyperbolic tangent memductance nonlinearity, from which coexisting infinitely many attractors are obtained. By utilizing voltage–current circuit model, a three-dimensional non-autonomous dynamical model is established, based on which the initial-dependent dynamics is explored by numerical plots and extreme multistability is thereby exhibited. To confirm that the improved non-autonomous memristive circuit operates in hidden oscillating patterns, an accurate two-dimensional non-autonomous dimensionality reduction model with initial-related parameters is further built by using incremental integral transformation, upon which stability analysis and bifurcation behaviors are elaborated. Because the equilibrium state of the dimensionality reduction model is always a stable node-focus, hidden extreme multistability with coexisting infinitely many attractors is truly confirmed. Finally, PSIM circuit simulations validate the initial-related hidden dynamical behaviors.

Breather and hybrid solutions for a generalized (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation for the water waves

Abstract: Water waves are one of the most common phenomena in nature, the study of which helps in designing the related industries. In this paper, a generalized ( $$3+1$$ )-dimensional B-type Kadomtsev–Petviashvili equation for the water waves is investigated. Gramian solutions are constructed via the Kadomtsev–Petviashvili hierarchy reduction. Based on the Gramian solutions, we construct the breathers. We graphically analyze the breather solutions and find that the breathers can be reduced to the homoclinic orbits. For the higher-order breather solutions, we obtain the mixed solutions consisting of the breathers and homoclinic orbits. According to the long-wave limit method, rational solutions are constructed. We look at two types of the rational solutions, i.e., the lump and line rogue wave solutions, and give the condition for the lumps being reduced to the line rogue waves. Taking another set of the parameters for the Gramian solutions, we also derive the kinky breather solutions which can be reduced to the kink solitons. For the higher-order kinky breather solutions, we obtain the mixed solutions consisting of the breathers and kink solitons. Combining the breather and rational solutions, we construct two kinds of the hybrid solutions composed of the breathers, lumps, line rogue waves and kink solitons. Characteristics of those hybrid solutions are graphically analyzed and the conditions for the generation of those hybrid solutions are given.

The amplitude, frequency and parameter space boosting in a memristor–meminductor-based circuit

Abstract: In this paper, a meminductor emulator and an active memristor emulator are designed to construct a new chaotic circuit. The initial-condition-triggered amplitude, frequency and parameter space boosting are investigated. The system owns homogenous, heterogeneous and extreme multistabilities at the same time. Various coexisting attractors with different offsets, amplitudes and frequencies are observed and analyzed. Furthermore, the presented circuit is implemented by analog circuit and DSP platform. The mentioned unique dynamic features are confirmed in the experiments. Experimental results indicate the presented system and its initial-condition-triggered features can be realized in DSP digital system. Since the system owns variable amplitude, frequency and parameter space, it has great potential value in encryption engineering fields.

Lie symmetry reductions and group invariant solutions of (2 + 1)-dimensional modified Veronese web equation

Abstract: The Lie symmetry method is successfully applied to compute group invariant solutions for (2 + 1)-dimensional modified Veronese web equation. The purpose of this present article is to study the modified Veronese web (mVw) equation and to obtain its infinitesimals, commutation table of Lie algebra, symmetry reductions and closed form analytical solutions. The obtained results are explicitly in the form of the functions $$f_1(y),f_2(t),f_3(x)$$ and $$f_4(x)$$ and hold numerous solitary wave solutions that are more helpful to describe dynamical phenomena through their evolution profile. The solutions are analysed physically via numerical simulation. Consequently, elastic behaviour multisolitons, line soliton, doubly soliton, parabolic wave profile, nonlinear behaviour of wave profile and elastic interaction soliton profile of solutions are demonstrated in the analysis and discussion section to make this study more praiseworthy.

Adaptive neural network tracking control for underactuated systems with matched and mismatched disturbances

Abstract: This paper studies neural network-based tracking control of underactuated systems with unknown parameters and with matched and mismatched disturbances. Novel adaptive control schemes are proposed with the utilization of multi-layer neural networks, adaptive control and variable structure strategies to cope with the uncertainties containing approximation errors, unknown base parameters and time-varying matched and mismatched external disturbances. Novel auxiliary control variables are designed to establish the controllability of the non-collocated subset of the underactuated systems. The approximation errors and the matched and mismatched external disturbances are efficiently counteracted by appropriate design of robust compensators. Stability and convergence of the time-varying reference trajectory are shown in the sense of Lyapunov. The parameter updating laws for the designed control schemes are derived using the projection approach to reduce the tracking error as small as desired. Unknown dynamics of the non-collocated subset is approximated through neural networks within a local region. Finally, simulation studies on an underactuated manipulator and an underactuated vibro-driven system are conducted to verify the effectiveness of the proposed control schemes.

One-soliton shaping and two-soliton interaction in the fifth-order variable-coefficient nonlinear Schrödinger equation

Abstract: One- and two-soliton analytical solutions of a fifth-order nonlinear Schrodinger equation with variable coefficients are derived by means of the Hirota bilinear method in this paper. Various scenarios of one-soliton shaping and two-soliton interaction and reshaping are investigated, using the obtained exact solutions and adjusting parameters of the underlying model. We find that widths of two colliding solitons can change without changing their amplitudes. Furthermore, we produce a solution in which two originally bound solitons are separated and are then moving in opposite directions. We also show that two colliding solitons can fuse to form a spatiotemporal train, composed of equally separated identical pulses. Moreover, we display that the width and propagation direction of the spatiotemporal train can change simultaneously. Effects of corresponding parameters on the one-soliton shaping and two-soliton interaction are discussed. Results of this paper may be beneficial to the application of optical self-routing, switching and path control.

Characteristics and interactions of solitary and lump waves of a (2 + 1)-dimensional coupled nonlinear partial differential equation

Abstract: A (2 + 1)-dimensional coupled nonlinear partial equation which possesses a Hirota bilinear form is introduced. Based on the Hirota bilinear form, two solitary waves are constructed. In the meanwhile, lump waves are derived by using a positive quadratic function. By combining an exponential function with a quadratic function, interaction solutions between a lump and a one-kink soliton, and between a bi-lump and a one-soliton solution are generated. Some special concrete interaction solutions are depicted in both analytical and graphical ways.

Nonlinear vibration of a slightly curved beam with quasi-zero-stiffness isolators

Abstract: Bending vibration of isolated structures has always been neglected when the vibration isolation was studied. Isolated structures have usually been treated as discrete systems. In this study, dynamics of a slightly curved beam supported by quasi-zero-stiffness systems are firstly presented. In order to achieve quasi-zero-stiffness, a nonlinear isolation system is implemented via three linear springs. A nonlinear dynamic model of the slightly curved beam with nonlinear isolations is established. It includes square nonlinearity, cubic nonlinearity, and nonlinear boundaries. Then, the mode functions and the frequencies of the curved beam with elastic boundaries are derived. The schemes of the finite difference method (FDM) and the Galerkin truncation method (GTM) are, respectively, proposed to obtain nonlinear responses of the curved beam with nonlinear boundaries. Numerical results demonstrate that both the GTM and the FDM yield accurate solutions for the nonlinear dynamics of curved structures with nonsimple boundaries. The multi-mode resonance characteristics of the curved beam affect the vibration isolation efficiency. The quasi-zero-stiffness isolators reduce the transmissibility of modal resonances and provide a promising future for isolating the bending vibration of the flexible structure. However, the initial curvature significantly increases the resonant frequency of the flexible structure, and thus the frequency range of the effective vibration isolation is narrower. Furthermore, the quadratic nonlinear terms in the curved beam make the dynamic phenomenon more complicated. Therefore, it is more challenging and necessary to investigate the isolation of the bending vibration of the initial curved structure.

Darboux transformation and analytic solutions for a generalized super-NLS-mKdV equation

Abstract: Darboux transformation is an efficient method for solving different nonlinear partial differential equations. In this paper, on the basis of a Lie super-algebras, a generalized super-NLS-mKdV equation is solved by the Darboux transformation. The analytic solutions are presented with the help of symbolic computation. Besides, two special cases are given to make the solution intuitive. Dynamic properties of solitons are also discussed.

Bursting oscillations and coexisting attractors in a simple memristor-capacitor-based chaotic circuit

Abstract: The design and analysis of a simple autonomous memristive chaotic circuit are important in theoretical, numerical, and experimental demonstrations of complex dynamics. In this paper, a simple autonomous memristive circuit is implemented, which only consists of an active second-order memristive diode bridge and a capacitor. Based on the available circuit, the mathematical model is established and its symmetry, dissipativity, and equilibrium stability are analyzed. Numerical simulations show that the proposed circuit exhibits complex behaviors of unipolar periodic and chaotic bursting oscillations along with coexisting attractors. It is worth noting that the circuit exhibits such a special bursting behavior previously unobserved in third-order autonomous memristive circuits. Moreover, spectral entropy complexities are calculated to provide an intuitive and effectual method for the circuit parameter configurations. The circuit simulations and hardware experiments verify the theoretical analyses and numerical simulations.

Rumor propagation dynamic model based on evolutionary game and anti-rumor

Abstract: In the online social network, the spreading process of rumor contains complex dynamics. The traditional research of the rumor propagation mainly studies the spreading process of rumor from the perspectives of rumor and participating user. The symbiosis and confrontation of rumor and anti-rumor information and the dynamic changes of the influence of anti-rumor information are not emphasized. At the same time, people’s profitability and herd psychology are also ignored. In view of the above problems, we fully consider the anti-rumor information and user’s psychological factors, construct a rumor propagation dynamics model based on evolutionary game and anti-rumor information, and provide a theoretical basis for studying the inherent laws in the spreading process of rumor. First of all, we analyze the interaction pattern and characteristic of rumor in social network. In allusion to the symbiosis of rumor and anti-rumor information and the dynamic changes of the influence of anti-rumor information, we constructed the SKIR rumor propagation model based on the SIR model. Secondly, due to rivalry between rumor and anti-rumor information, as well as the user’s profitability and herd psychology, we use evolutionary game theory to construct the driving force mechanism of information and explore the causes of user behavior in the spreading process of rumor. At the same time, we combine the behavior factors and external factors of the user to build the influence of information by multivariate linear regression method, which provides the theoretical basis for the driving force of information. Finally, combining the SKIR model proposed in this paper, we get a rumor propagation dynamics model based on evolutionary game and anti-rumor information. We have proved by experiments that the model can effectively describe the propagating situation of rumor and the dynamic change rule of the influence of anti-rumor information. On the other hand, it can also reflect the influence of people’s psychology on rumor propagation.

Analytical analysis of the vibrational tristable energy harvester with a RL resonant circuit

Abstract: In this paper, analytical analysis of the vibrational tristable energy harvester with a RL resonant circuit is presented. The analytical solutions of the steady-state response displacement and the steady-state output voltage are derived via the method of multiple scales. The influence mechanism of the excitation amplitude and frequency, the electromechanical coupling coefficient, the damping and the detuning parameters on the dynamic response characteristics and the output voltage is studied. In order to enhance the energy harvesting performance, the appropriate choice of the excitation amplitude and the electromechanical coupling coefficient is discussed.

Interaction and energy transition between the breather and rogue wave for a generalized nonlinear Schrödinger system with two higher-order dispersion operators in optical fibers

Abstract: A generalized nonlinear Schrodinger system is investigated, which can be used to describe the optical pulse propagation in inhomogeneous optical fibers with the fourth- and third-order dispersions operators. The Darboux transformation method is extended to construct a mixed breather and rogue wave solution for the system. The interaction behaviors between the breather and rogue wave are studied. As a novel result, the energy transition between the breather and rogue wave is observed. Furthermore, the impacts of the different operators on the mixed solution are analyzed.

New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions

Abstract: In the present course of study, we examine a family of Boussinesq equations of distinct structures and dimensions. We investigate the complete integrability of these equations via Painleve test. Real and complex multiple soliton solutions, for each considered model, are derived by mode of simplified Hirota’s method. Moreover, exponential expansion method has been employed to each equation, resulting into soliton solutions possessing rich spatial structure due to the presence of abundant arbitrary constants.

Coexisting multiple firing patterns in two adjacent neurons coupled by memristive electromagnetic induction

Abstract: The membrane potential difference between two adjacent neurons can induce an electromagnetic induction current that behaves like a memristor synapse effect using to couple these two neurons. Based on two-dimensional (2D) Hindmarsh–Rose (HR) neuron and non-ideal threshold memristor, this paper presents a five-dimensional (5D) neuron model of two adjacent neurons coupled by memristive electromagnetic induction. With the 5D memristor-coupled neuron model, the equilibrium states and their stabilities are investigated by qualitative analyses, and the coexisting phenomena of multiple firing patterns and initials-depending bifurcation routes along with extreme events are then uncovered by numerical simulations. Due to complex stabilities of the seven equilibrium states, the attraction basin of one neuron in the 5D memristor-coupled neuron model is not only related to the coupling strength of memristor synapse but also associated with another neuron initials, leading to that the coexisting multiple firing patterns are emerged from different regions in the attraction basin and the bifurcation routes are closely dependent to the initials. Furthermore, circuit syntheses using the off-the-shelf components and breadboard experiments for the proposed 5D memristor-coupled neuron model are executed so that the coexisting multiple firing patterns and extreme events are then conformed perfectly, which have not yet been previously reported in the coupled HR neuron model.

A nonlinear resonator with inertial amplification for very low-frequency flexural wave attenuations in beams

Abstract: Although elastic metamaterials in a subwavelength scale can control macroscopic waves, it is still a big challenge to attenuate elastic waves at very low frequency (a few tens Hz). The main contribution of this paper is to develop a high-static-low-dynamic-stiffness (HSLDS) resonator with an inertial amplification mechanism (IAM), which is able to create a much lower band gap than a pure HSLDS resonator. The nonlinear characteristics of a locally resonant (LR) beam attached with such new resonators are also explored. The band gap of this LR-IAM beam is revealed by employing transfer matrix method and validated by numerical simulations using Galerkin discretization. It is shown that a very low-frequency band gap can be formed by tuning the net stiffness of the resonator towards an ultra-low value. In addition, the nonlinearity, arising from the restoring force of the resonator, the damping force and effective inertia of the IAM, gives rise to an intriguing feature of amplitude-dependent wave attenuation, which could potentially act as a switch or filter to manipulate flexural waves.