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This paper is dedicated to identifying stable equilibrium positions of the tether systems attached to the L1 or L2 libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L1 or L2 libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L1 and L2 points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases.

We investigated the thermal buckling temperature and nonlinear free vibration of functionally graded fiber-reinforced composite laminated (FG-FRCL) beams. The governing nonlinear partial differential equations were derived from the Euler–Bernoulli beam theory, accounting for the von Kármán geometrical nonlinearity. Such equations were then reduced to a single equation by neglecting the axial inertia. Thus, the Galerkin method was applied to discretize the governing nonlinear partial differential equation in the form of a nonlinear ordinary differential equation, which was then solved analytically according to the He’s variational method. Three different boundary conditions were selected, namely simply, clamped and clamped-free supports. We also investigated the effect of power-index, lay-ups, and uniform temperature rise on the nonlinear natural frequency, phase trajectory and thermal buckling of FG-FRCL beams. The results showed that FG-FRCL beams featured the highest fundamental frequency, whereas composite laminated beams were characterized by the lowest fundamental frequency. Such nonlinear frequencies increase for an increased power index and a decreased temperature. Finally, it was found that FG-FRCL beams with [0/0/0] lay-ups featured the highest nonlinear natural frequency and the highest thermal buckling temperature, followed by [0/90/0] and [90/0/90] lay-ups, while a [90/90/90] lay-up featured the lowest nonlinear natural frequency and critical buckling temperature.

Short-circuit current analysis, especially transient current calculation, provides a basis for protection design. Because the control systems of inverter-interfaced renewable energy sources have a high degree of nonlinearity, it is difficult to analyze the transient current. To address this issue, in this work, a nonlinear differential equation of transient current of symmetrical fault is derived while considering the saturation characteristics of the current loop. Aiming at the solution of this nonlinear differential equation, an analysis method based on the phase plane is proposed, which makes use of the linear relationship between the current deviation and its derivative to simplify the direct determination of the nonlinear working state in the time domain. The analytical expression of the transient current can then be obtained by the inverse transform from the phase plane to the time domain. The proposed calculation method is verified by both simulations and field tests.

A modular electric vehicle platform enables ability to integrate different control systems in a flexible way. In this paper, a decentralized cooperative control framework is proposed to achieve the integration of active front steering system (AFS) and active suspension system (ASS) by applying a multi-constrained distributed model predictive control (MDMPC) approach, which aims to improve the vehicle lateral stability, ride comfort and roll safety during path tracking. First, a partly decoupled 6-degree-of-freedom half-vehicle dynamics model is constructed. Then, a multi-agent-system (MAS)-based framework is introduced to coordinate AFS and ASS, which allows for the multi-constraints within the information exchange between agents. Through minimizing linear convex combination of objective functions, the cooperative control strategy is ultimately solved by the Pareto-optimality theory. Moreover, vehicle lateral and roll motion stability region described by the phase plane is employed to bound the controllable limits and achieve the dynamic cooperation between AFS and ASS. The simulation and hardware-in-the-loop (HIL) test results show that the proposed framework is effective for coordinating AFS and ASS, thereby enhancing the vehicle lateral and vertical stability during path tracking.

Hovering is an effective and challenging way for asteroid exploration. In this paper, the adaptive quasi-periodic switching control method for body-fixed hovering of asteroids with low fuel consumption and strong robustness using constant thrust is proposed. By analyzing the active control process and the drift process of the spacecraft in the phase plane, constant thrust control is applied in the appropriate orientation to maintain the desired hovering state above the asteroids. Considering the uncertainty of the dynamic environments near the asteroids, the apex position of drift process in the phase plane is predicted, meanwhile, the compensation conditions and the calculation formula for the constant-thrust engine ignition are given. The simulation results show that the quasi-periodic switching control method can reduce the fuel consumption of the spacecraft compared with other methods, moreover, the adaptive quasi-periodic switching controller can effectively improve the robustness of the system.

The dynamics of neurons consist of oscillating patterns of a membrane potential that underpin the operation of biological intelligence. The FitzHugh−Nagumo (FHN) model for neuron excitability generates rich dynamical regimes with a simpler mathematical structure than the Hodgkin−Huxley model. Because neurons can be understood in terms of electrical and electrochemical methods, here we apply the analysis of the impedance response to obtain the characteristic spectra and their evolution as a function of applied voltage. We convert the two nonlinear differential equations of FHN into an equivalent circuit model, classify the different impedance spectra, and calculate the corresponding trajectories in the phase plane of the variables. In analogy to the field of electrochemical oscillators, impedance spectroscopy detects the Hopf bifurcations and the spiking regimes. We show that a neuron element needs three essential internal components: capacitor, inductor, and negative differential resistance. The method supports the fabrication of memristor-based artificial neural networks.

In ecology, the refuge protection of the prey plays a significant role in the dynamics of the interactions between prey and predator. In this paper, we investigate the dynamics of a non-smooth prey-predator mathematical model characterized by density-dependent intermittent refuge protection of the prey. The model assumes the population density of the predator as an index for the prey to decide on when to avail or discontinue refuge protection, representing the level of apprehension of the prey by the predators. We apply Filippov's regularization approach to study the model and obtain the sliding segment of the system. We obtain the criterion for the existence of the regular or virtual equilibria, boundary equilibrium, tangent points, and pseudo-equilibria of the Filippov system. The conditions for the visibility (or invisibility) of the tangent points are derived. We investigate the regular or virtual equilibrium bifurcation, boundary-node bifurcation and pseudo-saddle-node bifurcation. Further, we examine the effects of dispersal delay on the Filippov system associated with prey vigilance in identifying the predator population density. We observe that the hysteresis in the Filippov system produces stable limit cycles around the predator population density threshold in some bounded region in the phase plane. Moreover, we find that the level of apprehension and vigilance of the prey play a significant role in their refuge-dispersion dynamics.

In view of the problems related to vehicle-handling stability and the real-time correction of the heading direction, nonlinear analysis of a vehicle steering system was carried out based on phase plane theory. Subsequently, direct yaw-moment control (DYC) of the vehicle was performed. A four-wheel, seven-degree-of-freedom nonlinear dynamic model that included the nonlinear characteristics of the tire was established. The stable and unstable regions of the vehicle phase plane were divided, and the stable boundary model was established by analyzing the side slip angle–yaw rate ([Formula: see text]) and side slip angle–side slip angle rate [Formula: see text] phase planes as functions of the vehicle state variables. In the unstable region of the phase plane, taking the instability degree as the control target, a fuzzy neural network control strategy was utilized to determine the additional yawing moment of the vehicle required for stability restoration, which pulled the vehicle back from an unstable state to the stable region. In the stable region of the phase plane, a fuzzy control strategy was utilized to determine the additional yawing moment so that the actual state variables followed the ideal state variables. In this way, the vehicle responded rapidly and accurately to the steering motion of the driver. A simulation platform was established in MATLAB/Simulink and three working condition was tested, that is, step, sine with dwell, and sine amplification signals. The results showed that the vehicle handling stability and the instantaneous heading-direction adjustment ability were both improved due to the control strategy.

The article is devoted to solving the problem of analysis and synthesis of a control system with a fuzzy controller by the phase plane method. The nonlinear transformation, built according to the Sugeno fuzzy model, is approximated by a piecewise linear characteristic consisting of three sections: two piecewise linear and one piecewise constant. This approach allows us to restrict ourselves to three sheets of phase trajectories, each of which is constructed on the basis of a second-order differential equation. Taking this feature into account, the technique of "stitching" of three sheets of phase trajectories is considered and an analytical base is obtained that allows one to determine the conditions for "stitching" of phase trajectories for various variants of piecewise-linear approximation of the characteristics of a fuzzy controller. In view of the specificity of the approximated model of the fuzzy controller used, useful analytical relations are given, with the help of which it is possible to calculate the time of motion of the representing point for each section with the involvement of the numerical optimization apparatus. For a variant of the approximation of three sections, a technique for synthesizing a fuzzy controller is proposed, according to which the range of parameters and the range of input signals are determined, at which an aperiodic process and a given control time are provided. On the model of the automatic control system of the drive level of the mechatronic module, it is shown that the study of a fuzzy system by such an approximated characteristic of a fuzzy controller gives quite reliable results. The conducted studies of the influence of the degree of approximation on the quality of control show that the approximated characteristic of a fuzzy controller gives a slight deterioration in quality in comparison with the smooth characteristic of a fuzzy controller. Since the capabilities of the phase plane method are limited to the 2nd order of the linear part of the automatic control system, the influence of the third order on the dynamics of the system is considered using the example of a mechatronic module drive. It is shown that taking into account the electric time constant leads to overshoot within 5-10 %. Such overshoot can be eliminated due to the proposed recommendations for correcting the static characteristic of the fuzzy controller.

Phase plane analysis of nucleus-acoustic waves (NAWs) is explored for the Burgers equation in a white dwarf plasma system constituting degenerate electrons, heavy nuclear species and degenerate light nuclear species. Employing the reductive perturbation method (RPM), the Burgers equation is formulated. The Backlund transformation of the Burgers equation is deduced in polynomial form by exploiting the homogeneous balance principle (HBP) which eventually characterizes the integrability of the system in the sense of HBP. Exact solutions are also deduced from the Backlund transformation. The phase plane of the planar Hamiltonian system (PHS) is studied and the effects of physical parameters, such as, viscosity ( η ), total charge number density of lighter to heavier nucleus ( μ h ) and velocity of NAWs ( v ) are studied on the snoidal, kink and anti-kink NAW solutions of the Burgers equation. The outcomes of this work can be significant in discerning nucleus-acoustic snoidal, kink and anti-kink wave features in white dwarf.

This article presents a protection scheme based on the magnitude-phase plane of impedance difference (ID) using wide-area positive sequence components of current and voltage signals for the microgrid. The computation of the ID of the line is performed by extracting signals from the phasor measurement units complied with IEEE C37.118 standards. The composite magnitude-phase plane obtained from the magnitude of the ID and the angle of ID is considered as the key index for the detection of faults. The feasibility to include the proposed microgrid protection technique is examined by the extensive fault cases simulation with substantial variation in fault conditions, including types of fault and fault at different line length and for the grid-connected and islanding modes of microgrid operation. The fault study is carried out on a medium-voltage 15 bus test microgrid and a standard IEEE 34 bus microgrid testbed. The impact of several critical no-fault conditions is also investigated to analyze the possibility of false operation. The proposed ID-based scheme is extensively tested on MATLAB/Simulink platform and validated on the real-time digital simulator. The test results indicate that the magnitude-phase plane of the ID-based scheme can provide a dependable microgrid protection measure.

Stability and nonlinear dynamics of slightly curved viscoelastic pipe conveying fluid resting on linear and nonlinear viscoelastic foundations is investigated. The dissipation of the pipe material is modelled using the Kelvin–Voigt scheme. The equation of motion was derived using Hamilton’s principle. Three different trigonometry initial curvatures were considered corresponding to simply supported, clamped simply supported and clamped–clamped boundary conditions. Eigenfunction expansion method was employed to change the partial differential equation of motion to modal ordinary differential equations. The first four modes were considered and solved for natural frequency. The linear natural frequency analysis shows that an increase in viscoelasticity coefficient reduces the natural frequency for all the boundary conditions considered. An increase in viscoelasticity coefficient eliminates coupled mode flutter especially for the third and fourth mode and has no effect on first critical velocity. An increase in initial curvature increases the natural frequency, and the increase is more pronounced for the clamped simply supported boundary condition. The linear stiffness increases the critical velocity and natural frequency of the pipe. Bifurcation diagrams and phase plane plots are presented for various boundary conditions. Chaotic motions are observed without viscoelasticity but are dampened out as increase in viscoelasticity coefficient.

In this paper, Hamiltonian approach for solving vibration in the first approximation is extended into a high order Hamiltonian approach method for n order approximation. The method is applied for computation of the frequency of doubly clamped beam type N/MEMS subjected to the van der Waals attraction and the Duffing oscillator with seventh order nonlinearity. The obtained results are compared with numerically and other analytically obtained results. It is concluded that the solution in the third approximation gives the most appropriate solution in comparison to other analytical methods. The time history diagrams and the phase plane diagrams calculated analytically are on the top of the curves obtained numerically.

The exact and asymptotic analyses of the buckling of a quadratic-cubic model structure subjected to static loading are discussed. The governing equation is first solved using the phase plane method and next, using the method of asymptotics. In the asymptotic method, we discuss the possibilities of using regular perturbation method in asymptotic expansions of the relevant variables to get an approximate analytical solution to the problem. Finally, the two results are compared using numerical results obtained with the aid of Q-Basic codes. In the two methods discussed in this work, it is clearly seen that the static buckling loads decrease as the imperfection parameters increase. It is also observed that the static buckling loads obtained using the exact method are higher than those obtained using the method of asymptotics.

It has been shown that account of elastic nonlinearity during the propagation of an acoustic wave in a solid lead to the appearance of a quadratic nonlinearity, which in its turn leads to the possibility of generating a wave of double frequency, the interaction of harmonics is asymmetric. The conditions under which nonlinear stationary waves are formed are considered. A phase portrait is constructed, and the dependence of the wavenumber of a nonlinear wave on its amplitude is estimated.

This paper investigates the effects of dynamic loads and structural damping on the nonlinear behaviors of incompressible hyperelastic spherical shells modeled by the Yeoh strain energy function. Firstly, the dynamical modeling is formulated for the nonlinear behaviors of the shells by the variational principle, and the second order nonlinear ordinary differential equation describing the radially symmetric motion is obtained. Then, the dynamic behaviors, such as periodic, quasiperiodic and chaotic motions, are discussed under different loading types. Particularly, for constant loads, the first integral of the integrable Hamiltonian system without damping is given and it is numerically proved that there exists an asymmetric "∞" homoclinic orbit for the prescribed material parameters obtained in experiments; moreover, it is found that for different prestretches, the structural parameter has a completely different role on the nonlinearity of the system, and the basins of attraction are given with the structural damping. For periodic loads, there exist some interesting dynamic phenomena, i.e., quasiperiodic oscillation in the approximately integrable Hamiltonian system, limit cycles and chaos with the damping. The criterion for chaos is discussed by the Melnikov method combined with the numerical calculation and the chaos is further analyzed with the Poincare section and the phase plane.

In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence results of globally exponentially stable (positive) stationary solution of delayed reaction–diffusion cell neural networks under Dirichlet zero boundary value, including the global stability criteria in the classical meaning. Next, sufficient conditions are proposed to guarantee the global stability invariance of ordinary differential systems under the influence of diffusions. New theorems show that the diffusion is a double-edged sword in judging the stability of diffusion systems. Besides, an example is constructed to illuminate that any non-zero constant equilibrium point must be not in the phase plane of dynamic system under Dirichlet zero boundary value, or it must lead to a contradiction. Next, under Lipschitz assumptions on active function, another example is designed to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks via a Saddle point theorem. Finally, a numerical example illustrates the feasibility of the proposed methods. It is worth mentioning that some interesting mathematical problems are originally put forward in the last chapter.

In this paper, a new type of non-volatile locally active memristor with bi-stability is proposed by injecting appropriate voltage pulses to realize a switching mechanism between two stable states. It is found that the memristive parameters of the new memristor can affect the local activity, which has been rarely reported, and this phenomenon is explained based on mathematical analyses and numerical simulations. Then, a locally active memristive coupled neuron model is constructed using the proposed memristor as a connecting synapse. The parameter-associated dynamical behaviors are revealed by bifurcation plots, phase plane portraits and dynamical evolution maps. Moreover, the bi-stability phenomenon of the new coupled neuron model is disclosed by local attraction basins, and the periodic burster and multi-scroll chaotic burster are found if a multi-level pulse current is used to imitate a periodical external stimulus on the neurons. The Hamiltonian energy function is calculated and analyzed with or without external excitation. Finally, the neuronal circuit is designed and implemented, which can mimic electrical activity of the neurons and is useful for physical applications. The experimental results captured from the analog circuit are consistent well with the numerical simulation results.

We reconsider the multiplier–accelerator model of business cycles, first introduced by Samuelson and then modified by many authors. The original simple model, besides damped oscillations, also leads to divergent oscillations. To avoid this, we introduce two different types of governmental expenditures leading a two-dimensional continuous piecewise linear map that can generate sustained oscillations (attracting cycles). The map is defined by three different linear functions in three different partitions of the phase plane, and this peculiarity influences the overall dynamics of the system. We show that, similar to the classical Samuelson model, there is a unique feasible equilibrium as well as converging oscillations. However, close to the center bifurcation value the attracting equilibrium coexists with attracting cycles of different periods, which lose stability via a center bifurcation simultaneously with the equilibrium. Moreover, we show that attracting cycles of particular type also exist when the equilibrium becomes an unstable focus. For several families of attracting cycles, by introducing the symbolic representation, we obtain boundaries of the related periodicity regions, associated with border collision bifurcations.

In this paper, the discontinuous dynamic analysis of a two-degree-of-freedom system with bilateral rigid constraints in the coexistence of nonlinear friction and external excitations with negative feedbacks is presented through the theory of flow switchability and flow barrier in discontinuous dynamical systems, where the nonlinear friction force is approximated by a piecewise linear, kinetic friction model with static friction. Considering inequality of coefficients of static and kinetic friction, the flow barriers on velocity boundaries are introduced. Based on friction and collisions, different domains and boundaries are defined in phase plane. The analytical conditions of motion switching at separation boundaries are explored through the force criteria and the flow barrier’s criteria based on the analysis of vector fields in domains. Finally, the numerical simulations of several typical motions are given for a better understanding of switching conditions between different motions. The results obtained in this paper are of great significance to the optimization design of mechanical system with bilateral rigid constraints and negative feedbacks.

An asymmetric memristive diode-bridge (MDB) emulator is raised to imitate the asymmetric volt-ampere characteristic of a physical memristor. Then, an asymmetric MDB-based Jerk circuit is built and its state equation is derived, upon which the theoretical analysis, MATLAB-based numerical simulations, and hardware measurements are executed to reveal the asymmetric coexisting bifurcations and the phenomenon of multi-stability. The memristive Jerk circuit has three equilibrium points of a pair nontrivial equilibrium points of asymmetric unstable saddle-foci and a zero equilibrium point of unstable saddle-focus, which leads to the occurrence of asymmetric coexisting bifurcations and asymmetric local attraction basins. The asymmetrical bifurcations are numerically disclosed by 1-D/2-D bifurcation plots, Lyapunov spectrum, and phase plane trajectories. Multi-stability with asymmetric coexisting attractors under two sets of system parameters are demonstrated as examples by local attraction basins and phase plane trajectories. Thereafter, experimental circuit prototype employing discrete components is manually welded and hardware measurements are executed to validate the numerical simulations.

All differential equations students have encountered eigenvectors and eigenvalues in their study of systems of linear differential equations. The eigenvectors and phase plane solutions are displaye...

To address the time-optimal trajectory planning (TOTP) problem with joint jerk constraints in a Cartesian coordinate system, we propose a time-optimal path-parameterization (TOPP) algorithm based on nonlinear optimization. The key insight of our approach is the presentation of a comprehensive and effective iterative optimization framework for solving the optimal control problem (OCP) formulation of the TOTP problem in the (s,s˙)-phase plane. In particular, we identify two major difficulties: establishing TOPP in Cartesian space satisfying third-order constraints in joint space, and finding an efficient computational solution to TOPP, which includes nonlinear constraints. Experimental results demonstrate that the proposed method is an effective solution for time-optimal trajectory planning with joint jerk limits, and can be applied to a wide range of robotic systems.

We apply the Ericksen–Leslie dynamic model to investigate stationary solutions of planar shear flows for nematic liquid crystals. Nematic material is extremely rich and involves several parameters to characterize the basic local properties. The Ericksen–Leslie model of the nematic material includes Frank’s parameters for Oseen–Frank energy and Leslie’s dynamic parameters. Even in this simple setting of shear flows, the dynamics supported by the material exhibit quick rich behavior. With the aid of a Hamiltonian formulation and phase plane portraits, we are able to provide a rather complete picture about the possible solutions. In particular, we establish the existence of multiple solutions for the boundary value problem in many situations. We also try to explain the physical reasons for some of the results.

形態: カラー図版あり, Physical characteristics: Original contains color illustrations, Accepted: 2020-11-23, 資料番号: PA2110013000

In this paper, fast setpoint altitude tracking control for Hypersonic Flight Vehicle (HFV) satisfying Angle of Attack (AOA) constraint is studied with a two-loop structure controller, in the presence of parameter uncertainties and disturbances. For the outer loop, phase plane design is adopted for the simplified model under Bang-Bang controller to generate AOA command guaranteeing fast tracking performance. Modifications based on Feedback-Linearization (FL) technique are adopted to transform the phase trajectory into a sliding curve. Moreover, to resist mismatch between design model and actual model, Fast Exponential Reaching Law (FERL) is augmented with the baseline controller to maintain state on the sliding curve. The inner-loop controller is based on backstepping technique to track the AOA command generated by outer-loop controller. Barrier Lyapunov Function (BLF) design is employed to satisfy AOA requirement. Moreover, a novel auxiliary state is introduced to remove the restriction of BLF design on initial tracking errors. Dynamic Surface Control (DSC) is utilized to ease the computation burden. Rigorous stability proof is then given, and AOA is guaranteed to stay in predefined region theoretically. Simulations are conducted to verify the efficiency and superior performance of the proposed method.

In this paper, we study the competition between two firms whose outputs are quantities. The first firm considers maximization of its profit while the second firm considers maximization of its social welfare. Adopting a gradient-based mechanism, we introduce a nonlinear discrete dynamic map which is used to describe the dynamics of this game. For this map, the fixed points are calculated and their stability conditions are analyzed. This includes investigating some attracting set and chaotic behaviors for the complex dynamics of the map. We have also investigated the types of the preimages that characterize the phase plane of the map and conclude that the game’s map is noninvertible of type Z 4 − Z 2 .

Starting from the car suspension system, the nonlinear characteristics of a class of two-degree-of-freedom oscillators with strong nonlinearity under a periodic excitation are discussed by the switching theory of flow in discontinuous dynamical systems. Based on the discontinuous forces and different motions of the two masses, the phase plane of each mass is composed of stick domain, nonstick domain (or free domain) and separation boundary in absolute and relative coordinates, respectively. The swithching criteria between the stick and nonstick motions and the conditions of grazing motion in two different regions are developed via the G-functions and switching control laws. The mapping dynamics theory is used to give the four-dimensional transformation set and four-dimensional mapping, and the conditions for periodic motions are explored. In addition, the stick motions, two kinds of grazing motions, periodic motions for this system and a comparison of the velocities, accelerations (or forces responses) of the two masses under the two conditions of control force are simulated numerically. The results show that the stability and comfort of the vehicle can be improved by adjusting the control force, which is generated by the control unit of system or exerted by the external excitation. For further investigating the influence of system parameters on dynamical behaviors, the stick and grazing bifurcation scenarios varying with driving frequency or amplitude are also developed, which can provide useful information for parameter selection of vibration systems with clearance and the optimal design of vehicle suspension systems. This paper also has important reference value for practical applications in other industries or machinery with elastic impacts.

In this paper, a new third-order chaotic system which has extremely multistability is constructed by introducing the boosted control of cosine function. In comparison with other chaotic systems of extremely multistability, the proposed chaotic system can spontaneously generate the infinitely many coexisting attractors towards two directions of the phase plane. It indicates the proposed system can output more chaotic sequences of different amplitudes at the same time. This peculiar physical phenomena is very interesting and worth studying. Relative to original chaotic system, the chaos characteristic of the proposed system is obviously enhanced, the value of max Lyapunov exponent is increased significantly and the complexity value was higher. In particular, many periodic windows of the original chaotic system become chaos. It means the proposed chaotic system has better chaotic characteristics. If the new system is applied to the field of cryptography, it would be a better system model as a pseudo-random signal generator (PRSG). Then, the new image encryption algorithm is designed based on the proposed discrete system, and its safety performance is tested. The experimental results demonstrate the feasibility of its application in the field of cryptography.

In this paper, the control of chaotic vibrations in vehicle suspension system is studied via a fuzzy fast terminal sliding mode method. Therefore, the nonlinear equations of motion in the vehicle half model are derived initially through the Newton-Euler laws and then these equations are solved by the fourth-order Runge-Kutta method. For analysis of chaos, some techniques including bifurcation diagrams, frequency response, power spectrum, phase plane trajectories, and Poincare section are used to identify the chaotic behaviors. The chaotic zones are depicted with critical values on the uncontrolled model under the road surface force. In order to eliminate chaotic behavior, the control signals in the active suspension system are generated using the novel fuzzy fast terminal sliding mode control algorithm. The simulation results of the feedback system show that the suspension system can be stabilized the vibrations by efficient fuzzy fast terminal SMC while eliminating irregular chaotic behaviors.

In this paper, a robust adaptive fuzzy proportional-derivative inverse dynamics decoupling control scheme with fuzzy-based linear extended state observer (FLESO) is presented and applied to the trajectory tracking control of a two degree-of-freedom (2-DOF) spherical motion mechanism (SMM). The dynamics of the SMM has the characteristics of multivariable nonlinearity, uncertainties and strong coupling. Uncertainties like the modeling errors and external disturbances affect the tracking performance, and coupling increases the difficulty of controller design and reduces the tracking precision. Therefore, a novel hybrid control scheme that is composed of a fuzzy proportional-derivative (FPD) feedback control with varying gains, inverse dynamic model-based feed-forward decoupling term, FLESO with varying bandwidth, and robust term is developed. The novel control strategy combines the advantages of simplicity and easy design of the FPD control, the effectiveness of the FLESO to handle the modeling errors and external disturbances, and the robustness of the robust term to estimation errors of the FLESO. First, introduce the structure of the SMM and establish the dynamic model. Second, the feed-forward decoupling principle is derived based on the inverse dynamic model. Then the FPD control with two-inputs and two-outputs is designed, whose rule base is derived by the phase plane method. The linear extended state observer is designed, whose bandwidth is tuned via the fuzzy logic system. Furthermore, the asymptotic stability of the proposed controller is proved by the Lyapunov theorem. Finally, the high tracking performance of the proposed controller is validated via both simulation and experiment results.

Analytical modeling and solution of nonlinear vibration behaviors are vitally important for the optimization of vibratory actuators. In this paper, the nonlinear vibration characteristics of a self-excited electrostatic actuator are investigated. The actuator consists of two plate electrodes and a metal cantilever, which is fixed on an isolated insulating base and placed in the middle of the electrodes. When a DC voltage is applied to the electrodes, the cantilever is excited into reciprocating vibration autonomously and collides with the electrodes with strongly nonlinear features. During modeling, the collision is considered as a transient process and only the velocities of the cantilever before and after the collisions are concerned. By using the phase plane method, the limit cycle and boundary conditions of the self-excited oscillation of the cantilever are obtained and the strongly nonlinear problem of the entire system is converted into a weakly nonlinear problem of the cantilever moving between the electrodes. The nonlinear equations are then solved by using the averaging method, and the approximate analytical results demonstrate good consistency with the numerical and experimental results. Besides, the approximate analytical results are also adopted in the optimization of the actuator for enhanced power density and efficiency. The modeling and solution methods provide a framework for further optimization of the electrostatic actuator in the application field of centimeter-scale robotics.

In this paper, we investigate the elementary wave interactions for the Suliciu relaxation system and construct uniquely the solution by the characteristic analysis method in the phase plane. We find that the elementary wave interactions have a much simpler structure for the Temple class than the general systems of conservation laws. It is observed that the Riemann solutions of the Suliciu relaxation system are stable under the small perturbation on the Riemann initial data.

We study the solutions to the following Dirichlet boundary problem: \begin{equation*}\frac{d^2x(t)}{dt^2}+\lambda f(x(t))=0,\end{equation*} where $x \in \mathbb{R}$, $t \in \mathbb{R}$, $\lambda \in \mathbb{R}^+$, with boundary conditions: \begin{equation*} x(0)=x(1)=A \in \mathbb{R}. \end{equation*} Especially we focus on varying the parameters $\lambda$ and $A$ in the case where the phase plane representation of the equation contains a saddle loop filled with a period annulus surrounding a center. We introduce the concept of mixed solutions which take on values above and below $x=A$, generalizing the concept of the well-studied positive solutions. This leads to a generalization of the so-called period function for a period annulus. We derive expansions of these functions and formulas for the derivatives of these generalized period functions. The main result is that under generic conditions on $f(x)$ so-called S-shaped bifurcations of mixed solutions occur. As a consequence there exists an open interval for sufficiently small $A$ for which $\lambda$ can be found such that three solutions of the same mixed type exist. We show how these concepts relate to the simplest possible case $f(x)=x(x+1)$ where despite its simple form difficult open problems remain.

The elementary wave interactions for the Chapman-Jouguet model with combustion are investigated. We obtain the unique solution of the initial value problem under the global entropy conditions. We analyze the elementary wave interactions in the phase plane and construct uniquely the solution of this initial value problem. It is found that the combustion wave solution of the corresponding Riemann may be extinguished after perturbation which shows that the unburnt gas is unstable.

This article proposes a model that generalizes the well-known equations of the nonlinear theory of oscillations (Van der Pol and Rayleigh equations), the limit cycles of which are the curves of the phase plane, determined by the total energy of oscillations in the absence of dissipation/energy inflow into the system. By changing the parameters of the system and the type of force action, it is possible to set the oscillation characteristics that are resistant to disturbances. The phase portraits of the system are considered, including the case of multiply connected limit cycles.

Local blackout occurs when the transient stability limit is exceeded after a transmission line fault. As to global blackout, a recording of the 10 August 1996-WECC blackout shows that it was accompanied negative damping. A turbine-generator swinging against an infinite bus is used to approximate the dynamics leading to the WECC blackout. There is no analytic solution when non-linearity is included. Small-signal linearisation shows that negative damping can occur but cannot produce the waveform recorded in the WECC blackout. Therefore, the graphical phase-plane method, which is easy to use, is resorted to. The study shows that the kernel of non-linearity consists of the gradients which produce limit cycles. Positive or negative damping is produced by shifting the gradients of the limit cycles to the right or left in the direction of the trajectory. The study makes a contribution by showing that phase plane can be used to study the impact of multiple independent controllers. A worked example shows when transient stability limit is exceeded, local blackout occurs. The swing equation of local blackout is applied to the WECC blackout. As proof that negative damping was due to non-linearity, a waveform having the frequency of the recorded WECC blackout is presented.

Information has both theoretical and natural importance. At it’s roots from both a theoretical and quantum mechanical perspective, information is a dynamical process that is governed by a coupled relationship between energy and entropy. This research will investigate means for synthesizing information that is consistent with both quantum mechanical and dynamical processes. Such an approach is expected to be more consistent with naturally occurring systems, such as chemical and biological ones. The precise relationship between multiple stable equilibriums, energy quanitzation and stable information bearing patterns in the phase plane are examined. Relationships to information theory are a logical application of these results. The theorem provided allows for physical realizations of actual continuous-time quantized computational devices that are consistent with atomic and molecular dynamical processes. This pursuit can be thought of as the development of an electronic device that behaves according to quantum dynamical behaviors similar to molecular orbit phenomenology. The motivation for such devices is derived from an interest in developing a dynamical process that adheres to what will be described as physical means for representing the dynamical process of information. Results from actually synthesized physical devices are also provided. Collectively, these results represent the first known characterization and synthesis of a quantum mechanical facsimile for a naturally occurring information bearing dynamical system.

Investigation of nonlinear dust-acoustic waves (DAWs) in a four-component unmagnetized electron–ion dusty plasma composed of fluid dust species, the Maxwellian positive, and negative ions with $q$ -nonextensive electrons is performed. The dust charge variation is considered for both adiabatic and nonadiabatic cases. For the nonadiabatic case, the Korteweg-de Vries–Burgers (KdV-B) equation is obtained utilizing the reductive perturbation technique (RPT). Implementing the traveling wave transfiguration, the KdV-B equation is transformed into a planar dynamical system (PDS). Implementing the phase plane theory of PDS, qualitative phase portrait profiles of a stable spiral for the corresponding system are presented. The effects of relevant physical parameters are shown on the DAW behavior. Dynamical features, such as quasiperiodic and chaotic motions of the system, are presented via a numerical investigation in the presence of an extraneous periodic force. Sensitivity analysis and the box-counting dimensions for chaotic profiles are presented.

The object of this study is an autonomous van der Pol system on a real plane. The subject of the study is the properties of the limit cycle of this system. The main purpose of this paper is to find the localization of the limit cycle on the phase plane and establish its shape for various values of the real parameter of the van der Pol system. Our approach is based on the use of transverse curves related to the Dulac – Cherkas functions and approximating the location of the limit cycle. As the first step, five topologically equivalent systems, including systems with a parameter rotating the vector field, as well as singularly perturbed systems are determined for the van der Pol system. Then, applying the previously elaborated method, we constructed two polynomial Dulac – Cherkas functions for each of three systems from the considered ones in the phase plane for all real nonzero values of the parameter. Using them, transverse curves forming the boundaries of the localization regions of the limit cycle for the van der Pol system are found. Thus, the constructed Dulac – Cherkas functions allow us to determine the location of the limit cycle on the basis of algebraic curves for all real parameter values, including values close to the bifurcation of a limit cycle from the center ovals, the Andronov – Hopf bifurcation, and the bifurcation from a closed trajectory related to a discontinuous periodic solution.

The traditional method of matched field processing is based on comparing the complex amplitudes of the measured and calculated sound fields in an underwater waveguide. Because of the high sensitivity of the wave field to variations in environmental parameters, the use of this approach requires accurate knowledge of the ocean-acoustic environment. In this paper, it is shown that under conditions of uncertain environment, instead of comparing the depth dependencies of complex field amplitudes, it is advisable to compare their field amplitude distributions in the phase plane “grazing angle–depth.” Such distributions are calculated using the coherent state expansion borrowed from quantum mechanics. Due to the absence of multipath in the phase space, the amplitudes of the coherent states are less sensitive to variations in the environmental parameters than the total wave field. This makes it possible to construct the similarity coefficients of measured and calculated fields that almost “do not notice” the differences of the compared fields caused by weak sound-speed variations, and “react” only to differences caused by strong changes in the sound-speed field and/or source position.

We describe the application of the phase plane method to analyze the behavior of stationary harmonic plane electromagnetic waves. The phase plane method allows one to describe qualitatively all possible solutions of the Helmholtz nonlinear differential equation without performing a numerical integration. All types of phase portraits for waves in the media with self-action are presented and their physical interpretation is discussed. The consideration is carried out on the model of cubic nonlinearity without specifying the physical mechanism of this nonlinearity. The features of the nonlinear case are discussed. The phenomenon of the self-limitation of the transmitted wave power in a medium with negative nonlinearity is described.

In this paper, we investigated the nonlinear oscillations, stability and resonance of a cantilever beam system which carrying an intermediate lumped mass. The response of amplitudes and phases of the differential equation governing is derived applying the multiple scale perturbation (MSP) method. The nonlinear vibration control is analyzed using a new nonlinear modified positive position feedback (MPPF) approach. The nonlinear approximate solutions for this system up to and inclusive the second order approximation are solved. The influence of various parameters on the system response is studied. The numerical solutions and amplitude response of this system are also studied and investigated. The stability and bifurcation of the cantilever beam system is determined applying the frequency response curves and phase plane technique. Comparison between numerical and analytical solutions is illustrated. Results demonstrated that controller structure has the most salient role in the suppression performance. Utilizing the nonlinear MPPF approach, the vibration amplitude is suppressed at the exact resonant frequency sufficiently and the subsequent peaks of the frequency range is reduced significantly compared to the positive position feedback (PPF) approach.

In this paper, asymmetric equations for an elliptical boundary layer in the vicinity of the conditional front of Rayleigh surface waves, which occurs in shells of revolution under shock end impacts of normal type are constructed. The technique of asymptotic derivation of these equations, based on the use of the symbolic Lurie method and the introduction of special coordinates that distinguish a small frontal zone required to reduce the original problem to an equivalent problem for an infinite shell by isolating a particular solution. The considered boundary layer complements the full description of the considered type of stress-strain state (SSS) in all sections of the phase plane. It also uses a quasi-static boundary layer of the Saint-Venant type in a small vicinity of the butt end, a parabolic boundary layer according to the two-dimensional Kirchhoff–Love theory, a quasi-plane short-wave component, and a hyperbolic boundary layer in a small neighborhood of the shear wave front. In conclusion, an example of constructing an elliptical boundary layer under shock action on the butt end of a cylindrical shell is considered.

In this paper, the flow switching theory of discontinuous dynamical systems is used as the main tool to study the dynamical behaviors of a two-degree-of-freedom friction oscillator with elastic impacts, where considering that the inequality of static and kinetic friction forces results in the existence of flow barriers at the relative velocity between the mass and conveyor belt tending to zero. The negative feedback is also taken into account, $$\mathrm {i.e.}$$ , when the direction of the relative velocity between the mass and conveyor belt changes, the constant force in the periodic excitation force will also change to adapt to the variation in speed. Due to the discontinuity resulted from the friction and non-smoothness resulted from impact, the motion of the two masses can be divided into the following kinds: non-sliding motion (or free motion), sliding motion, non-sliding–stick motion and sliding–stick motion, and the phase space in system is divided into different domains and boundaries, where the vector field is continuous in each region but discontinuous on the velocity boundary. The corresponding normal vectors and G-functions on the separation boundaries are introduced to acquire the switching conditions for all possible motions such as passable motion, sliding motion, grazing motion and stick motion. The mapping theory provides an effective method for the discussion of periodic motion in discontinuous dynamical systems. The four-dimensional switching sets are defined by the means of the form of direct product of two-dimensional switching sets, and the mapping structures are introduced based on the switching sets to describe periodic motions. Eventually, in order to explain more intuitively the above motions in such a friction–impact system, the velocity, displacement, G-functions time histories and the trajectories in phase plane for the masses are illustrated by simulation numerically.

Generally, neural networks involve the change with time of the neuron activities and that in strength of the synapses between neurons. This paper investigates the global multistability and analog circuit implementation of a two-dimensional adapting synapse-based neuron model in depth. The neuron model is non-autonomous and possesses periodically switchable equilibrium states associated with the externally imposed input closely. In every full periodic cycle of the input, the equilibrium stability has complex dynamical transitions between stable and unstable points via Hopf/fold bifurcations, resulting in the emergence of the global multistability that was not yet reported previously. Complex dynamics of the global coexisting multiple firing activities are demonstrated by multiple numerical measures, such as bifurcation plot, dynamical map, phase plane plot, and basin of attraction. Furthermore, an off-the-shelf discrete component-based circuit design is optimized to implement the neuron model and the outputs agree with the numerical results well.

New methods for retrieving localized diagnostic information from cyclic signals of complex shape are proposed. The advantages of an alternative method of estimating the shape of an averaged cycle based on transition from a scalar signal to its mapping on the phase plane are shown. Original methods are proposed for estimating the dynamics of parameters characterizing the shape of informative fragments of the signal, based on construction of the convex hull of the phase portrait of permutation entropy and the Levenshtein distance.

The results of computational modeling of relay control of a vibrating screen drive system in the phase plane are presented.

To better understand the dynamic behaviors of cable-stayed bridges, this study investigates the dynamic behaviors of a cable-stayed shallow arch subjected to two external harmonic excitations using the analytical approach. First, dimensionless planar vibration equations of the system are obtained by applying the Hamilton principle, and three ordinary differential equations of the arch and the two cables are obtained by using the Galerkin discretization method. Second, modulation equations involving the amplitude and phase of the dynamic response of the system are derived by applying the method of multiple scales. Third, three simultaneous resonance cases are considered. Finally, parametric study results are illustrated through frequency responses, amplitude responses, phase plane and bifurcation diagrams. Chaos phenomenon is also detected and presented. To validate the developed analytical solutions, numerical simulations are conducted by applying the Runge–Kutta method to integrate the original ordinary differential equations. The results demonstrate that acceptable consistency is reached in the results obtained from the analytical solutions and the Runge–Kutta method in the three simulated cases. The obtained results show that the system’s dynamic responses in the three simulated cases exhibit similarities in their frequency and amplitude responses, while some qualitative differences exist in the phase plane portraits (e.g., period-1, period-2, period-3 solutions) and their bifurcation diagrams.

Under consideration is some mathematical model of a clock frequency generator, a device of the MEMS class (microelectromechanical systems). We numerically study the solution of the corresponding second-order ordinary differential equation with nonlinear right-hand side and show that there is a region of the model parameters in which the bounded solutions tend to a stable limit cycle in the phase plane and, therefore, the periodic oscillations are stable with respect to the external perturbations. To determine the boundary of the region, we use the parameter continuation method of the solution of the boundary value problem defining the limit cycle. The model leads to the numerical identification of the region of generator operability.