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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.

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, 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.

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.

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.

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, 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.

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.

Nonlinear oscillations of graphene resonators are unavoidable due to enhancing the mass sensitivity of graphene-based mass sensors and the nonlinear behavior of the systems provides the route to chaos. In this paper, the nonlinear and chaotic behavior of the graphene-based mass sensor is investigated. The nano-mechanical sensor includes an electrostatically actuated fully clamped single-graphene sheet as a nano-resonator with an attached concentrated mass. By neglecting the rotary inertia, the equation of motion of the nano- resonator and the attached mass is derived using the nonlocal strain gradient theory of elasticity. The nano-resonator is modeled as a Kirchhoff nano-plate with the von Karman-type geometric nonlinearity. Applying the Galerkin decomposition method to the partial differential equation of motion leads to the ordinary differential equation. Based on the Melnikov's integral method two analytical criteria are derived which provide necessary conditions that determine the chaotic region of the system. The chaotic dynamics of the system are also scrutinized and verified through plotting the Lyapunov exponent diagram, phase plane trajectories and Poincare maps.

In this paper, a three-dimensional novel chaotic system and its projective synchronisation are investigated. The proposed chaotic system has no equilibria. The topological structure of proposed chaotic system is different form Lorenz, Rossler and Chen systems. Different qualitative and quantitative tools such as time series, phase plane, Poincare section, bifurcation plot, Lyapunov exponents, Lyapunov spectrum, and Lyapunov dimension are used to evidence the chaotic behaviour of the proposed system. Further, the projective synchronisation between the proposed chaotic systems is achieved using nonlinear active control. Active control laws are designed, by using sum of the relevant variables of the proposed chaotic systems, to ensure the convergence of error dynamics. The required global asymptotic stability condition is derived using Lyapunov stability theory. Simulation is done in MATLAB environment to verify the theoretical approach. Simulation results reveal that the objectives of the paper are achieved successfully.

For the nonlinear Schrodinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity, by using the method of dynamical systems, we investigate the bifurcations and exact traveling wave solutions. Because obtained traveling wave system is an integrable singular traveling wave system having a singular straight line and the origin in the phase plane is a high-order equilibrium point. We need to use the theory of singular systems to analyze the dynamics and bifurcation behavior of solutions of system. For \begin{document}$ m>1 $\end{document} and \begin{document}$ 0 , corresponding to the level curves given by \begin{document}$ H(\psi, y) = 0 $\end{document} , the exact explicit bounded traveling wave solutions can be given. For \begin{document}$ m = 1 $\end{document} , corresponding all bounded phase orbits and depending on the changes of system's parameters, all exact traveling wave solutions of the equation can be obtain.

The hair bundles of inner hair cells in the auditory nervous exhibit spontaneous oscillations, which is the prerequisite for an important auditory function to enhance the sensitivity of inner ear to weak sounds, otoacoustic emission. In the present paper, the dynamics of spontaneous oscillations and relationships to steady state are acquired in a two-dimensional model with fast variable X (displacement of hair bundles) and slow variable $$X_a$$ . The spontaneous oscillations are derived from negative stiffness modulated by two biological factors (S and D) and are identified to appear in multiple two-dimensional parameter planes. In (S, D) plane, comprehensive bifurcations including 4 types of codimension-2 bifurcation and 5 types of codimension-1 bifurcation related to the spontaneous oscillations are acquired. The spontaneous oscillations are surrounded by supercritical and subcritical Hopf bifurcation curves, and outside of the curves are two cases of steady state. Case-1 and Case-2 steady states exhibit Z-shaped (coexistence of X) and N-shaped (coexistence of $$X_a$$ ) X-nullclines, respectively. In (S, D) plane, left and right to the spontaneous oscillations are two subcases of Case-1, which exhibit the stable equilibrium point locating on the upper and lower branches of X-nullcline, respectively, resembling that of the neuron. Lower to the spontaneous oscillations are 3 subcases of Case-2 from left to right, which manifest stable equilibrium point locating on left, middle, and right branches of X-nullcline, respectively, differing from that of the neuron. The phase plane for steady state is divided into four parts by nullclines, which manifest different vector fields. The phase trajectory of transient behavior beginning from a phase point in the four regions to the stable equilibrium point exhibits different dynamics determined by the vector fields, which is the basis to identify dynamical mechanism of complex forced oscillations induced by external signal. The results present comprehensive viewpoint and deep understanding for dynamics of the spontaneous oscillations and steady states of hair bundles, which can be used to well explain the experimental observations and to modulate functions of spontaneous oscillations.

The objective of this paper is to investigate the convergence of coupling-parameter expansion-based solutions to the Ornstein–Zernike equation in liquid state theory. The analytically solved Baxter’s adhesive hard sphere model is analyzed first by using coupling-parameter expansion. It was found that the expansion provides accurate approximations to solutions—including the liquid-vapor phase diagram—in most parts of the phase plane. However, it fails to converge in the region where the model has only complex solutions. Similar analysis and results are obtained using analytical solutions within the mean spherical approximation for the hardcore Yukawa potential. However, numerical results indicate that the expansion converges in all regions in this model. Next, the convergence of the expansion is analyzed for the Lennard-Jones potential by using an accurate density-dependent bridge function in the closure relation. Numerical results are presented which show convergence of correlation functions, compressibility versus density profiles, etc., in the single as well as two-phase regions. Computed liquid-vapor phase diagrams, using two independent schemes employing the converged profiles, compare excellently with simulation data. The results obtained for the generalized Lennard-Jones potential, with varying repulsive exponent, also compare well with the simulation data. Solution-spaces and the bifurcation of the solutions of the Ornstein–Zernike equation that are relevant to coupling-parameter expansion are also briefly discussed. All of these results taken together establish the coupling-parameter expansion as a practical tool for studying single component fluid phases modeled via general pair-potentials.

In this paper, a flux controlled memristor-based novel complex-valued chaotic system and its projective synchronisation is investigated. The proposed complex-valued chaotic system has line and plane of equilibria, i.e. an infinite number of equilibria. Different qualitative and quantitative tools such as time series, phase plane, Poincare section, Lyapunov exponents, Lyapunov spectrum, and Lyapunov dimension are used to evidence the chaotic behaviour of the proposed complex-valued system. Further, the projective synchronisation between the proposed complex-valued chaotic systems is achieved using nonlinear active control. Active control laws are designed, by using sum of the relevant variables of the proposed complex-valued chaotic systems, to ensure the convergence of error dynamics. The required global asymptotic stability condition is derived using Lyapunov stability theory. Simulation is done in MATLAB environment to verify the theoretical approach. Simulation results reveal that the objectives of the paper are achieved successfully.

The patterns of discharge activity are closely associated with the physiological manifestation of neurons, and hence, it is particularly significant to understand the dynamic behavior of the patterns. In this paper, a Henon-like map (generalized map) system is reported, which is derived from the discrete-time FitzHugh–Nagumo model that is a simplification of the classical Hodgkin–Huxley neuron model. Our main contributions are as follows: (i) proposing the Henon-like map; (ii) discovering the fractal structure and the chaos of the map; and (iii) presenting the horseshoe structure of the subsequence of the map. Specifically, it is observed that bifurcation structure in the process of period-doubling cascade to chaos is compatible with the largest Lyapunov exponent spectrum. Moreover, some basic properties of the generalized map are also analyzed with repeated numerical simulation, such as phase plane, bifurcation diagram, power spectrum and chaotic attractor. In addition, it is confirmed that the hidden horseshoe structure also exists in the generalized map. The potential connection between decomposition of chaotic behavior and period-3 is logically prospected. The behavior of period-doubling cascade to chaos and Smale horseshoe coexists in the map. Furthermore, the relationship between the internal mechanisms of the map and external performance may be discovered. Finally, the effects of forced external factors—delayed feedback, noise and electromagnetic field on the system—are evaluated. It may be the first time to study discrete neuron models by using Simulink simulation in the field of neurodynamics. In the above process, a circuit simulation is built in the Simulink package, which opens up a new path for the detection of discrete neuron network.

We study the dynamics of a delayed diffusive hematopoiesis model with two types of Dirichlet boundary conditions. For the model with a zero Dirichlet boundary condition, we establish global stability of the trivial equilibrium under certain conditions, and use the phase plane method to prove the existence and uniqueness of a positive spatially heterogeneous steady state. We further obtain delay-independent as well as delay-dependent conditions for the local stability of this steady state. For the model with a non-zero Dirichlet boundary condition, we show that the only positive steady state is a constant solution. Results for the local stability of the constant solution are also provided. By using the delay as a bifurcation parameter, we show that the model has infinite number of Hopf bifurcation values and the global Hopf branches bifurcated from these values are unbounded, which indicates the global existence of periodic solutions.

A complex logistic-type map is considered in the present work. The dynamic behavior of the underlined map is discussed in two different cases: the first case is when the parameter of the map being real, and the second case is when the parameter being complex. Existence and stability of fixed points are derived. The conditions for existence of a pitchfork bifurcation, flip bifurcation and Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Numerical simulations including Lyapunov exponent, phase plane, bifurcation diagrams is carried out using matlab to ensure theoretical results and to reveal more complex dynamics of the map. The results show that expressing the logistic map in terms of complex variables leads to more distinguished behaviors, which could not be achieved in the logistic map with real variables. In addition, considering the control parameter as a complex parameter shows more interesting dynamics if compared to the case when considering it as a real parameter. Existence of a snapback repeller is proved in the sense of Marotto. Finally, chaos is controlled using the OGY feedback method.

This paper puts forward the microscopic chaos control in the deterministic dynamics of the chemical reactor system. First, the dynamic behavior of the chemical reactor system is explored for some of the parameters and chaotic behavior is investigated. Phase plane, bifurcation plots and Lyapunov exponents are presented to verify the chaotic behavior. Second, nonlinear active plus proportional integral sliding mode control (NA-PISMC) is proposed to control microscopic chaos in the chemical reactor system. A proportional integral switching surface is proposed to achieve the stability condition of the error dynamics and controller is designed by using the relevant variables of the chemical system. Unlike the open loop and open plus closed loop control techniques, the design of proposed controller does not require the parameter perturbation. The required stability condition is derived based on Lyapunov stability theory. Simulation is done in MATLAB environment. Numerical simulation results are presented in order to show the effective performances of the proposed controller design. Simulation results correspond that the objectives of chaos existence and chaos control are achieved successfully.

Introduction: The models of vector random processes are characterized by a variety of practical applications and, at the same time, by the complexity of studying the detailed probabilistic structure. Such features often lead to considering only separate components of vector functions, forcing the shift to scalar process analysis. All this significantly reduces the overall information content of the research, increasing the relevance of the search for promising approaches to probabilistic analysis of vector processes. Purpose: To present vector random processes in a phase space of states, and to apply the general theory of level crossings of random functions to the study of probabilistic structure of phase trajectories. Results: We have identified the characteristic features of displaying random processes on a phase plane, and introduced numerical characteristics for the description and analysis of probabilistic behavior of sample functions. A probabilistic analysis has been performed for the characteristics of the «level crossing» type for vector processes with various definitions of the areas of acceptable values. We used the typical models of two-dimensional Gaussian processes and «signal plus noise» models to demonstrate how the probabilistic structure of the phase trajectories depends on the given threshold levels, major distribution parameters and spectral-correlation properties of the studied processes. Practical relevance : The article contributes to the research information content and its visualization in the analysis of probabilistic behavior of vector random processes. It combines general methods of phase space of vector processes, methods of phase plane in the analysis of scalar processes and the theory of level crossings of random functions. The classical approaches to the visual description of phase trajectories are supplemented by the methods of quantitative analysis of detailed probabilistic structure of random functions.

In this study a polydyne cam and a roller follower with different cam rotational speeds are analyzed. The chaos has been investigated based on positive Lyapunov exponent value. The effect of follower guide’s clearance on chaos is considered. The contact between cam and follower is simulated by using Solidworks program. The power spectrum of Fast Fourier Transform and phase plane have been examined the follower non-periodicity during follower motion in the y-direction. Rosenstein program is used to calculate largest Lyapunov exponent. The experimental setup has been implemented by using OPTOTRAK/3020 through a 3-D infrared markers to track follower motion. The signal of follower motion in the y-direction has been processed depend upon data acquisition technique. The system with follower guide’s clearance C = 2 mm is more chaotic than the others at cam rotational speed N = 800 rpm and N = 1200 rpm. The simulation and experimental results are compared and verified for largest Lyapunov exponent.

The aim of this work is to evaluate the dynamics of the quasi-zero stiffness vibration isolator, with particular emphasis on the density distribution of Poincare cross-section points for different conditions of the system's excitation. The research was mainly focused on model tests carried out to examine the structures representing the geometry of strange attractors of a quasi-zero stiffness vibration isolator. The analysed mechanical system consists of one main spring and two compensation springs. Energy losses are caused by friction at the connection points of the compensation springs and vibroisolated mass. On the basis of a formulated non-linear mathematical model, the ranges of variation of physical parameters of an external dynamic input for which the system motion is chaotic are identified. Taking into account three selected values of the dynamic input amplitude, a bifurcation diagram and a diagram defining the number of phase stream intersections (NPSI) with the abscissa of the phase plane are generated. Based on the diagrams, a solution is selected. The diagram proposed by us (NPSI) is an alternative method of identifying areas in which the system moves chaotically. The classic Poincare cross-section combined by us with information about the density of point distribution on the trajectory intersection with the control plane serve as a basis for the assessment of evolution of the geometric structure of strange attractors. The evaluation of the density distribution of Poincare cross-section points provides important information regarding the evolution of geometrical structures of strange attractors. It has been shown that in relation to large ranges of changes in the control parameter, the geometric structure of the strange attractor is stretched and curved. However, in the area of small changes in the control parameter, the evolution of the attractor’ geometric structure can only be observed by analysing the density of point distribution on the Poincare cross-section by the probability density function (PDF). The areas with the highest density of the Poincare cross-section are usually located in places where the strange attractor is curved. Taking into account the practical application of research results, operational guidelines are formulated.

Chatter, the vibration between the workpiece and cutting tool, is a common phenomenon during the milling process. Traditionally, the Fourier transform analysis is mainly used to extract the features and determine whether chatter occurs. In this paper, an innovative and practical chatter identification method combining fractional order chaotic system and extension theory is proposed. A lathe spindle with embedded sensors is used in this study. The boundary of chattering state of the lathe spindle is decided by the center of gravity of phase plane of dynamic errors. The boundaries are then fed into extension model and relational function calculation is performed. In this way, chatter identification can be easily achieved based on the position of the chaotic center of gravity. The three chaotic systems, i.e. Lorenz, Chen-Lee, and Sprott, of different fractional orders are used and their results are compared. Compared with the traditional methods, the Fourier transform is time-consuming in terms of mathematical operations and adverse to the establishment of a real-time system. This paper uses the characteristics of the chaotic systems sensitive to input signals in order to more capably detect the boundary state from normal cutting to chattering cutting and more efficiently identify the chatter. The experiment results indicate that the Chen-Lee system (93.5%) exhibits have better chatter diagnosis rate than Lorenz (92.75%) and Sprott (69%) systems. The Chen-Lee system even reaches a diagnosis rate of 100% for orders 0.5 ~ 0.7. Therefore, the method presented in this paper has a very high diagnosis rate and is thus very effective for chatter identification of machine tools.

The oscillation, rotatability, and wandering characteristic indicators of Lyapunov type are studied for two-dimensional linear homogeneous differential systems determining rotations of the phase plane. A complete set of order relations between them is obtained. For each of those indicators it is established whether it is continuous or discontinuous as a function of the coefficients of the system.

We study the generalized Riemann problem of the Chapman-Jouguet model for an ideal combustible Chaplygin gas. By analyzing the wave curves in the phase plane, we construct uniquely the solution of the generalized Riemann problem under the global entropy conditions. We find that although there is no combustion wave of the corresponding Riemann solution, the combustion wave may occur after perturbation which reveals the instability of the unburnt gas.

This paper puts forward the analysis, design, implementation a new chaotic system and observer design of new chaotic systems. Different tools like phase plane, time series, Lyapunov spectrum, Poincare map, Lyapunov exponents and Lyapunov dimension are exercised to analyses different behaviors of the proposed system. Proposed chaotic system is considered for the observer-based synchronization and linear matrix inequality (LMI) approach is utilized to obtain non-convex inequality as well as to solve the optimal value of observer gain. Simulation is achieved in MATLAB environment to demonstrate the effectiveness and Multisim circuit design platform to validate the numerical approach and simulation results. Finally, achieved circuit implementation results disclose that aims are accomplished satisfactorily.

This paper puts forward a 4D autonomous chaotic system, synchronisation and anti-synchronisation using nonlinear active control technique. The 4D chaotic system is analyzed by using available qualitative and quantitative tools like phase plane analysis, time series plot, root locus, Lyapunov exponent and spectrum analyses. The proposed 4D chaotic system has seventeen equilibria and coexisting attractors of two and single wings and shows symmetric phase plane behaviour with respect to origin and coordinate plane. The proposed 4D autonomous dynamic system reflects sensitive dependence on initial conditions and exploring different types of synchronisation between 4D chaotic systems is a challenging task. Synchronisation and anti-synchronisation between proposed 4D chaotic systems are accomplished using nonlinear active control technique. Control laws are designed by employing sum of pertinent variables of 4D drive and response chaotic systems. Lyapunov direct stability theorem is used and global asymptotic stability condition is derived to confirm the stability of error dynamics. Simulation is finished in MATLAB environment and reveals the accomplishment of objectives.

This paper posit a novel three dimensional autonomous chaotic system. A six dimensional system is derived from the proposed three dimensional autonomous chaotic system by considering its real state variables as complex state variables. The derived system has nine equilibria and shows hyperchaotic 3-torus behaviour. Basic dynamical properties like types of equilibrium point, symmetry, invariance, dissipation, Lyapunov exponent, Lyapunov dimension are calculated and analysed for both the real and complex valued systems. Different qualitative and quantitative tools such as time series, phase plane, Lyapunov exponent spectrum and Lyapunov dimension are used to analyse different dynamic behaviours of both real and complex valued system. MATLAB simulation environment is used for the effective verification of the numerical or theoretical approach. Simulation results reveal that objectives are achieved successfully.

It is assumed as Euler-Bernoulli beam for flexible arm in large-scale rotation, and the nonlinear dynamics mode of beam is built through applying Galerkin method and Hamilton principle. The system phase trajectories is obtained before and after instability by phase plane method. Poincare cross sections of beams at different perturbations are obtained by numerical simulation. Then the disturbance threshold and parameter conditions for chaos behavior are defined. The research has shown that when the disturbance which is caused by the coupling of longitudinal and transverse vibration exceeds the critical threshold at under damping, the system will become unstable and generate sudden self-excited oscillation. Its amplitude reaches 1.73mm. We adjust the damping to restrain the fast oscillation. The system produces chaotic motion in over damping state. Then the disturbance increases to 20 times of the critical threshold, the frequency of aperiodic vibration increased obviously, and the amplitude of high frequency is distributed at −0.32mm~0.32mm. It provides important theoretical basis and data support for the design of nonlinear chaos suppression controller on the basis of above data.

The purpose of this paper is to highlight certain features of a dynamic optimisation problem in an economic growth model with environmental negative externalities that gives rise to a two-dimensional dynamical system. In particular, it is demonstrated that the dynamics of the model, which is based on a production function with perfect substitutability (perfect substitution technologies), admits a locally attracting equilibrium with a basin of attraction that may be considerably large, as it can extend up to the boundary of the system phase plane. Moreover, this model exhibits global indeterminacy because either equilibrium of the system can be selected according to agent expectation. Formulas for the calculation of the bifurcation coefficients of the system are derived, and a result on the existence of limit cycles is obtained. A numerical example is given to illustrate the results.

A coupled torsional-transition nonlinear dynamic model of a two-stage helical gear (TSHG) reduction system for electric vehicles (EVs) is presented in this paper. The model consists of 16 degrees of freedom (DOF), which includes factors such as the nonlinearity of backlash, time-varying mesh stiffness (TVMS), mesh damping, supporting bearings, static transmission error (STE), and the torsional damping and stiffness of the intermediate shaft, in which the fourth-order Runge–Kutta numerical integration method was applied to solve the differential equations. With the help of bifurcation diagrams, time-domain histories diagrams, amplitude-frequency spectrums, phase plane diagrams, Poincaré maps, root-mean-square (RMS) curves, peak-peak values (PPVs), and Lyapunov exponents, the effects of pinion rotational speed, backlash, torsional stiffness, and torque fluctuation on the dynamic behavior of TSHG system are investigated. The stability properties of steady-state responses are investigated using Lyapunov exponents. The results reveal various types of dynamic evolution mechanisms and nonlinear phenomena such as periodic-one responses, quasiperiodic responses, jumps phenomena, and chaotic responses. The research presents useful results and information to vibration control and dynamic design of the TSHG transmission system used in EVs.

The exact solution of the Riemann problem associated to Ripa model is constructed. A primitive variable formulation of the model is used instead of conserved variable formulation. This lead to establish the solutions to Riemann problem without calculating the flux terms. The closed form solution is obtained in the case of all rarefaction wave. The phase plane curves and corresponding characteristics curves have been plotted for different cases including rarefaction waves and shock waves connecting left and right known states. Many numerical methods for solving nonlinear hyperbolic conservation laws are based on conserved variable formulation, hence the conversion from primitive to conserved variables is required within algorithm. This method gives direct calculation of the exact solution in primitive variable formulation of the model. This method can be effectively applied to design more robust numerical solvers for hyperbolic conservation laws.

Purpose The purpose of this paper is to construct a proportion-integral-type (PI-type) memristor, which is different from that of the previous memristor emulator, but the constructing memristive chaotic circuit possesses line equilibrium, leading to the emergence of the initial conditions-related dynamical behaviors. Design/methodology/approach This paper presents a PI-type memristor emulator-based canonical Chua’s chaotic circuit. With the established mathematical model, the stability region for the line equilibrium is derived, which mainly consists of stable and unstable regions, leading to the emergence of bi-stability because of the appearance of a memristor. Initial conditions-related dynamical behaviors are investigated by some numerically simulated methods, such as phase plane orbit, bifurcation diagram, Lyapunov exponent spectrum, basin of the attraction and 0-1 test. Additionally, PSIM circuit simulations are executed and the seized results validate complex dynamical behaviors in the proposed memristive circuit. Findings The system exhibits the bi-stability phenomenon and demonstrates complex initial conditions-related bifurcation behaviors with the variation of system parameters, which leads to the occurrence of the hyperchaos, chaos, quasi-periodic and period behaviors in the proposed circuit. Originality/value These memristor emulators are simple and easy to physically fabricate, which have been increasingly used for experimentally demonstrating some interesting and striking dynamical behaviors in the memristor-based circuits and systems.

This paper studies generation of robust periodic solutions in a class of nonlinear discrete-time system. The sustained oscillations, with the desired frequency and amplitude, are achieved through the creation of the appropriate elliptic limit cycle in the phase plane of the uncertain closed-loop discrete-time system. In the first step, the nominal control law is designed to enforce the trajectories of the nominal closed-loop system to converge to the desired limit cycle. Next, considering uncertain terms, an additional robustifying term is designed. This term is added to the nominal controller to sustain the desirable stable oscillations in the presence of uncertain terms. The resulted robust controller brings the trajectories of the uncertain closed-loop discrete-time system to a boundary layer (with adjustable width) around the desired limit cycle. Moreover, the domain of attraction of the limit cycle and also the ultimate boundary layer around it are calculated via the Lyapunov analysis. Additi...

In this paper, a new two-probe surface heat flux calibration integral equation method (CIEM) is proposed for resolving nonlinear inverse heat conduction problem possessing temperature-dependent thermophysical properties and an unknown time-varying back surface boundary condition. Compared to the one-probe CIEM, the two-probe CIEM can be applied to a more general class of one-dimensional inverse heat conduction problems that can be used for accurately predicting the front surface heat flux without any restrictions on the back surface boundary condition. Temperature-dependent thermophysical properties and probe locations are not specified a priori but are implicitly accounted through a calibration campaign. The final mathematical formulation involves resolving a Volterra integral equation of the first kind for the unknown surface heat flux. A first kind Chebyshev expansion possessing undetermined coefficients, is used for approximating the introduced property transform function. Next, three well-defined calibration tests are used for estimating the undetermined coefficients associated with the Chebyshev expansion. A time sequential study of the coefficients is shown to yield the optimal truncation of the series expansion. A future-time method is applied for stabilizing the ill-posed first kind Volterra integral equation. Phase plane and cross-correlation analyses are employed for estimating the optimal regularization parameter (i.e., the future-time parameter) from a spectrum of chosen values. The effectiveness of the proposed approach is verified by numerical simulation using stainless steel as the host material. The calibration-based or “parameter free” inverse approaches significantly reduce time and costs in the resolution of the surface condition as there is no need to experimentally determine the thermophysical properties, probe locations, and probe characteristics. These system properties are implicitly included in the proposed calibration procedure. Many applications, especially in aerospace sciences where exotic materials are often used, do not have replicable properties among samples. Conventional inverse methods rely on accurately specifying the input parameters for forming the predictions. These “parameter required” methods are solely based on numerical algorithms and thus their predictions are strongly relied on accurate input of parameters and measured probe data.

This paper presents a novel third-order autonomous memristive diode bridge-based oscillator with fast-slow effect. Based on the modeling of the presented memristive oscillator, stability of the equilibrium point is analyzed by using the eigenvalues of the characteristic polynomial, and then symmetric periodic bursting behavior is revealed through bifurcation diagrams, phase plane plots, time sequences, and 0–1 test. Furthermore, bifurcation mechanism of the symmetric periodic bursting behavior is explored by constructing the fold and Hopf bifurcation sets of the fast-scale subsystem with the variations of the system parameter and slow-scale variable. Consequently, the presented memristive oscillator is always unstable and exhibits complex dynamical behavior of symmetric periodic bursting oscillations with a symmetric fold/Hopf cycle-cycle burster. In addition, experimental measurements are performed by hardware circuit to confirm the numerical simulations.

We concern with the Riemann problem the compressible fluid flow with the generalized Chaplygin gas. With the analysis on the phase plane, we rigorously confirm the occurrence of delta shock wave with Dirac delta function in density. Then the formation mechanism, generalized Rankine–Hugoniot relation and entropy condition for the delta shock wave are clarified. Based on these preparations, five kinds of exact solutions are obtained. Finally, the corresponding numerical results are also presented to illustrate our analysis.

Image registration is a critical process for the various applications in the remote sensing community, and its accuracy greatly affects the results of the subsequent applications. Image registration based on phase correlation has been widely concerned due to its robustness to gray differences and efficiency. After calculating the normalized cross-relation matrix $Q$ , the most commonly used approach is fitting the 2-D phase plane that passes through the origin, but it needs to remove contaminated spectrum carefully and the corresponding parameters are empirical. In fact, the phase correlation matrix is rank one for a noise-free translation model. This property simplifies the matching problem to finding the best rank-one approximation of the normalized cross-relation matrix. We develop a novel algorithm that performs the rank-one matrix factorization on the phase correlation matrix by assuming its noise as mixture of Gaussian (MoG) distributions. The MoG model is a general approximator for any continuous distribution, and hence is able to model a wide range of noise distribution. The parameters of the MoG model can be evaluated under the framework of maximum likelihood estimation by using an expectation-maximization method, and the subspace is calculated with standard methods. The advantages of the algorithm, high accuracy, and robustness to aliasing, noise, gray difference, and occlusions are illustrated by a series of simulated and real-image experiments.

This paper presents a novel third-order autonomous memristive chaotic oscillator, which is accomplished by parallelly coupling a simple memristive diode bridge emulator into a Sallen–Key low-pass filter (LPF). With the modeling of this oscillator, stability analyses of the equilibrium point and numerical simulations of the phase plane orbit, time-domain sequence, bifurcation diagram, and finite-time Lyapunov exponent spectrum are performed, from which period, quasi-period, chaos, and quasi-period to chaos route are found. Particularly, two types of dynamical phenomena of quasi-periodic behavior and point-cycle chaotic bursting that are further identified by using 0–1 test are observed in such a third-order autonomous memristive oscillator, which have been rarely reported in the previous literatures. Additionally, hardware experiments are implemented and the quasi-periodic behavior and point-cycle chaotic bursting are well confirmed.

A theoretical analysis of quasi-one-dimensional, steady, inviscid, compressible channel flow at a low magnetic Reynolds number with variable cross-section is performed in this paper. A second-order nonlinear dynamical system describing the variation of physical parameters is investigated in the phase plane. The characteristics of all possible channel flow with a constant electromagnetic field are obtained by the phase plane and the isomorphism of labeled graphs. It is revealed that the magnetic interaction number novelly derived from the variation rate of the cross-sectional area could significantly affect the phase trajectory in the phase plane of dimensionless velocity and Mach number. Meanwhile, the phase trajectory is only dependent on this magnetic interaction number. Further analysis of the second-order dynamical system discovers five critical values in the range of the magnetic interaction number. These five critical values can divide the whole range into eight subsets under the isomorphism of labeled graphs, forming eight equivalence classes of the phase plane. The study of such eight equivalence classes reveals that the variable cross-section flow in the phase plane can be viewed as a linear superposition of constant cross-section magnetohydrodynamic flow and isentropic flow, the weight ratio being exactly the magnetic interaction number. Consequently, for the divergent channel, the acceleration region in the supersonic zone is larger than that of the constant cross-section flow, while for the convergent channel, only the acceleration region in the subsonic zone is larger than that of the constant cross-section flow.

In this report we study the mathematical model for mutually penetrating continua. This model consists of the wave equation describing the carrying medium and the equation for oscillators forming the oscillating inclusion. Prescribing the constitutive equation of the carrying medium and kinetics of oscillator’s dynamics for model in question, the cubic nonlinearity is accounted. We are interested in the structure of wave solutions obeying the dynamical system of Hamiltonian type. This allows us to determine the peculiarities of the phase space of dynamical system, namely, the relation describing the homoclinic trajectories, the division of phase plane into the parts with equivalent orbits’ behavior, the conditions of bifurcations. To simulate the wave dynamics, we construct the three level finite-difference numerical scheme and study the evolution of solitary waves, their pair interactions and stability. Propagation of periodic waves is modeled as well.

The chaotic motion of a suspension system with fractional order differential under random excitation is studied. The critical condition of chaos in the mean square sense of suspension system is derived by using random Melnikov method. The function relationship between the parameters of suspension system and chaos threshold is established. The boundary curve of chaos is obtained. The influence of fractional differential parameters on chaos boundary curve is studied. The numerical simulation of fractional order suspension system is carried out, and the time domain diagram and frequency of the system are calculated the spectrum, phase plane, Poincare section and the maximum Lyapunov exponent were obtained. The results show that there is chaotic motion in the suspension system with fractional differential under random road excitation, and the coefficient and order of fractional differential term will change the boundary conditions of chaos.

In this work, we have studied stability and Hopf bifurcation analysis for use in a delayed diffusive logistic population equation in spatially heterogeneous environments. The solutions of the 1-D reaction-diffusion equation are considered using the Galerkin technique. Full maps of Hopf bifurcation are determined for the parameters of maturation time, diffusion coefficient and growth rate. In addition, the effects of the free parameters in this model have been examined with the consequence that they can destabilize or stabilize the solution. The Hopf bifurcations for proliferation rate decreased as the maturation time increased while the diffusion coefficient grew. Furthermore, bifurcation diagrams and examples of periodic limited cycle solutions and 2D phase plane maps have been constructed. The comparisons between the numerical simulations with the analytical solutions provided confirmatory evidence and the validation of the technique used, with an excellent agreement compared for all the examples shown. Lindstedt–Poincare in perturbation theory was applied to calculate the asymptotic results around the Hopf bifurcation point for both the one and two-term analytical systems.

The present paper analyzes an ODE model of a certain class of (open) enzymatic reactions. This type of model is used, for instance, to describe the interactions between messenger RNAs and microRNAs. It is shown that solutions defined by positive initial conditions are well defined and bounded on [Formula: see text] and that the positive octant of [Formula: see text] is a positively invariant set. We prove further that in this positive octant there exists a unique equilibrium point, which is asymptotically stable and a global attractor for any initial state with positive components; a controllability property is emphasized. We also investigate the qualitative behavior of the QSSA system in the phase plane [Formula: see text]. For this planar system we obtain similar results regarding global stability by using Lyapunov theory, invariant regions and controllability.

A class of generic spatially extended fractional reaction-diffusion systems that modelled predator-prey interactions is considered. The first order time derivative is replaced with the Caputo fractional derivative of order γ ∈ (0, 1). The local analysis where the equilibrium points and their stability behaviours are determined is based on the adoption of qualitative theory for dynamical systems ordinary differential equations. We derived conditions for Hopf bifurcation analytically. Most significantly, existence conditions for a unique stable limit cycle in the phase plane are determined analytically. Our analytical findings are in agreement with the numerical results presented in one and two dimensions. The system of fractional nonlinear reaction-diffusion equations has demonstrated the usefulness of understanding the dynamics of nonlinear phenomena.

Solutions of a nonlinear hyperbolic system of equations describing weakly nonlinear quasitransverse waves in a weakly anisotropic elastic medium are studied. The influence of small-scale processes of dissipation and dispersion is investigated. The small-scale processes determine the structure of discontinuities (shocks) and a set of discontinuities with a stationary structure. Among the discontinuities with a stationary structure, there are special ones that, in addition to relations following from conservation laws, satisfy additional relations required for the existence of their structure. In the phase plane, the structure of such discontinuities is represented by an integral curve joining two saddles. Special discontinuities lead to nonunique self-similar solutions of the Riemann problem. Asymptotics of non-self-similar problems for equations with dissipation and dispersion are found numerically. These asymptotics correspond to self-similar solutions of the problems.

In this paper, some considerations regarding a ground vehicle oscillating system based on chaotic behaviors are studied. The vehicle system is modeled as a full nonlinear seven-degree freedom with an additional degree of freedom for each passenger. Roughness of the road surface is considered as sinusoidal waveforms with time delays for the tires. The governing di erential equations are extracted under Newton-Euler laws andsolved via numerical methods. The dynamic behavior of the system is investigated by special nonlinear techniques such as bifurcation diagram, time series, phase plane portrait, power spectrum, Poincar e section, and maximum Lyapunov exponents. The time delays between the tires are used as a control parameter. First, the vehicle behavior is investigated and the chaotic regions are detected. Then, the damping and sti ness coe cients are used to return to the regular behavior. Results show that by changing the system parameters andselecting the appropriate values, one can minimize vibrations as well as eliminate chaotic behavior. The comparison of the results obtained from the proposed model and those from the vehicle without passengers show the great di erences in the dynamic behaviors of the two models.

We propose a time-delayed hyperchaotic system with a single-humped nonlinearity that can be implemented in a controlled way using off-the-shelf electronic circuit elements. The proposed system is simple in design yet complex in its dynamical behavior. A rigorous stability analysis reveals that the system gives birth to a limit cycle via a supercritical Hopf bifurcation and also theoretical analysis predicts the occurrence of higher periodic cycles with increasing time delay. The complexity of the system is characterized by phase plane plots, bifurcation diagram and Lyapunov exponent spectrum. The system is implemented in an electronic circuit, and a data acquisition system is used to control the relevant circuit parameter to visualize the experimental bifurcation diagram. Experimental observations qualitatively support the analytical and numerical results. We believe that the present study will improve our understanding of the dynamical behavior of time-delayed systems with single-humped nonlinearity, which are very much relevant in physiological systems.

Recently there has been interest in studying a new class of elastic materials, which is described by implicit constitutive relations. Under some basic assumption for elasticity constants, the system of governing equations of motion for this elastic material is strictly hyperbolic but without the convexity property. In this paper, all wave patterns for the nonclassic nonlinearly elastic materials under Riemann data are established completely by separating the phase plane into twelve disjoint regions and by using a nonnegative dissipation rate assumption and the maximally dissipative kinetics at any stress discontinuity. Depending on the initial data, a variety of wave patterns can arise, and in particular there exist composite waves composed of a rarefaction wave and a shock wave. The solutions for a physically realizable case are presented in detail, which may be used to test whether the material belongs to the class of classical elastic bodies or the one wherein the stretch is expressed as a function of the stress., 28 pages, 7 figures

In this article, we study the three-dimensional rotational motion of a symmetric rigid body (gyro) about a fixed point under the influence of a gyrostatic moment vector about the principal axes of inertia according to Kovalevskaya’s conditions. The six nonlinear differential equations of motion beside their first integrals are reduced to two differential equations from second order composed an autonomous system with one first integral. It is taken into consideration that the gyro spins with high angular velocity about one of the orthogonal principal axes. The small parameter method is used to obtain the analytical solutions of the governing system. The attained solutions are represented graphically through several plots concerning with the different values of the gyrostatic moment vector to evaluate the performance of these values on the gyro’s motion. Euler’s angles are estimated to describe the motion at any instant. The fourth-order Runge–Kutta algorithms through Matlab packages are used to obtain the numerical solutions for the mentioned problem and to represent these solutions graphically. A comparison between the analytical and the numerical solutions shows high consistency between them which indicates the powerful of the presented analytical technique. The phase plane plots are presented in order to give a full description about the regular motion of the gyro. The considered subject has a great interest in many applications such as submarines, satellites and aircrafts.