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Perturbation methods are developed to solve nonlinear differential equations. Therefore, it is essential to review differential equations and methods of phase space in which we study the behavior of differential equations. Applied differential equations, and phase plane are the main topics in this chapter. These topics are essential to study perturbation methods.

The article provides a complete bifurcation analysis of the mathematical model of the dynamic system “Supply and demand” proposed by V.P. Milovanov. The behavior of trajectories at infinity is studied using the Poincare transform. All phase portraits of the system were obtained using theoretical analysis and numerical experiment in the Matlab+Simulink package. The system was rough in the open first quarter of the phase plane. A system of additive control of both monetary and commodity flows is constructed by analytical design of aggregated regulators to achieve a given dynamic equilibrium from an arbitrary initial state. A class of acceptable achievable States is highlighted. The numerical experiment shows the stability of this state in General. This model allows you to predict the development of the process for any pre-set initial state of the system, as well as manage the system parameters for the design of a pre-set dynamic equilibrium.

Switching policy has been considered in many biological systems and can exhibit rich dynamical behaviors which include different types of the bifurcations and deterministic chaos. The Chialvo neuron model analyzed in this article illustrates how bifurcations and multiple attractors can arise from the combination of the switching mechanism acting on membrane potential. The elementary dynamics of the system without the switching policy are analyzed firstly using phase plane methods. The comparisons of the bifurcation analysis with or without switching mechanism near the fixed points are provided. It can be concluded that the switching policy can be prone to give rise to the coexistence of multiple periodic attractors, which indicates there exist abundant firing modes in the switching system with the same system parameters and different initial values. More complex bifurcation and dynamical behaviors can be observed since applying the switching policy.

A spatial two link chain with revolute joints is considered. The equations of motion for the system are developed using Kane’s dynamical equations. Phase plane and return maps of the kinematic chain for different initial condition are developed. The motion of the mechanical system in non-periodic.

Travelling wave solutions to reaction-diffusion systems are considered from the standpoint based on chaos functions. Firstly, the Fisher-KPP equation, which describes a model for the propagation of gene as nonlinear dynamics, is introduced and is transformed into a two-dimensional (2-D) system of nonlinear differential equations. Then, 2-D solvable chaos maps for the 2-D system are derived from chaos functions, and the bifurcation diagrams are numerically calculated to find a system parameter for limit cycles with discrete and chaotic properties. Finally, the chaotic dynamics are discussed by presenting the so-called entrainment and synchronization, and by illustrating the propagation of limit cycles as travelling waves on a phase plane corresponding to the original plane.

The article provides a complete bifurcation analysis of the mathematical model of the dynamic system “Emergence of planned regulation” proposed by V. P. Milovanov. The behavior of trajectories at infinity is studied using the Poincare transform. With the help of theoretical analysis and numerical experiment the phase portrait of the system is obtained in Matlab package. The system turned out to be a lip in the open first quarter of the phase plane. The system of additive control of both cash and commodity flows to achieve a given dynamic equilibrium from an arbitrary initial state is constructed by the method of analytical design of aggregated regulators. Dedicated class a valid reachable States. The numerical experiment shows the stability of this state as a whole. This model allows you to predict the development of the process for any predetermined initial state of the system, as well as to control the parameters of the system to design a predetermined dynamic equilibrium.

This paper contains elements of a comprehensive research devoted to the dynamic behavior of radial ball bearings in real working conditions. The general motivation for this topic comes from the requirements for high performance operation of bearings within complex mechanical systems, defined in many industrial branches during the last decades. The discussion of the fundamental postulates of the approach used for analyzing the vibration response of rolling ball bearings in order to select the optimal working conditions is given. The certain simplifications and reductions used for analyzing the radial ball bearings are explained. The developed procedure can be used for research of influence of different damages and variable operation conditions on the rolling bearings dynamics. The detail analyses of the dynamic behavior of rolling bearings are performed for particular types of radial ball bearings in two case studies: For the damaged outer raceway surface in accordance with real fatigue damage shapes and dimensions and, for variable working temperature. Obtained results are shown by comparative diagrams of vibration and phase plane portraits. Presented results could be a base for more widely research of nonlinear dynamics of radial ball bearings with different damages and for the application of phase plane analysis in order to choose the optimal operation conditions.

The main peculiarity of the Leslie–Gower type models is the predator growth equation is the logistic type, in which the environmental carrying capacity is proportional to the prey population size. This assumption implies the predators are specialists. Considering that the predator is generalist, the environmental carrying capacity is modified adding a positive constant. In this work, the two simple classes of Leslie–Gower type predator-prey models are analyzed, considering a non-usual functional response, called Rosenzweig or power functional responses, being its main feature that is non-differentiable over the vertical axis. Just as Volterra predator-prey model, when the Rosenzweig functional response is incorporated, the systems describing the models have distinctive properties from the original one; moreover, differences between them are established. One of the main properties proved is the existence of a wide set of parameter values for which a separatrix curve, dividing the phase plane in two complementary sectors. Trajectories with initial conditions upper this curve have the origin or a point over the vertical axis as their \(\omega \)-limit. Meanwhile those trajectories with initial conditions under this curve can have a positive equilibrium point, or a limit cycle or a heteroclinic curve as their \(\omega \)-limit. The marked differences between the two cases studied shows as a little change in the mathematical expressions to describe the models can produce rich dynamics. In other words, little perturbations over the functions representing predator interactions have significant consequences on the behavior of the solutions, without change the general structure in the classical systems.

Stabilization of motion of a wheeled robot with constrained control resource by means of a continuous feedback linearizing the closed-loop system in a neighborhood of the target path is considered. The problem of selection of the feedback coefficients is set and discussed. In the case of a straight target path, the desired feedback coefficients are defined to be those that result in the partition of the phase plane into two invariant sets of the nonlinear closed-loop system while ensuring the greatest asymptotic rate of the deviation decrease. A hybrid control law is proposed that ensures the desired properties of the phase portrait and minimal overshooting and is stable to noise. The proposed techniques are extended to the case of circular target paths.

This work presents the comparison of three friction models for the analysis of a rotor hitting the housing. The analytical solution are compared with experimental measurements of the friction force. The measurements are taken form a test rig that represents a rotor fractioning the housing. All the data were analyzed using the Continuous Wavelet Transform, the phase plane and the Recurrence Plot. The three models provided different solutions, and they represents parts of the phenomenon. The models represent the friction but the dynamic response is unable to predict the behaviour observed in the field.

The generation of dynamic spirals under conditions of excitability is presented. After a short description of some basic principles of nonlinear dynamics illustrated in the phase plane, we explain what excitable systems are, how excitation waves propagate, and why external forces influence rotating or moving spirals. Some images of such dynamic spirals are exhibited.

Problem statement: the group use of emerging SEMS devices, i.e., robotic assembly and servicing space modules (RASSM), involves solving the task of synthesizing a program motion for each RASSM, taking into account its own inertial motions in terms of internal mobility degrees. It was previously shown for the model case of the plane motion of an RASSM equipped with an one degree of freedom manipulator with a load in the gripper that in the absence of the forces and moments external to the “base-manipulator-load” system in the study of a controlled relative motion, it is appropriate to introduce a given system into consideration. For the initial system’s nonzero kinetic moment determined by the initial conditions and remaining constant due to the momentum conservation principle, the given system is a nonlinear oscillatory system qualitatively similar to a mathematical pendulum. In this case, the kinetic energy of the initial system can be interpreted as the total energy of the given system, which is the sum of the kinetic and potential components. In absence of a control moment in the hinge, there is an energy integral of the equation for the given system, i.e., the equation of a phase trajectory family. The problems of the synthesis of program motions for an RASSM-associated nonlinear oscillating given system have been investigated. In particular, the pulse control option, which provides a transition to the required point of a phase plane by transferring the image point to the corresponding phase trajectory over a negligible time range, has been studied. Practical significance: the results obtained are of interest from the viewpoint of implementing the important principle of organizing the movement of robots—coordination of free and forced manipulator movements—in the synthesis of control for a robotic space module as an element of an assembly and servicing system for emerging orbital facilities.

Phase plane analysis is one of the most important techniques for studying the behavior of dynamic systems, especially in the nonlinear case. Recent research shows that transient stability problem of a power system following a large disturbance such as a fault can be solved with greater efficiency based on phase plane analysis.

This paper addresses the yaw stability analysis of articulated vehicles using the phase trajectory method. The goal of this work is to ascertain the dynamic conditions that the articulated vehicle can assume without the occurrence of instability events such as jackknife and rollover. The study focuses on the vehicle configuration composed by one tractor unit and a driven unit such as, for instance, a tractor semi-trailer combination. The system consists of a nonlinear tire model and a nonlinear articulated bicycle model with four degrees of freedom. The analysis presented in this paper illustrates the convergence regions of equilibrium points obtained through numerical integration of the equations of motion of the model for different initial conditions in the phase plane. In addition, the changes in the obtained regions are presented as a function of the tractor speed and the position of the articulation point between the two units.

This study presents subject specific analysis at complementary time, frequency and phase plane domains of entire lower limb joint angular kinematic series during normal and modified gait. Healthy adult subject case study was assessed at human movement lab during normal gait, stiff knee gait and slow running. In vivo and non-invasive assessment was performed with eight camera Qualisys system at 100 Hz acquisition of cartesian coordinates of skin reflective markers at lower limb anatomical selected point. Inverse kinematics was performed using AnyGait v.0.92 and TLEM model to match the size and joint morphology of the stick-figure. Entire series of subject specific hip, knee and ankle joint angular displacements (θ), angular velocities (ω) and angular accelerations (α) at the sagittal plane were assessed at the time, frequency and phase plane domains providing useful information for development of subject specific rehabilitation equipment towards gait restoring. The main novelty and contribution of this study when compared with available literature consists on complementary approach of continuous entire signal analysis, time, frequency and phase analysis which can contribute for prescription of requisites for design of rehabilitation equipment towards normal gait of subjects with stiff knee gait and slow running.

A preliminary mathematical model of diabetes has been proposed in [4], in which the evolution of the size of a population of diabetes mellitus patients and the number of patients with complications, has been modeled by second order system of nonlinear differential equations. The model, has already been analyzed for the linear local stability of the equilibria of the system. However, the global behavior of the flow of the nonlinear system has not been studied. The present article analyzes the global behavior of the trajectories of the population growth using Lyapunov stability analysis. Toward this, we construct a suitable Lyapunov function corresponding to an interior equilibrium point and show that it is asymptotically stable within the entire open first quadrant of the planar state space which is the region of interest. Further, transient or incremental stability in the phase plane has been studied via Lyapunov exponent analysis. The stability analysis has also been verified through numerical simulations, under various parameters. A physical interpretation of the parametric dependence of the flows of the nonlinear system is provided from the point of view of diabetic population dynamics.

The influence of exposure to electric and magnetic fields in pancreatic islets are still scarce and controversial, and it is difficult to conduct a comparison of existing studies due to the different research methods employed. Here, computational simulations were used to study the burst patterns in pancreatic beta cell exposure to constant voltage pulses. Results show that burst patterns in pancreatic beta cells are dependent on the applied voltage and that some voltages may even inhibit this phenomenon. There are critical voltages, such as 2.16 mV, in which the burst change from a medium oscillation to a slow oscillation phase or 3.5 mV that induces transition in the burst from slow to fast oscillation phase. Voltage pulse higher than 3.5 mV leads to the extinction of bursts and, therefore, inhibits the process of insulin secretion. These results are reforced by phase plane analysis.

In spite of important technological developments in the medical field and particularly in neuroscience one, epilepsy remained a serious pathology that could affect the human brain. In this work, we modeled a healthy and an epileptic cerebral activity in rest state. We used, the virtual brain TVB toolbox to simulate the two states based on FHN and epileptor model. We compared phase plane spaces, electrophysiological time series (electroencephalogram EEG, magnetoencephalogram MEG and intracerabral EEG), specter of eigenvalues transition matrix and topographic maps for healthy and epileptic rest state. There is a unique metastable state for healthy cerebral dynamics convergence which disappears in epileptic cerebral dynamics. Epileptic rest state time series depicts several transitory activities that vanish in the normal state. Normal rest state topographic maps illustrate a limited dipolar activity; which is more extended in epileptic model. These prominent differences would have an important impact on real cerebral activities analysis.

Theoretical results regarding two-dimensional ordinary-differential equations (ODEs) with second-degree polynomial right-hand sides are summarized, with an emphasis on limit cycles, limit cycle bifurcations, and multistability. The results are then used for construction of two reaction systems, which are at the deterministic level described by two-dimensional third-degree kinetic ODEs. The first system displays a homoclinic bifurcation, and a coexistence of a stable critical point and a stable limit cycle in the phase plane. The second system displays a multiple limit cycle bifurcation, and a coexistence of two stable limit cycles. The deterministic solutions (obtained by solving the kinetic ODEs) and stochastic solutions [noisy time-series generating by the Gillespie algorithm, and the underlying probability distributions obtained by solving the chemical master equation (CME)] of the constructed systems are compared, and the observed differences highlighted. The constructed systems are proposed as test problems for statistical methods, which are designed to detect and classify properties of given noisy time-series arising from biological applications.

This paper presents vehicle stability analysis for a nonlinear vehicle model based on phase plane behavior theorem. Vehicle models under pure lateral slip, constant velocity, and constant front steer were developed. Four-wheel, two-track vehicle model was evolved from the bicycle model and were extended to include vehicle roll dynamics. For improving the stability and rollover prevention, then the height of center gravity (CG) is the important parameter must be taken into account. Through the phase plane behavior, the stability zone and instability zone of the vehicle are completely determined. For illustrating the effect of CG height on vehicle stability, extensive simulations with different CG heights are created. Finally, the analysis results would be applied to design the real model or mobile rack for enhancing the stability of the system.

In this work, we analyze a predator–prey model derived from the Leslie–Gower type model considering two modifications: a generalized Holling type III functional response and a weak Allee 430054755 effect on prey, which is described by an autonomous bidimensional ordinary differential equation system. Conditions for the existence of the equilibrium points or singularities and their nature are determined. The existence of separatrix curves on the phase plane dividing the behavior of the trajectories are also shown. Thus, two closed solutions but in different sides of this separatrix curve can have different ω-limit sets; therefore, there exist trajectories highly sensitive to initial conditions. The existence of constraints on the parameter values for which the unique equilibrium point at the first quadrant is unstable and surrounded by a unique limit cycle in the phase plane is also proven. Computer simulations are also given in order to support our conclusions.

The paper brings phase plane design of a constrained Proportional Derivative (PD) controller for undamped second order oscillatory plants. The approach is focused on achieving the possibly best tracking dynamics considering the control signal saturation. The linear pole assignment control and relay minimum time control are included as its limit solutions. The paper shows that such a task has infinitely many solutions. The existing degree of freedom follows from a free choice of a distance definition in evaluating deviation of the operating point from the considered invariant set specified by the required reference braking trajectory. It may be used to choose the simplest controller equations, or to modify the loop performance in the case of an unmodeled dynamics.

Problems that can be written as singularly perturbed systems of first order differential equations can be solved using approaches that combine matched asymptotic expansions with phase plane analysis. The \(\varepsilon \rightarrow 0\) limit yields a separation of time-scales that reduces the overall system to different forms for the fast and slow dynamics over different intervals of time. Asymptotic matching is used to connect the fast and slow solutions and can be interpreted in terms of the geometric structure of nullclines and the structure of the phase plane.

This paper investigates bounded domains of the solution to a higher-order model in the density-velocity phase plane, according to the monotonicity of two characteristic variables. The basic principle is that the evolution of any phase states in a domain can be confined within this domain which is embraced by four isolines of two characteristic variables, if the evolution towards these isolines is impossible or against the monotonicity of the corresponding characteristic variable. The study provides more information than the classical linear analysis regarding the stability of solution to a higher-order model.

This chapter introduces the idea of state space, including the special (two dimensional) case of the phase plane. I discuss the general solution of the state space formulation for mechanical systems, including eigenvaluesand eigenvectors. I discuss linearization and linear stability in the context of state space. I illustrate the topics of this chapter using examples from previous chapters and introduce a new system, the inverted pendulum on a cart, which is the overhead crane turned upside down

A dynamical state of an autonomous system is completely determined by the generalized coordinates y i (t) and the generalized velocities \(\dot{y}_{i}(t)\) (i = 1, 2, …, n, where n is the number of degrees of freedom). Treating time t as a parameter, a point of the coordinates \((y_{i},\dot{y}_{i})\) will be a point of 2n-dimensional phase space. Motion of this point describes a phase trajectory as time increases. In the case of n = 1 a vibrating system has one degree-of-freedom and the phase space reduces to the phase plane.

Consider a class of autonomous continuous-time dynamical systems whose state at any time can be unambiguously given by a variable x and its derivative \(y =\mathrm{ d}x/\mathrm{d}t\). The phase space of such a system is the phase plane (x, y). Thus, the phase space dimension is N = 2 and the number of degrees of freedom is \(N/2 = 1\).

In the first part of this study an exponential integration scheme for computing solutions of large stiff systems is introduced. It is claimed that the integrator is particularly effective in large-scale problems with localized nonlinearity when compared with the general purpose methods. A brief literature review of different integration schemes is presented and theoretical aspect of the proposed method is discussed in detail. Computational efficiency concerns that arise in simulation of large-scale systems are treated by using an approximation of the Jacobian matrix. This is achieved by combining the proposed integration scheme with the developed methods for model reduction, in order to treat the large nonlinear problems. In the second part, geometric and structural properties of the presented integrator are examined and the preservation of these properties such as area in the phase plane and also energy consistency are investigated. The error analysis is given through small scale examples and the efficiency and accuracy of the proposed exponential integrator is investigated through a large-scale size problem that originates from a moving load problem in railway mechanics. The superiority of the proposed method in sense of computational efficiency, for large-scale problems particularly system with localized nonlinearity, has been demonstrated, comparing the results with classical Runge–Kutta approach.

In the first part of this study an exponential integration scheme for computing solutions of large stiff systems was presented. It was claimed that the integrator is particularly efficient in large-scale problems with localized nonlinearity when compared to general-purpose methods. Theoretical aspects of the proposed method were investigated. The method computational efficiency was increased by using an approximation of the Jacobian matrix. This was achieved by combining the proposed integration scheme with the developed methods for model reduction, in order to treat the large nonlinear problems. In this second part geometric and structural properties of the presented integration algorithm are examined and preservation of these properties such as area in the phase plane and also energy consistency are investigated. The error analysis is given through small scale examples and the efficiency and accuracy of the proposed exponential integrator is investigated through a large-scale size problem that originates from a moving load problem in railway mechanics. The superiority of the proposed method in sense of computational efficiency, for large-scale problems particularly system with localized nonlinearity, has been demonstrated, comparing the results with classical Runge–Kutta approach.

The objective of this study is to quantify and investigate nonlinear motion of the metacarpal and proximal inter-phalange human finger joints, using tools of nonlinear analysis. The second objective is to compare the human finger movement with the cyclic movement of the artificial finger of an anthropomorphic hand-forearm system. The kinematic data of the flexion–extension angles for human and artificial finger were analyzed for two tests which differ by frequency. A time series was obtained for each test. For each time series the angular diagrams are obtained. The main spatio-temporal parameters of the performed tests are presented. Profiles of the phase plane portraits are determinate. For artificial finger joints the dominant frequency of both flexion–extension tests are calculated. For all time series, the human joints motion was characterized with the correlation dimension as nonlinear measure. The calculation of the correlation dimension was performed using the Chaos Data Analyzer software.