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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 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.

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.

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, 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.

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.

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.

Pyramidal space trusses are currently used in many present-day engineering applications, either as main parts or as a constitutive element. The present work studies, using a corotational finite element formulation considering large displacements and rotations and an elastoplastic material behavior with isotropic strain hardening, the nonlinear dynamic behavior of a pyramidal space truss. A bilinear hysteretic model with kinematic hardening is used to represent the material nonlinearity. An appropriate strain measure in such processes is the logarithmic strain. However, the engineering strain and the quadratic strain measures have been usually used to perform the elastic stability analysis of such structures. Thus, in the present work results using both the quadratic and the logarithmic strain measures are compared. First, the static nonlinear behavior is investigated in order to understand the influence of nonlinearities and static preload on the potential energy landscape, which controls the dynamics of the truss. These structures exhibit a two-well potential function that leads to coexisting in-well and cross-well solution branches, resulting in several coexisting periodic, quasi-periodic and chaotic attractors. The influence of the constitutive law, strain measure, truss geometry and support flexibility on the static stability is then investigated. Based on these results, a detailed parametric analysis of the nonlinear truss response under vertical load is conducted using bifurcations diagrams of the Poincare map to investigate the effect of the constitutive law, strain measure, truss geometry and load control parameters on the bifurcation scenario, coexisting attractors and dynamic buckling loads. The effect of the strain hardening parameter is particularly emphasized, allowing a more comprehensive picture of the dynamical behavior exhibited by the bistable system. Properties of the response are further illustrated by samples of time and phase plane responses and the related Poincare section plots. In addition to known behaviors, a rich class of solutions and bifurcations, including jump phenomena, symmetry-breaking, period-doubling cascades, fold and chaos is detected.

We propose that a system is in a gravitational field and that the rolling of a heavy ball, over a sphere, is activated by a force of proper weight and initial kinetic and potential energies given to the ball at the initial moment. For mathematical description of the rolling without slipping of the heavy rigid homogeneous ball over the sphere on both the inside and the outside of the sphere surface, spherical coordinates are used: angle in circular and angle in meridional directions, and angle of the ball self-rotation about radial direction. Ordinary nonlinear differential equations are derived. The angle coordinate in the circular direction is a cyclic coordinate, and an integral for the circular-cyclic coordinate is derived. Integral constant that depends on initial conditions is determined. The main nonlinear differential equation is expressed through a meridional angle coordinate, and a corresponding first integral is derived. The equation of the first integral is an equation of phase trajectory. By using this equation and the corresponding set of initial conditions, phase trajectory portraits are graphically presented. An elliptic integral is derived. By using new Hedrih’s results in theory of collision between two rolling bodies, geometry, kinematics and dynamics of successive collisions of two rolling balls over the surface of a sphere are analyzed and a methodology for investigation of vibro-impact nonlinear dynamics of vibro-impact system with rolling bodies over the sphere surface is described. A mathematical analogy between differential equations and phase trajectory portraits of dynamics of a rolling heavy ball and a material heavy mass particle moving along sphere surface is identified. New original results are visible in the first part of abstract of the manuscript. Let’s, again, point out in short list the main new original author’s results, different than published results of a number of mathematicians: (1) new nonlinear differential equations as description of the rolling, without slipping, of the heavy rigid homogeneous ball over the sphere, on both the inside and the outside, of the sphere surface, presented in spherical coordinates, (2) first integral of new nonlinear differential equation as description of the rolling, without slipping, of the heavy rigid homogeneous ball over the sphere, on both the inside and the outside, of the sphere surface, presented in spherical meridional coordinate, (3) first graphical presentation of a series of qualitative phase trajectory portraits on phase plane with phase coordinate meridional angle coordinate and meridional angular velocity, (4) proof that in the rolling, without slipping, of the heavy rigid homogeneous ball over the sphere, on both the inside and the outside, of the sphere surface, the circular angle coordinate is cyclic coordinate, and obtained cyclic integral which shows the nonlinear dynamics of a rolling heavy ball over the sphere is with constant sectorial velocity (it can be accepted as a theorem), (5) application of the new Hedrih’s results in theory of collision between two rolling bodies, in geometry, kinematics and dynamics of successive collisions of two rolling balls over the surface of a sphere, (6) a new methodology for investigation of vibro-impact nonlinear dynamics of vibro-impact system with rolling bodies over the sphere surface, (7) a mathematical analogy between nonlinear differential equations and phase trajectory portraits of dynamics of a rolling heavy ball and a material heavy mass particle moving along sphere surface is identified. Previously listed new original results of nonlinear dynamics of a ball rolling over a sphere suitable for inclusion in a university monograph on dynamics for postgraduate study are presented. Main importance of listed results is applicability in investigation of the nonlinear dynamics of the numerous engineering systems.

Rub–impact phenomenon occurring in hydrodynamic journal bearings is one of the main malfunctions in rotating machines and causes undesirable dynamic behavior. In order to investigate such a phenomenon, nonlinear dynamics due to rub–impact within tilting pad journal bearings supporting a flexible rotor is studied. In simulating rub–impact between the journal and the associated pads, this paper employs mixed lubrication theory along with elasto-plastic asperity contact model between pads and journal. Periodic, quasiperiodic and chaotic vibrational behaviors of system are studied by varying the unbalanced load magnitude and the rotational speed of the rotor as control parameters. Phase plane orbits, waterfall frequency response spectra and bifurcation diagrams are used to show various dynamic responses of rotor when the control parameters are changed. The Poincare maps are used to determine the onset of irregular motions. Presented results provide better understanding of strongly nonlinear vibrations occurring due to rub–impact in TPJBs supporting industrial rotating machines.

This paper studied the nonlinear vibration and resonance of a Cartesian manipulator system carrying an intermediate end effector under mixed excitations. The multiple scales method is applied to get the approximate solutions of this system of the second-order differential equation. Furthermore, the analytical solution obtained the amplitudes and phases of the response from the first-order differential equation governing. We extracted all worst resonance cases and studied it numerically. The numerical solutions and response amplitude of this system are also studied and discussed. We analyzed the stability of the steady-state solution of a Cartesian manipulator system using frequency response equations and phase plane technique at the worst resonance cases. Comparison between analytical and numerical solutions is obtained. We determined both bifurcation diagrams and stability using Poincare maps. Also, the numerical results are obtained using MAPLE and MATLAB algorithms.

Rauscher method becomes the matter of interest because in combination with the method of nonlinear normal vibration modes it allows to calculate steady forced vibrations in the system with multiple degrees of freedom (DOF) via reduction in the number of DOFs. However, modern realizations of that approach have drawbacks such as iterative nature and the need to have initial approximation for the solution. The primary principle of Rauscher method is in obtaining periodic solutions of a non-autonomous system via studying some equivalent autonomous one. In the paper, a new non-iterative variant of Rauscher method is considered. In its current statement, the method can be used in analysis of forced harmonic oscillations in a nonlinear system with one degree of freedom. The primary goals of the study were to find out what kind of equivalent autonomous systems could be built for a given non-autonomous one and how they can be used for the construction of periodic solutions and/or periodic phase plane orbits of the initial system. It is shown that three different types of equivalent autonomous dynamical systems can be built for a given 1-DOF non-autonomous one. The system of 1st type is a fourth-order dynamical system. Technically it can be considered as a 2-DOF system where additional “DOF” is explicitly “responsible” for forced oscillations. The system of 2nd type is a third-order dynamical system. Its periodic orbits are exactly the same as in the initial system. Using the invariant manifold of the system of 1st type, the system of 2nd type can be reduced to the form $$W(x,x')=0$$ (which is called here the equivalent system of the 3rd type). It is important that the function $$W(x,x')$$ can be built a priori. Once $$W(x,x')$$ is found: (i) one can obtain different periodical orbits corresponding to forced oscillations in the initial system; (ii) one can estimate amplitudes of vibrations for these regimes; (iii) one can track bifurcations of periodical regimes of the initial system with respect to change in amplitude of external excitation f. As shown in the paper, periodical orbits of the initial non-autonomous system can be obtained via two different approaches: (i) as set of points on phase plane satisfying the condition $$W(x,x')=0$$ ; (ii) via the application of harmonic balance method to the equivalent system of 1st type using system’s energy level as a continuation parameter. This approach has advantage over application of harmonic balance method to initial system because the latter requires good initial guess for expansion coefficients, while the new approach does not and always starts from zero initial guess.

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.

In this paper, two simple 3D chaotic systems are constructed by introducing nonlinear functions into Sprott A system, which all have no equilibrium, and multi-scroll hidden attractors can be obtained. In the case of simple sine function without restricting its nonlinear dynamical range, the number of multi-scroll attractors has nothing to do with the system equilibria, which is only determined by the transient simulation time. In the other case, the introduced nonlinear function is composed of nonlinear part and linear part; thus, the nonlinearity of improved Sprott A system is limited in certain range, as a result, the number of multi-scroll hidden attractors can be selected arbitrarily within finite transient simulation time. Furthermore, the dynamical properties of two systems are studied through phase plane, time series, Poincare map and frequency spectra. Finally, an electronic circuit of improved Sprott A system is implemented in Pspice, and the results of electronic circuit are consistent with that of the numerical simulation.

In this paper, a novel method is proposed for producing pseudo random numbers from differential equation-based chaotic systems. The proposed method utilizes a Lu-like chaotic system capable of exhibiting both Lorenz-like and Chen-like chaotic system behaviors for different parameter values. A parameter switching process controlled by a linear feedback shift register is used to obtain the transition between Lorenz-like and Chen-like behaviors. The system has a new initial condition for each transition. Thus, both the stabilization effect and the propagation of the integration errors that occur in the discretization of the differential equation-based chaotic systems are prevented. Additionally, the more complex behavior of the pseudo orbits in the phase plane makes future predictions difficult. The pseudo random numbers produced by the proposed method are tested with NIST SP 800-22 and TestU01 randomness test suites, and all tests are passed. The field programmable gate array implementation of the pseudo random number generator based on the proposed method is also presented, and the implementation results are compared with the implementation of a chaotic map-based pseudo random number generator.

The nonlinear dynamics of bimodular laminated composite conical panels subjected to harmonic excitation is investigated using Bert’s constitutive model and first-order shear deformation theory based finite element. The governing equations of motion are solved to obtain the state vector representing the periodic response by employing shooting technique coupled with Newmark time marching and arc length/pseudo-arc length continuation algorithms. The influence of bimodularity, geometric nonlinearity and curvature on the steady state response characteristics of bimodular material laminated composite conical panels is analyzed for the first time. The through the thickness variations of normal strain/stress, steady state stress history, phase plane plots and frequency spectra are presented to explore the nonlinear periodic response characteristics of bimodular conical panels. The frequency response curves reveal increase in hardening nonlinearity with the increase in bimodularity ratio, load amplitude and semi-cone angle. The frequency response curves tend toward softening nonlinear behavior with the increase in curvature and/or decrease in semi-cone angle.

A class of models of predator–prey interaction with Allee effect on the prey population is presented. Both the Allee effect and the functional response are modelled in the most simple way by means of general terms whose conveniently chosen mathematical properties agree with, and generalise, a number of concrete Leslie–Gower-type models. We show that this class of models is well posed in the sense that any realistic solution is bounded and remains non-negative. By means of topological equivalences and desingularization techniques, we find specific conditions such that there may be extinction of both species. In particular, the local basin boundaries of the origin are found explicitly, which enables one to determine the extinction or survival of species for any given initial condition near this equilibrium point. Furthermore, we give conditions such that an equilibrium point corresponding to a positive steady state may undergo saddle-node, Hopf and Bogdanov–Takens bifurcations. As a consequence, we are able to describe the dynamics governed by the bifurcated limit cycles and homoclinic orbits by means of carefully sketched bifurcation diagrams and suitable illustrations of the relevant invariant manifolds involved in the overall organisation of the phase plane. Finally, these findings are applied to concrete model vector fields; in each case, the particular relevant functions that define the conditions for the associated bifurcations are calculated explicitly.

This study elucidates the bifurcation structure causing chaos disappearance in a four-segment piecewise linear Bonhoeffer–van der Pol oscillator with a diode under a weak periodic perturbation. The parameter values of this oscillator are chosen such that stable focus and stable relaxation oscillation can coexist in close proximity in the phase plane if no perturbation is applied. Chaos disappearance occurs through a previously unreported novel and unconventional bifurcation mechanism. To rigorously analyze these phenomena, the diode in this oscillator is assumed to operate as a switch. In this case, the governing equation is represented as a constraint equation, and the Poincare map is constructed as an one-dimensional map. By analyzing the Poincare map, we clearly demonstrate why the stable relaxation oscillation that exists when no perturbation is applied disappears via chaotic oscillation when an extremely weak perturbation is applied.

The global homoclinic bifurcation and transition to chaotic behavior of a nonlinear gear system are studied by means of Melnikov analytical analysis. It is also an effective approach to analyze homoclinic bifurcation and detect chaotic behavior. A generalized nonlinear time varying (NLTV) dynamic model of a spur gear pair is formulated, where the backlash, time varying stiffness, external excitation, and static transmission error are included. From Melnikov method, the threshold values of the control parameter for the occurrence of homoclinic bifurcation and onset of chaos are predicted. Additionally, the numerical bifurcation analysis and numerical simulation of the system including bifurcation diagrams, phase plane portraits, time histories, power spectras, and Poincare sections are used to confirm the analytical predictions and show the transition to chaos.

In this paper we study the existence of limit cycles in a one-parameter family of discontinuous piecewise linear differential systems with two zones in the plane. It is characterized for all the parameter values the number of non-sliding limit cycles of the family studied, detecting a rather non-generic bifurcation leading to the simultaneous generation of three limit cycles.

The bifurcation phenomena of a compound K(m,m) equation are investigated in this paper. Three singular lines have been found in the associated topological vector field, which may evolve in the phase trajectories of the system. The influence of parameters as well as the singular lines on the properties of the equilibrium points has been explored in details. Transition boundaries have been obtained to divide the parameter space into regions associated with different types of phase trajectories. The existence conditions and related discussions for different traveling wave solutions have been presented in the end.

We study the nature of motion in a 3D potential composed of perturbed elliptic oscillators. Our technique is to use the results obtained from the 2D potential in order to find the initial conditions generating regular or chaotic orbits in the 3D potential. Both 2D and 3D potentials display exact periodic orbits together with extended chaotic regions. Numerical experiments suggest, that the degree of chaos increases rapidly, as the energy of the test particle increases. About 97% of the phase plane of the 2D system is covered by chaotic orbits for large energies. The regular or chaotic character of the 2D orbits is checked using the S(c) dynamical spectrum, while for the 3D potential we use the S(c) spectrum, along with the P(f) spectral method. Comparison with other dynamical indicators shows that the S(c) spectrum gives fast and reliable information about the character of motion., Published in Nonlinear Dynamics (NODY) journal

The two-degree-of-freedom (2-DOF) airfoil system with freeplay nonlinearity in pitch is investigated numerically. The relation between eigenvalues and flutter speed has been analyzed. The effect of parameters of the freeplay nonlinearity on the system responses is obtained. The probability density function (PDF) and phase plane of the deterministic system have been studied and the results show that the amplitude of limit cycle oscillation (LCO) grows with mean airspeeds increasing. Marginal PDFs, bidimensional PDFs, random bifurcation, and the largest Lyapunov exponent are used in investigation of the random system. The results show that, for low and intermediate level turbulences, the marginal PDFs of system exhibit different characters at different airspeed ranges. However, for high level turbulence the marginal PDFs are similar in the whole airspeed region. The bidimensional PDF has different shapes in low level turbulence at pre- and post-flutter speeds, but the PDF keeps similar shape in high level turbulence. The random bifurcation analysis indicates the P-bifurcation can happen at both pre- and post-flutter speeds but the D-bifurcation never occurs. Numerical simulations approve the results.

A method for identifying a piecewise-linear approximation to the nonlinear forces acting on a system is presented and demonstrated using response data from a micro-cantilever beam. It is based on the Restoring Force Surface (RFS) method by Masri and Caughey, which is very attractive when initially testing a nonlinear system because it does not require the user to postulate a form for the nonlinearity a priori. The piecewise-linear fitting method presented here assures that a continuous piecewise-linear surface is identified, is effective even when the data does not cover the phase plane uniformly, and is more computationally efficient than classical polynomial based methods. A strategy for applying the method in polar form to sinusoidally excited response data is also presented. The method is demonstrated on simulated response data from a cantilever beam with a nonlinear electrostatic force, which highlights some of the differences between the local, piecewise-linear model presented here and polynomial-based models. The proposed methods are then applied to identify the force-state relationship for a micro-cantilever beam, whose response to single frequency excitation, measured with a Laser Doppler Vibrometer, contains a multitude of harmonics. The measurements suggest that an oscillatory nonlinear force acts on the cantilever when its tip velocity is near maximum during each cycle.

Time-optimal feedback control is obtained that brings a pendulum to the upper unstable equilibrium position. The solution is based on the maximum principle and involves analytical investigations combined with numerical computations. As a result, the switching and dispersal curves that bound the domains in the phase plane corresponding to different values of the optimal bang-bang control are constructed for various values of the maximal admissible control torque.

The longitudinal braking dynamics of a two-wheel vehicle model on an incline are considered using techniques from nonlinear dynamics. The model is planar and incorporates the coupled dynamics of two independently braked wheels and the vehicle body, and takes into account the slip dynamics of each wheel. By using the wheel slip values and the vehicle speed as dynamic states, it is shown that the qualitative behavior of the system can be completely captured by studying a relatively simple phase plane problem described in terms of the slip values. A systematic bifurcation analysis is carried out in which the brake torques of the two wheels are varied, and it is shown how the system transitions from stable braking to the possibility of lockup in one or both wheels, to guaranteed lockup in both wheels. In this manner a quite complete picture of the dynamic behavior is obtained as a function of the two brake torques, including regions with multiple possible steady braking outcomes, depending on the initial conditions. This analysis provides new insights into the dynamics of vehicle braking, and it provides a correction to the standard result for the critical values of the brake torques at which the wheels undergo lockup. This approach may also prove useful for evaluating brake proportioning schedules, or for investigating anti-lock braking systems and other methods of traction control.

The pull-in range (ωp) of a phase-locked loop (PLL) is defined as the maximum value of loop detuning ω0s for which pull-in occurs from anywhere on the PLL's phase plane. That is, pull-in is guaranteed from anywhere on the phase plane if ω0s

A strategy is proposed to solve the fixed final state optimal control problem using the simple cell mapping method. A non-uniform time step simple cell mapping is developed to create a general database from which solutions of various optimal control problems can be obtained. A two-stage backward search algorithm is proposed to eliminate degenerated paths often associated with the simple cell mapping. The proposed method can accurately delineate the switching curves and eliminate false limit cycles in the solution. The method is applied to two optimal control problems with bang-bang control. The well-known minimum time control problem of moving a point mass from any initial condition to the origin of the phase plane is studied first. This example has exact solutions available which provide a yardstick to examine the accuracy of the method. The cell size dependence of the solution accuracy is studied numerically. The second example is a variable stiffness feedback control problem with tuning range saturation. The strategy proposed is able to provide the switching curves in the phase plane. This result has not been obtained before.

In this paper the dynamic stability of greyhound gait was analyzed within the framework of nonlinear dynamics theory. A video based motion analysis system was utilized to obtain the kinematic data of the hindlimb joints of greyhounds at trot. Phase plane portraits and first return maps for the coxofemoral, femorotibial, and the tarsal joints were calculated from averaged kinematic data for each dog. The analysis was based on the assumption that the steady state dog locomotion could be represented as a nonlinear periodic system. Using the Floquet theory, the dynamic stability of gait was quantified by computing the characteristic multipliers from experimental data. A stability index based on multipliers was used for comparison between normal and arthritic dogs. Phase plane portraits and first return maps of the dogs with transient synovitis were compared with the averaged portraits of the normal dogs. It was observed that the coxofemoral angle exhibited the maximum difference while the femorotibial and the tarsal joints showed little or no difference from those of the normals. Comparison of the Floquet multipliers indicated that the dogs with synovitis had a less stable gait than that of the normal dogs.

Invariant tori of solutions for nonlinearly coupled oscillators are generalizations of limit cycles in the phase plane. They are surfaces of aperiodic solutions of the coupled oscillators with the property that once a solution is on the surface it remains on the surface. Invariant tori satisfy a defining system of nonlinear partial differential equations. This case study shows that with the help of a symbolic manipulation package, such as MACSYMA, approximations to the invariant tori can be developed by using Galerkin's variational method. The resulting series must be manipulated efficiently, however, by using the Poisson series representation for multiply periodic functions, which makes maximum use of the list processing techniques of MACSYMA. Three cases are studied for the single van der Pol oscillator with forcing parameter e=0.5, 1.0, 1.5, and three cases are studied for a pair of nonlinearly coupled van der Pol oscillators with forcing parameters e=0.005, 0.5, 1.0. The approximate tori exhibit good agreement with direct numerical integrations of the systems.

This work investigates nonlinear dynamic response of circular rings rotating with spin speed which involves small fluctuations from a constant average value. First, Hamilton's principle is applied and the equations of motion are expressed in terms of a single time coordinate, representing the amplitude of an in-plane bending mode. For nonresonant excitation or for slowly rotating rings, a complete analysis is presented by employing phase plane methodologies. For rapidly rotating rings, periodic spin speed variations give rise to terms leading to parametric excitation. In this case, the vibrations that occur under principal parametric resonance are analyzed by applying the method of multiple scales. The resulting modulation equations possess combinations of trivial and nontrivial constant steady state solutions. The existence and stability properties of these motions are first analyzed in detail. Also, analysis of the undamped slow-flow equations provides a global picture for the possible motions of the ring. In all cases, the analytical predictions are verified and complemented by numerical results. In addition to periodic response, these results reveal the existence of unbounded as well as transient chaotic response of the rotating ring.

Two alternative criteria for predicting the onset of chaos are presented. Both are based on the notion that it is the interaction between a stable and nearby unstable limit cycle pair in the phase space that disrupts the stable motion, thereby producing chaotic behavior. The first criterion is based upon an intersection of the unstable and stable limit cycle orbits in the phase plane. The second criterion proposes that an energy equivalence between the stable and unstable limitcycles may be responsible for the loss of periodicity of the stable motion. Both criteria are tested numerically using three distinct softening spring oscillators and their predictive capabilities are discussed. The results of this study, particularly for the energy criterion, are encouraging.

The purpose of this paper is to examine a highly nonlinear model of a slender beam which yields chaotic solutions for some forcing amplitudes. The study is unique in that the governing partial differential equations are solved directly, and that the model lends itself to a more physical analysis of the beam than traditional chaotic models. In addition, the analysis will provide proof that a beam experiencing moderate deformations without stops or an initial axial force can exhibit chaotic motion. The model represents a simply-supported. Euler-Bernoulli beam subjected to a transverse load. The forcing function is sinusoidally distributed in space with an amplitude which also varies sinusoidally in time and is assumed to reach a maximum sufficient to allow nonlinearities associated with finite deformations to become important. During motion, even though displacements are large, the beam is assumed to attain only small strain levels and thus is assumed to be linearly elastic. The results indicate that for most levels of the forcing function the response of the beam is periodic. However, the steady state motion is not sinusoidal in time and in fact exhibits some bifurcated motions. At a certain level of the forcing amplitude, an asymmetry is observed and the periodicity of the motion breaks down as the beam experiences a period doubling cascade which culminates in a chaotic motion. The progression from periodic to chaotic motion is presented through a series of phase plane and Poincane plots, and physical variables such as bending moment are examined.

We investigate the system\(\ddot x - x\cos \varepsilon 1 + x^3 = 0\) in which e≪1 by using averaging and elliptie functions. It is shown that this system is applicable to the dynamics of the familiar rotating-plane pendulum. The slow foreing permits us to envision an ‘instantancous phase portrait’ in the\(x - \dot x\) phase plane which exhibits a center at the origin when cos e1≤0 and a saddle and associated double homoclinic loop separatrix when cos ɛ 1 > 0. The chaos in this problem is related to the question of on which side (left (=L) or right (=R)) of the reappearing double homoclinic loop separatrix a motion finds itself. We show that the sequence of L's and R's exhibits sensitive dependence on initial conditions by using a simplified model which assumes that motions cross the instantancous separatrix instantancously. We also present an improved model which ‘patches’ a separatrix boundary layer onto the averaging model. The predictions of both models are compared with the results of numerical integration.