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We consider a wide class of randomly perturbed systems subjected to stationary Gaussian processes and show that chaotic orbits exist almost surely under some nondegenerate condition, no matter how small the random forcing terms are. This result is very contrasting to the deterministic forcing case, in which chaotic orbits exist only if the influence of the forcing terms overcomes that of the other terms in the perturbations. To obtain the result, we extend Melnikov's method and prove that the corresponding Melnikov functions, which we call the Melnikov processes, have infinitely many zeros, so that infinitely many transverse homoclinic orbits exist. In addition, a theorem on the existence and smoothness of stable and unstable manifolds is given and the Smale–Birkhoff homoclinic theorem is extended in an appropriate form for randomly perturbed systems. We illustrate our theory for the Duffing oscillator subjected to the Ornstein–Uhlenbeck process parametrically.

Classical Hamiltonian systems with balanced loss and gain are considered in this review. A generic Hamiltonian formulation for systems with space-dependent balanced loss and gain is discussed. It is shown that the loss-gain terms may be removed completely through appropriate co-ordinate transformations with its effect manifested in modifying the strength of the velocity-mediated coupling. The effect of the Lorentz interaction in improving the stability of classical solutions as well as allowing a possibility of defining the corresponding quantum problem consistently on the real line, instead of within Stokes wedges, is also discussed. Several exactly solvable models based on translational and rotational symmetry are discussed which include coupled cubic oscillators, Landau Hamiltonian etc. The role of PT-symmetry on the existence of periodic solution in systems with balanced loss and gain is critically analyzed. A few non-PT-symmetric Hamiltonian as well as non-Hamiltonian systems with balanced loss and gain, which include mechanical as well as extended system, are shown to admit periodic solutions. An example of Hamiltonian chaos within the framework of a non-PT-symmetric system of coupled Duffing oscillator with balanced loss-gain and/or positional non-conservative forces is discussed. It is conjectured that a non-PT-symmetric system with balanced loss-gain and without any velocity mediated interaction may admit periodic solution if the linear part of the equations of motion is necessarily PT-symmetric -- the nonlinear interaction may or may not be PT-symmetric. Further, systems with velocity mediated interaction need not be PT-symmetric at all in order to admit periodic solutions. Results related to nonlinear Schrodinger and Dirac equations with balanced loss and gain are mentioned briefly. A class of solvable models of oligomers with balanced loss and gain is presented., 33 pages, 12 figures, review article; expanded the earlier version with results and discussions

We study stochastic resonance in an over-damped approximation of the stochastic Duffing oscillator from a random dynamical systems point of view. We analyse this problem in the general framework of random dynamical systems with a nonautonomous forcing. We prove the existence of a unique global attracting random periodic orbit and a stationary periodic measure. We use the stationary periodic measure to define an indicator for the stochastic resonance.

In this paper, the characterization of mechanical system that behaves as a hardening Duffing oscillator is presented. This mechanical system comprises a mass attached to a tensioned wire which exhibits a hardening stiffness behavior when the displacement of the mass is large. Firstly, the equation of motion of the system is derived to provide the relationship between the applied static force and the resulting displacement. Then, the effect of initial tension, and number of the wires on the force-displacement relationship are analyzed. It has been found that a higher tension will produce higher linear stiffness, whilst having a negligible effect on cubic stiffness. Moreover, the nonlinearity is less sensitive for small inequality between the length of wire on the left and right side of the mass. The results presented herein provide an insight of the system behavior for its application as a vibration isolator.

Conventional parameter estimation methods for pseudo-random binary code-linear frequency modulation (PRBC-LFM) signals require prior knowledge, are computationally complex, and exhibit poor performance at low signal-to-noise ratios (SNRs). To overcome these problems, a blind parameter estimation method based on a Duffing oscillator array is proposed. A new relationship formula among the state of the Duffing oscillator, the pseudo-random sequence of the PRBC-LFM signal, and the frequency difference between the PRBC-LFM signal and the periodic driving force signal of the Duffing oscillator is derived, providing the theoretical basis for blind parameter estimation. Methods based on amplitude method, short-time Fourier transform method, and power spectrum entropy method are used to binarize the output of the Duffing oscillator array, and their performance is compared. The pseudo-random sequence is estimated using Duffing oscillator array synchronization, and the carrier frequency parameters are obtained by the relational expressions and characteristics of the difference frequency. Simulation results show that this blind estimation method overcomes limitations in prior knowledge and maintains good parameter estimation performance up to an SNR of –35 dB.

Due to machining accuracy, installation error and other factors, the rotor has various eccentricity, which will cause adverse effects on the operation of generator of certain mobile power station and even lead to all systems failure. In order to investigate the generator rotor’s vibration characteristics, taking a military mobile power station generator as the research object, a Jeffcott rotor model is established which considering rotor eccentricity with cubic term in this paper. It is consistent with the Duffing equation in any transverse motion equation. It is simplified to Duffing equation for solving. The analytical solution is solved by the Runge-Kutta method. By studying the effects of various types mechanical parameters on the transverse vibration, the rotor system’s main resonance is studied: the stiffness has extremely few effect on the amplitude change, however, the stability of the system changes greatly with them. While the amplitude only changes and the mass eccentricity reaches a certain value, the rotor system is prone to rub. Resonance amplitude, the response and phase of the system are all affected by the damping, therefore, damper is one of the most effective ways to suppress vibration of generator.

State-dependent time-impulsive perturbations to a two-dimensional autonomous flow with stable and unstable manifolds are analysed by posing in terms of an integral equation which is valid in both forwards- and backwards-time. The impulses destroy the smooth invariant manifolds, necessitating new definitions for stable and unstable pseudo-manifolds. Their time-evolution is characterised by solving a Volterra integral equation of the second kind with discontinuous inhomogeniety. A criteria for heteroclinic trajectory persistence in this impulsive context is developed, as is a quantification of an instantaneous flux across broken heteroclinic manifolds. Several examples, including a kicked Duffing oscillator and an underwater explosion in the vicinity of an eddy, are used to illustrate the theory.

In this paper, we consider a method for the simple exact analytical solution of autonomous nonlinear oscillator equations. While the approach can be used to solve nonlinear oscillator equations with smooth solutions (and we demonstrate this with an application of the approach to an autonomous Duffing equation), our primary interest will be on solving equations with non-smooth yet continuous solutions. To this end, we consider the second-order pseudo-oscillator equation yy″ + 1 = 0 used as a simple model of the path taken by an electron in an electron beam injected into a plasma tube. In recent results of Gadella and Lara, the authors claim the non-existence of periodic solutions to this equation, but actually show that there are no smooth periodic solutions. We show that although there are no smooth solutions to this equation, there does exist a type of continuous periodic solution on the whole problem domain, hence periodic solutions do indeed exist. These periodic solutions can be constructed to have any arbitrary positive period, and the amplitude of these solutions increases as the period is increased. The approach allows one to construct periodic solutions to a variety of nonlinear oscillator equations, even if the solutions are not smooth, and hence could be a useful tool for those interested in physical applications in which nonlinear oscillator models arise.

We have investigated nonlinear dynamic characteristics of piezotransistive GaN microcantilevers, with integrated AlGaN/GaN heterostructure field effect transistor as a highly sensitive deflection transducer. When excited with a piezochip actuator, both softening and hardening type of nonlinear behavior were exhibited by the GaN microcantilevers in their first flexural mode. The nonlinear behavior was found to strongly depend on the dimensions of these microcantilevers. While the hardening behavior was found to be enhanced with increase in length of the cantilever for a fixed width, the nonlinear behavior changed much more sharply with width, exhibiting a clear switchover from hardening to softening type, with an increase in width of the cantilever for a constant length. The nonlinear characteristics of the cantilevers were modeled using a Duffing equation with excellent agreement observed between the theoretical model and experimental data for both nonlinearity types and frequency sweep directions.

According to observations of the waves in 2006 in the harbor of Kholmsk, chaotic vibrations were detected. Studies have shown that with an increase in the amplitude of external excitation, wind waves and swell coming to the entrance of the harbor, the energy of oscillations in the system also increases. For incoming waves higher than 3.5 m the spectrum changes, the energy of movements in the harbor increases significantly in a wide range of periods and reaches the level of energy of the harbor's eigen oscillations, which in this case are practically not allocated. The authors suggested that at large amplitudes of waves at the entrance of the harbor the fluctuations become chaotic. Verification of this assumption was performed on the basis of numerical simulation using the Duffing equation with external periodic excitation. The simulation showed the possibility of chaotic vibrations in the harbor. The system has intermittent chaotic and periodic modes of oscillations with increasing amplitude of external excitation for a wide range of periods and increasing the energy of chaotic vibrations.

In the port harbors there are reoccurring a reciprocating movements of water, which a connected with strong waves at sea. Threatening the safety of ships and berthing facilities, these movements, called harbor oscillations, are caused by the infragravity waves. They occur when their periods are close to the periods of eigen resonant oscillations of the water area. As a result of the observations, it was found that the seiches modulated with the frequency of tidal waves. This phenomenon cannot be explained by analogy with the modulation of infragravity waves. Observations show that the primary reason is an enhancement of energy loss over the low-tide surfzone bottom profile, while in the port harbour significant currents are observed only in the harbor entrance zone. It is shown that the reason of modulation is a change in the water level in the harbor, which occurs nonlinearly. Using the Duffing equation, the dynamics of a nonlinear dynamic system a liquid oscillating in a port under the external excitation by tidal waves and swell, is analyzed. It is shown that under the influence of swell at the harbor entrance can lead to chaotic vibrations in the harbor, which can be dangerous for ships.

The Duffing oscillator remains a key benchmark in nonlinear systems analysis and poses interesting challenges in nonlinear structural identification. The use of particle methods or sequential Monte Carlo (SMC) is becoming a more common approach for tackling these nonlinear dynamical systems, within structural dynamics and beyond. This paper demonstrates the use of a tailored SMC algorithm within a Markov Chain Monte Carlo (MCMC) scheme to allow inference over the latent states and parameters of the Duffing oscillator in a Bayesian manner. This approach to system identification offers a statistically more rigorous treatment of the problem than the common state-augmentation methods where the parameters of the model are included as additional latent states. It is shown how recent advances in particle MCMC methods, namely the particle Gibbs with ancestor sampling (PG-AS) algorithm is capable of performing efficient Bayesian inference, even in cases where little is known about the system parametersa priori. The advantage of this Bayesian approach is the quantification of uncertainty, not only in the system parameters but also in the states of the model (displacement and velocity) even in the presence of measurement noise.

Nonlinear equations are able to present many behaviours of physical systems better than linear equations. Analytical solutions to nonlinear vibration equations have intractability characteristics, while limitations in computational software resources make it difficult to study systematically the phenomena in many systems. In this paper, the Euler-Cromer method is used to solve numerically the vibration equation nonlinear. The nonlinear vibrations of the equation of harmonic, Van der Pol, and Duffing oscillator motion are used as the physical case models to solve. Simulation results show that the Euler-Cromer method provides a numerical solution that is easy to implement and accurate.

In view of the problem of blind area in detecting weak iso-frequency signals of Duffing oscillator, an anova technique is proposed to detect the signal in blind area.By analyzing the condition of detecting blind area, the range of phase of the signal to be measured with the same frequency is obtained, and the variance calculation of the chaotic system and the signal to be measured is realized.The feasibility of the method is proved by experiments.It can also detect the existence of difference frequency signal.The results show that this method is a signal processing method with a lower threshold of SNR, which provides a basis for weak signal detection.

This paper intends to explore the application of the Reverse Multiple Input Multiple Output (R-MISO) technique in the system identification of nonlinear structural behaviour. This paper introduces two concepts in improving the identification estimates of multiple inputs and nonlinear systems, which is the conditioned spectral estimation technique and the inversion of the frequency response function formulas. In this paper, a linear equation of motion with a nonlinear Duffing oscillator is modelled and tested using this technique in order to model the typical nonlinear stiffness characteristics of materials and their design limit. The results indicate a good fit between the predicted physical parameters and the model properties. The nonlinearities were successfully isolated in a non-iterative procedure and the parameters were correctly identified. Multiple coherency was used as a tool to describe the various contributions of the inputs to the system outputs.

In this paper, a kind of novel analytical approximate solution was constructed to the Duffing oscillator via a modified homotopy analysis method (HAM), namely the piecewise perturbation method (PPM). The validity and efficient of the obtained PPM solutions were investigated by comparing with the traditional HAM solution. The main advantage of the PPM is that it possesses higher accuracy and faster convergence rate than the traditional HAM.

We study the nonlinear oscillations of a forced and weakly dissipative spring–magnet system moving in the magnetic fields of two fixed coaxial, hollow induction coils. As the first coil is excited with a dc current, both a linear and a cubic magnet-position dependent force appear on the magnet–spring system. The second coil, located below the first, excited with an ac current, provides the oscillating magnetic driving force on the system. From the magnet–coil interactions, we obtain, analytically, the nonlinear motion equation of the system, found to be a forced and damped cubic Duffing oscillator moving in a quartic potential. The relative strengths of the coefficients of the motion equation can be easily set by varying the coils’ dc and ac currents. We demonstrate, theoretically and experimentally, the nonlinear behaviour of this oscillator, including its oscillation modes and nonlinear resonances, the fold-over effect, the hysteresis and amplitude jumps, and its chaotic behaviour. It is an oscillating system suitable for teaching an advanced experiment in nonlinear dynamics both at senior undergraduate and graduate levels.

The max-min approach is applied to mathematical models of some nonlinear oscillations. The models are regarding to three different forms that are governed by nonlinear ordinary differential equations. In this context, the strongly nonlinear Duffing oscillator with third, fifth, and seventh powers of the amplitude, the pendulum attached to a rotating rigid frame and the cubic Duffing oscillator with discontinuity are taken into consideration. The obtained results via the approach are compared with ones achieved utilizing other techniques. The results indicate that the approach has a good agreement with other well-known methods. He's max-min approach is a promising technique and can be successfully exerted to a lot of practical engineering and physical problems.

A bead sliding, under the sole influence of its own weight, on a rigid wire shaped in the fashion of a plane curve, will describe (generally anharmonic) oscillations around a local minimum. For given shapes, the bead will behave as a harmonic oscillator in the whole range, such as an unforced, undamped, Duffing oscillator, etc. We also present cases in which the effective potential acting on the bead is not analytical around a minimum. The small oscillation approximation cannot be applied to such pathological cases. Nonetheless, these latter instances are studied with other standard techniques.

Nonautonomous forces appear in many applications. They could be periodic, quasiperiodic and almost periodic in time; or they could take the form of a sample path of a random forcing driven by a stochastic process, which is without any periodicity in time. In this paper, we study the chaotic behaviour of differential equations driven by a general nonautonomous forcing without assuming any periodicity in time, aiming at applications to systems driven by a bounded random force. As a direct application, we prove that, for the Duffing equation driven by a bounded stationary stochastic process induced by a Brownian motion, chaotic dynamics exist almost surely. We also obtain various chaotic behaviour that are exclusively associated with equations driven by nonautonomous forcing without any periodicity in time. It has turned out that, unlike the systems driven by a periodic or almost periodic forcing, the transversal intersections of the stable and unstable manifolds are neither necessary nor sufficient for chaotic dynamics to exist. Finally, we apply all our results to the Duffing equation.

Nonlinear response of the driven Duffing oscillator to periodic or quasi-periodic signals has been well studied. In this paper, we investigate the nonlinear response of the driven Duffing oscillator to non-periodic, more specifically, chaotic time series. Through numerical simulations, we find that the driven Duffing oscillator can also show regular nonlinear response to the chaotic time series with different degree of chaos as generated by the same chaotic series generating model, and there exists a relationship between the state of the driven Duffing oscillator and the chaoticity of the input signal of the driven Duffing oscillator. One real-world and two artificial chaotic time series are used to verify the new feature of Duffing oscillator. A potential application of the new feature of Duffing oscillator is also indicated.

Firstly, by using the Liouville formula, we prove that the Jacobian matrix determinants of splitting methods are equal to that of the exact Sow. However, for the explicit Runge–Kutta methods, there is an error term of order p + 1 for the Jacobian matrix determinants. Then, the volume evolution law of a given region in phase space is discussed for different algorithms. It is proved that splitting methods can exactly preserve the sum of Lyapunov exponents invariable. Finally, a Poincare map and its energy distribution of the Duffing equation are computed by using the second-order splitting method and the Heun method (a second-order Runge–Kutta method). Computation illustrates that the results by splitting methods can properly represent systems' chaotic phenomena.

In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter e. Firstly, based on the gradient-Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms (SPAs) for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge–Kutta methods, this paper finds that there is an error term of order p+1 for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method (a second-order Runge–Kutta method) are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when e is small or equal to zero.

We study the stability and exact multiplicity of periodic solutions of the Duffing equation from the global bifurcation point of view and show that the Duffing equation with cubic nonlinearities has at most three T-periodic solutions under a strong damped condition. More precisely, we prove that the T-periodic solutions form a smooth S-shaped curve and the stability of each T-periodic solution is determined by Floquet theory.

This paper presents a novel approach to extract the periodic signals masked by a chaotic carrier. It verifies that the driven Duffing oscillator is immune to the chaotic carrier and sensitive to certain periodic signals. A preliminary detection scenario illustrates that the frequency and amplitude of the hidden sine wave signal can be extracted from the chaotic carrier by numerical simulation. The obtained results indicate that the hidden messages in chaotic secure communication can be eavesdropped utilizing Duffing oscillators.

Stochastic period-doubling bifurcation is explored in a forced Duffing system with a bounded random parameter as an additional weak harmonic perturbation added to the system. Firstly, the biharmonic driven Duffing system with a random parameter is reduced to its equivalent deterministic one, and then the responses of the stochastic system can be obtained by available effective numerical methods. Finally, numerical simulations show that the phase of the additional weak harmonic perturbation has great influence on the stochastic period-doubling bifurcation in the biharmonic driven Duffing system. It is emphasized that, different from the deterministic biharmonic driven Duffing system, the intensity of random parameter in the Duffing system can also be taken as a bifurcation parameter, which can lead to the stochastic period-doubling bifurcations.

The chaotic dynamics of a Duffing oscillator with a parametric force is investigated. By using the direct perturbation technique, we analytically obtain the general solution of the 1st-order equation. Through the boundedness condition of the general solution we get the famous Melnikov function predicting the onset of chaos. When the parametric and external forces are strong, numerical simulations show that increasing the amplitude of the parametric or external force can lead the system into chaos via period doubling.

A technical framework of constructing a linear controller for chaotic synchronization by utilizing the stability theory of cascade-connected system is presented. Based on the method developed in the paper, two simple and linear feedback controllers, as examples, are derived for the synchronization of Liu chaotic system and Duffing oscillator, respectively. This method is quite flexible in constructing a control law. Its effectiveness is also illustrated by the simulation results.

The stability of the periodic solution of the Duffing oscillator system in the periodic phase state is proved by using the Yoshizaw theorem, which establishes a theoretical basis for using this kind of chaotic oscillator system to detect weak signals. The restoring force term of the system affects the weak-signal detection ability of the system directly, the quantitative relationship between the coefficients of the linear and nonlinear items of the restoring force of the Duffing oscillator system and the SNR in the detection of weak signals is obtained through a large number of simulation experiments, then a new restoring force function with better detection results is established.

We study the synchronization in a ring of mutually coupled electromechanical devices. Each device consists of an electrical Duffing oscillator coupled magnetically with a linear mechanical oscillator. By varying the coupling coefficient, we find the ranges for cluster and complete synchronization, either in the regular state or in the chaotic one. Effects of this coupling parameter show various types of bifurcation sequences.

This paper presents a numerical method for solving the controlled Duffing oscillator. The method can be extended to nonlinear calculus of variations and optimal control problems. The method is based upon compactly supported linear semi-orthogonal B-spline wavelets. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Piezoelectrically actuated and sensed microscale resonators have been widely studied for filtering, communication and sensing applications. In this effort, to characterize these resonators, the nonlinear frequency-response behavior of these resonators is examined and parametric identification is carried out. A nonlinear beam model is used with a single-mode approximation to produce a forced Duffing oscillator. Nonlinear analysis is used to obtain the frequency-response equation, and this equation is used along with a least-squares minimization scheme to identify the linear and nonlinear parameter values in oscillator models describing the microscale structures. A linearized analytical model of the stepwise axially varying resonator is also used to obtain additional system parameters. The experimentally identified parameter values are found to be in agreement with predicted values obtained from a nonlinear beam model. Parameter values obtained from multiple sets of data for PZT and AlGaAs microresonators are used to observe trends with respect to a variety of operating conditions.

We present a simple electrical circuit that consists of a Van der Pol oscillator coupled to a Duffing type oscillator. The equations of the model are derived. The stability of the system is investigated. Approximate analytical solutions are obtained and a quenching phenomenon is observed. Finally, synchronization of two identical such systems is investigated.

At the present time, the ultrasonic Doppler measuring means has been extensively used in the human body's bloodstream speed measuring. The ultrasonic Doppler measuring means can achieve the measuring of liquid flux by detecting Doppler frequency shift of ultrasonic in the process of liquid spread. However, the detected sound wave is a weak signal that is flooded in the strong noise signal. The traditional measuring method depends on signal-to-noise ratio. Under the very low signal-to-noise ratio or the strong noise signal background, the signal frequency is not measured. This article studied on chaotic movement of Duffing oscillator and intermittent chaotic characteristic on chaotic oscillator of Duffing equation. In the light of the range of the bloodstream speed of human body and the principle of Doppler shift, the paper determines the frequency shift range. An oscillator array including many oscillators is designed according to it. The reflected ultrasonic frequency information can be ascertained accurately by the intermittent chaos quality of the oscillator. The signal-to-noise ratio of −26.5 dB is obtained by the result of the experiment. Compared with the tradition the frequency method compare, the dependence to signal-to-noise ratio is lowered consumedly. The measuring precision of the bloodstream speed is heightened.

The extended Schrodinger equation for the Kirchhoff elastic rod with noncircular cross section is derived using the concept of complex rigidity. As a mathematical model of supercoiled DNA, the Schrodinger equation for the rod with circular cross section is a special case of the equation derived in this paper. In the twistless case of the rod when the principal axes of the cross section are coincident with the Frenet coordinates of the centreline, the Schrodinger equation is transformed to the Duffing equation. The equilibrium and stability of the twistless rod are discussed and a bifurcation phenomenon is presented.

An analysis of the chaos suppression of a nonlinear elastic beam (NLEB) is presented. In terms of modal transformation the equation of NLEB is reduced to the Duffing equation. It is shown that the chaotic behaviour of the NLEB is sensitively dependent on the parameters of perturbations and initial conditions. By adjusting the frequency of parametric perturbation to twice that of the periodic one and the amplitude of parametric perturbation to the same as the periodic one, the chaotic region of the nonlinear elastic beam driven by periodic force can be greatly suppressed.

An open plus nonlinear closed loop control law is presented for chaotic oscillations described by a set of non-autonomous second-order ordinary differential equations. It is proven that the basins of entrainment are global when the right-hand sides of the equations are given by arbitrary polynomial functions. The forced Duffing oscillator and the forced van der Pol oscillator are treated as numerical examples to demonstrate the applications of the method.

Vibration of a liquid droplet is easily controlled by the Rayleigh surface acoustic wave (SAW). In the vibration of a liquid droplet, two cases are considered: free vibration and forced vibration when it is excited without and with SAW excitation, respectively. For the free vibration, the theoretical and experimental results have already been reported. In the forced vibration, when input excitation was increased, we had determined experimentally that vibration of a liquid droplet shows two nonlinear phenomena. One is, with the increase in SAW input voltage, resonance frequency decreases and vibration amplitude exhibits jump phenomenon. The other, in addition to the fundamental frequency, second and third harmonics, and 1/2 and 3/2 subharmonics are observed. In this report, we explain the two nonlinear phenomena qualitatively using the Duffing equation including the term of a nonlinear spring.

This article proposes the generalized model of Van der Pol — Duffing equation for describing the relaxation oscillations in local brain hemodynamics. This equation connects the velocity and pressure of blood flow in cerebral vessels. The equation is individual for each patient, since the coefficients are unique. Each set of coefficients is built based on clinical data obtained during neurosurgical operation in Siberian Federal Biomedical Research Center named after Academician E. N. Meshalkin. The equation has solutions of different structure defined by the coefficients and right side. We investigate the equations for different patients considering peculiarities of their vessel systems. The properties of approximate analytical solutions are studied. Amplitude-frequency and phase-frequency characteristics are built for the small-dimensional solution approximations.

Nonlinear phenomena in science and engineering such as a periodically forced oscillator with nonlinear elasticity are often modeled by the Duffing oscillator (Duffing equation). The Duffling oscillator is a type of nonlinear higher order differential equation. In this research, a numerical approximation for solving the Duffing oscillator directly is introduced using a variable order stepsize (VOS) algorithm coupled with a backward difference formulation. By selecting the appropriate restrictions, the VOS algorithm provides a cost efficient computational code without affecting its accuracy. Numerical results have demonstrated the advantages of a variable order stepsize algorithm over conventional methods in terms of total steps and accuracy.

This paper is concerned with a finding of an approximate solution to the Duffing oscillator with a linear external force using a modified differential transform method (MDTM) and Runge-Kutta-Nystrom (RKN) method. The results obtained from the two methods are then numerically and graphically compared. We found that both results are close to one another when an amplitude and frequency of external forces are very small. However, both results are quite different when the amplitude and frequency of external forces are large. Their results are in agreement with the phase plane diagram which demonstrates a stable equilibrium point for both methods in case of small amplitude and frequency of external force. However, for large amplitude and frequency of external forces, only MDTM shows the stable equilibrium point.

A general method is discussed concerning reduction of homoclinic and heteroclinic instabilities for a wide class of dissipative systems subjected to two small harmonic perturbations (one chaos-inducing and the other chaos-suppressing) which verify an ultrasubharmonic resonance condition: Ω/ω = p/q, q > 1 (p ≠ q), p,q positive integers and Ω(ω) the chaos-suppressing (inducing) frequency. The relevance of the theoretical findings on steady-chaos suppression is confirmed by means of Lyapunov exponent calculations of a Duffing oscillator.

Equations of motion for an anharmonic interface separating two dispersionless 1-d media are shown to be equivalent to those of the Duffing oscillator coupled with an overdamped harmonic oscillator. Transmission and reflection coefficients as well as trapping times are numerically determined for the most representative sets of the model parameters in a large range of frequencies. Abrupt thresholds are found in the transmission and reflection for strong enough anharmonicity. Irregular frequency dependence of the transmission coefficient is found in regions of chaos and almost no frequency dependence in the windows of periodic motion.

The homoclinic orbits of the integrable nonlinear Schrodinger equation are of interest because of the chaotic behaviour of the equation when it is damped and driven near such an orbit (McLaughlin D W and Overman E A 1995 Whiskered tori for integrable Pde's: chaotic behaviour in near integrable Pde's Surveys in Applied Mathematics vol 1 ed Keller et al (New York: Plenum) ch 2). It is known that exact expressions can be obtained for these homoclinic orbits via Backlund transformations using squared eigenfunctions of the associated linear operator (see above reference). In this paper, the stationary equations of the NLS hierarchy are used to separate the temporal and spatial flows into completely integrable finite-dimensional systems which define the saturation of a given linear instability of a plane wave. Near the homoclinic orbits satisfying even boundary conditions, the phase space is a product of two-dimensional phase planes similar to the phase plane of the Duffing oscillator. The near homoclinic orbits of the plane wave are realized numerically by a simple Runge-Kutta scheme that solves the two systems along the characteristic directions of space and time.

The narrow-tube quasienergy quantization allows an identification of part of the spectrum due to stable elliptic islands of the period map. With its transparent connection to the classical dynamics it also sets an absolute scale to the understanding of the Brillouin zone spectrum of quasienergies. In combination with a recent time-dependent normal-form expansion we show that the applicability and accuracy of the narrow-tube quantization is improved. Its analytic form helps us understand the quasiregular motion centred near periodic responses. We numerically consider harmonic and subharmonic elliptic islands of the weakly nonlinear Duffing oscillator.