Behavior Of Nonlinear Oscillators Research Articles

Nonlinear behavior of encapsulated bubbles relevant to ultrasound mediated gene transfection applications

Ultrasound contrast agents (UCA) have been widely used therapeutic applications in medical research and clinical practice. Recently, UCA have been shown to be effective in vitro in ultrasound mediated sonoporation and gene transfection and treating vascular thrombosis. The possible primary reasons proposed for the therapeutic applications of UCA are acoustic cavitation [W. J. Greenleaf et al., Ultrasound Med. Biol. 24, 587–595 (1998)] and acoustic microstreaming surrounding acoustically driven UCA [J. Wu, Ultrasound Med. Biol. 28, 125–129 (2002)]. Our study has shown that low frequency (0.6 MHz) is more effective than high frequency (2 MHz) in effects related to the acoustic cavitation, and the opposite is true for the microstreaming due to the nonlinear oscillation behavior of encapsulated bubbles of UCA.

Prediction of the nonlinear transient and oscillatory rheological behavior of flour suspensions using a strain-separable integral constitutive equation

The evaluation and development of validated models for the nonlinear viscoelastic (VE) behavior of materials is an important area of research, which has impact on a number of industrial processes including those in the food industry. Various nonlinear VE models have been developed over the years and evaluated for petroleum-based polymers; however, our understanding of the nonlinear VE behavior of biopolymers of industrial import lags our understanding of synthetic polymers. In the work reported herein, the nonlinear VE behaviors of defatted oat flour, oat bran, barley flour, and oat flour suspensions were investigated. The rheological properties were measured using a Rheometrics Series IV controlled-strain rheometer equipped with a cone and plate fixture. The measurements were conducted at 23±0.5°C. The rheological data were interpreted using a strain-separable K-BKZ type (Wagner) model with a damping function evaluated from stress relaxation data. The Wagner model was found to provide an accurate description of the rheological behavior of the suspensions produced from defatted oat flour, barley flour, and oat bran. For suspensions produced from oat flour, the model predictions deviated from the experimental data to a greater degree than the other materials.

A LOCK-FREE MATERIAL FINITE ELEMENT FOR NON-LINEAR OSCILLATIONS OF LAMINATED PLATES

The objective of the present paper is to propose an efficient, accurate and robust four-node shear flexible composite plate element with six degrees of freedom per node to investigate the non-linear oscillatory behavior of unsymmetrical laminated plates. The degrees of freedom considered are three displacement (u, v, w) along x-, y- and z -axis, two rotations (θx, θy) abouty - and x -axis and twist θxy. The elementc employs coupled displacement field, which is derived using moment-shear equilibrium and in-plane equilibrium of composite strips along the x - andy -axis. The displacement field so derived not only depend on the element co-ordinates but are a function of extensional, bending–extensional, bending and transverse shear stiffness coefficients as well. A bi-cubic polynomial distribution with 16 generalized undetermined coefficients for the transverse displacement is assumed. The element stiffness and mass matrices are computed numerically by employing 3×3 Gauss Legendre product rules. The element is found to be free of shear locking and does not exhibit any spurious modes. The element is found to be free of shear locking and does not exhibit any spurious modes. In order to compute the non-linear frequencies, linear mode shape corresponding to fundamental frequency is assumed as the spatial distribution and non-linear finite element equations are reduced to a single non-linear second order ordinary differential equation. This equation is solved by employing direct numerical integration method. A series of numerical examples is solved to demonstrate the efficacy of the proposed material finite element.

Bubble dynamics of ultrasound contrast agents

The bubble dynamics involved with microbubble contrast agents under insonification is investigated. The acoustic field of an ATL HDI-3000 diagnostic ultrasound system in a contrast specific harmonic imaging mode is reviewed first, and its major features that are related with microbubble behavior are discussed. Issues relating to sound attenuation, mechanical index, and bubble destruction are addressed. The nonlinear oscillatory behavior of contrast microbubbles is modeled with the Gilmore equation. The acoustic pressure field of a short pulse utilized in harmonic imaging is measured with a hydrophone and used as the driving pressure of the Gilmore model. Radius-time [R(t)] and bubble wall velocity-time [U(t)] curves are shown. Frequency domain analysis of U(t) indicates transient resonance characteristics in both the fundamental and second harmonic components that are somewhat different from what one would expect with a continuous-wave steady-state response. The times for complete solution of microbubbles in water are calculated and correlated to observations seen in ultrasound images with contrast agents. Radio frequency (rf) data of scattered pulses from contrast agent microbubbles in an in-vitro experiment were collected with a phased array. This data is used to support and explain the contrast microbubble behavior.

Nonlinear oscillations of thick asymmetric cross-ply beams

The axial and transverse motion of asymmetrically laminated cross-ply beams is coupled due to (i) the non-coincidence of reference and mid-planes and (ii) finite amplitudes. To derive the equations that govern the nonlinear oscillatory behavior of these laminated beams, a spatial deformation satisfying geometric constraints is substituted in the kinetic potential and Hamilton's principle is employed. This leads to three coupled nonlinear second order ordinary differential equations. These equations have quadratic nonlinear terms in addition to usual linear and cubic terms. The existence of these terms does not allow a straight forward solution in the presence of axial and rotatory inertia for nonlinear frequencies/periods. The objective of the present paper is to propose two possible solution methods for computing nonlinear frequencies/periods of asymmetrically laminated cross-ply beams. In both the methods, energy of positive and negative deflection cycles is maintained equal. A few numerical examples are solved to validate the methods. The effect of axial and rotatory inertia on the large amplitude oscillatory behavior is investigated.

GLOBAL BEHAVIOR OF A BIASED NON-LINEAR OSCILLATOR UNDER EXTERNAL AND PARAMETRIC EXCITATIONS

In this work we expand our research on the global behavior of non-linear oscillators under external and parametric excitations. We consider a non-linear oscillator simultaneously excited by parametric and external functions. The oscillator has a bias parameter that breaks the symmetry of the motion. The example that we use to illustrate the problem is the rolling oscillation of a biased ship in longitudinal waves, but many mechanical systems display similar features. The global behavior of the system is characterized by bifurcation diagrams that identify the instabilities that appear when one of the excitations is slowly varied. The locus of these instabilities provides the stability boundaries of the system in a parameter space of physical significance. We found that the dynamics of the system significantly depends on the bias parameter, which confirms previous experimental observations. We also found a very interesting effect, which appears to result from the interaction between the parametric and external responses in a nonlinear manner and causes the primary response to lose stability. All results were obtained through analog simulation of the governing equation.

Analysis of Nonlinear Oscillations by Gabor Spectrograms

We examine the potential of the Gabor spectrogram as a tool for the study of the temporal behavior of nonlinear oscillators. Numerical simulations of a snap-through system and homogeneous Duffing equation are analyzed by this method. Gabor spectrograms show dynamic properties of nonlinear oscillators in situations where the conventional Fourier analysis is not appropriate.

The Mathematical Collaboration of M. L. Cartwright and J. E. Littlewood

- Balthasar van der Pol's experiments on electrical circuits during the 1920s and 1930s opened an interesting chapter in the history of dynamics. The need for advancements in radio technology made van der Pol's work pertinent and his research stimulated mathematical interest in nonlinear oscillators. In particular, van der Pol's work caught the attention of Cambridge mathematicians M. L. Cartwright and J. E. Littlewood. Topology and Poincare's transformation theory provided a key to analyzing behavior of nonlinear oscillators and dissipative systems. Resulting mathematical techniques have played a significant role in the development of the modern theory of dynamical systems and chaos. In addition, nonlinear oscillator theory has led to the development of radio, radar, and laser technology. The collaboration between Cartwright and Littlewood began just before World War II and lasted approximately ten years. They published four joint papers, and individually published several other papers based on joint work. Their collaboration produced some of the earliest rigorous work in the field of large parameter theory. They were among the first mathematicians to recognize that topological and analytical methods could be combined to efficiently obtain results for various problems in differential equations, and their results helped inspire the construction of Smale's horseshoe diffeomorphism.l Their names can be included with those mathematicians, including Levinson, Lefschetz, Minorsly, Liapounov, KIyloff, Bogolieuboff, Denjoy, Birkhoff, and Poincare, whose work provided an impetus to the development of modern dynamical theory. Cartwright knew G. H. Hardy at Oxford before she met Littlewood. Cartwright joined a special group when she began to attend Hardy's Friday class at New College in January, 1928. The participants included Gertrude Stanley, John Evelyn, E. H. Linfoot, L. S. Bosenquet, Frederick Brand, and Tirukkannapuram Vijayarhagavan. Nearly all completed their D. Phils by the summer of 1928. The universal feeling among the participants was that they were studying under a very great man who had not yet been recognized fully. Cartwright appreciated Hardy's stylE and philosophy of mathematics. She recalled that he took immense trouble with his students whether they were good, bad, or indifferent. Once, when she had produced an obviously fallacious result, Hardy remarked, Let's see, there's always hope when you get a sharp contradiction.2 Hardy became Cartwright's initial thesis advisor and she finished with E. C. Titschmarsh when Hardy went on leave to Princeton. Cartwright first met Littlewood in June of 1930 when he went to Oxford to examine her for her Doctor of Philosophy degree. The following October she went to Girton College, Cambridge on a three-year research fellowship. She attended a

Nonlinear dynamical behavior of thermionic low pressure discharges. II. Experimental

Strongly nonlinear relaxation oscillations of discharge current and plasma potential are investigated in a magnetized thermionic plasma discharge. The quasi-one-dimensional electron motion allows a direct comparison with one-dimensional models and computer simulations. Two different stable discharge modes can be established, the low-current space charge limited and the high-current temperature limited mode. Time resolved probe measurements of the plasma potential distribution demonstrate that the current oscillations result from a strongly nonlinear instability of the potential structure in the weak current discharge mode. This confirms the model based on particle-in-cell simulations [F. Greiner et al., Phys. Plasmas 2, 1810 (1995)]. The oscillation process consists of three distinct phases. The sequence of events and the observed parameter dependencies of the oscillation frequency is in accordance with the model. The periodically driven system shows the characteristic behavior of nonlinear oscillators: quasiperiodicity, mode-locking, and period doubling sequences towards chaos. It is possible to link the complex dynamical behavior to the details of the trigger mechanism that are revealed by the simulation.

An Algorithm for Higher Order Hopf Normal Forms

Normal form theory is important for studying the qualitative behavior of nonlinear oscillators. In some cases, higher order normal forms are required to understand the dynamic behavior near an equilibrium or a periodic orbit. However, the computation of high-order normal forms is usually quite complicated. This article provides an explicit formula for the normalization of nonlinear differential equations. The higher order normal form is given explicitly. Illustrative examples include a cubic system, a quadratic system and a Duffing-Van der Pol system. We use exact arithmetic andfind that the undamped Duffing equation can be represented by an exact polynomial differential amplitude equation in a finite number of terms. © 1995 John Wily & Sons, Inc.

An Algorithm for Higher Order Hopf Normal Forms

Normal form theory is important for studying the qualitative behavior of nonlinear oscillators. In some cases, higher order normal forms are required to understand the dynamic behavior near an equilibrium or a periodic orbit. However, the computation of high-order normal forms is usually quite complicated. This article provides an explicit formula for the normalization of nonlinear differential equations. The higher order normal form is given explicitly. Illustrative examples include a cubic system, a quadratic system and a Duffing–Van der Pol system. We use exact arithmetic and find that the undamped Duffing equation can be represented by an exact polynomial differential amplitude equation in a finite number of terms.

Long time behavior of nonlinear stochastic oscillators: The one-dimensional Hamiltonian case

The long time behavior of nonlinear, nondissipative systems, which are perturbed by a white noise force are discussed herein. Considering special nonlinear forces and an appropriate scaling, a stochastic convergence theorem is proven. In particular the convergence of the energy process of the system to a limit diffusion is discussed. This corresponds to convergence of the system to a stationary distribution. Furthermore, the limit process is investigated and an explicit formula for its transition probability density is given. An analytic approach to the convergence theorem in terms of a singular perturbation theorem for semigroups is also presented.

Definition of a laser threshold.

We propose that the threshold of a laser is more appropriately described by the pump power (or current) needed to bring the mean cavity photon number to unity, rather than the conventional ``definition'' that it is the pump power at which the optical gain equals the cavity loss. In general the two definitions agree to within a factor of 2, but in a class of microcavity lasers with high spontaneous emission coupling efficiency and high absorption loss, the definitions may differ by several orders of magnitude. We show that in this regime the laser undergoes a transition from a linear (amplifier) behavior to a nonlinear (oscillatory) behavior at our proposed threshold pump rate. The photon recycling resulting from the high spontaneous emission coupling efficiency and high absorption may in this case result in lasing without population inversion, and coherent light is generated via ``loss saturation'' instead of gain saturation. This mechanism for lasing without inversion is very different from lasing without inversion using a radiation trapped state.

On the Quasistochastic Behavior of Nonlinear Magnetization Oscillations in Orthorhombic Ferromagnets Induced by a Magnetic Pulse Field

AbstractIt is shown that large oscillations of magnetization (induced by fast changes of external magnetic field) in ferromagnets, of orthorhombic symmetry can show quasistochastic features: a slight deviation in initial state essentially changes the magnetization behavior for t → ∞. In contrast to the simplest uniaxial anisotropy case (where the quasistochastic behavior should be expected only after switching the in‐plane field on), the presence of orthorhombic anisotropy creates the new possibility of quasistochastic excitation after switching the external field off. This anisotropy component may also considerably change the ranges of quasistochastic behavior.

The behaviour of nonlinear oscillators subjected to ramped forcing

This paper considers the behaviour of a mechanical oscillator with cubic non-linearity subjected to a forcing excitation whose frequency remains constant while the amplitude is ramped, growing until it reaches a predetermined value. We concentrate on the nature of the basins of attraction whose size indicates the stability of the system, in a structural sense. The reduced level of forecing at the initial stages of ramping produces a delay in bifurcational events when compared to the constant sinusoidally forced counterpart. Preliminary results show that for some parameter values the area of basin does not increase monotonically as the length of ramping is varied.

Voltage-Clamp Frequency Domain Analysis of Nmda-Activated Neurons

1. Voltage and current-clamp steps were added to a sum of sine waves to measure the tetrodotoxin-insensitive membrane properties of neurons in the intact lamprey spinal cord. A systems analysis in the frequency domain was carried out on two types of cells that have very different morphologies in order to investigate the structural dependence of their electrophysiological properties. The method explicitly takes into account the geometrical shapes of (i) nearly spherical dorsal cells with one or two processes and (ii) motoneurons and interneurons that have branched dendritic structures. Impedance functions were analysed to obtain the cable properties of these in situ neurons. These measurements show that branched neurons are not isopotential and, therefore, a conventional voltage-clamp analysis is not valid. 2. The electrophysiological data from branched neurons were curve-fitted with a lumped soma-equivalent cylinder model consisting of eight equal compartments coupled to an isopotential cell body to obtain membrane parameters for both passive and active properties. The analysis provides a quantitative description of both the passive electrical properties imposed by the geometrical structure of neurons and the voltage-dependent ionic conductances determined by ion channel kinetics. The model fitting of dorsal cells was dominated by a one-compartment resistance and capacitance in parallel (RC) corresponding to the spherical, non-branched shape of these cells. Branched neurons required a model that contained both an RC compartment and a cable that reflected the structure of the cells. At rest, the electrotonic length of the cable was about two. Uniformly distributed voltage-dependent ionic conductance sites were adequate to describe the data at different membrane potentials. 3. The frequency domain admittance method in conjunction with a step voltage clamp was used to control and measure the oscillatory behavior induced by N-methyl-D-aspartate (NMDA) on lamprey spinal cord neurons. Voltage-clamp currents and impedance functions were measured at different membrane potentials. The impedance functions had a voltage-dependent resonance and phase shift characteristic of a negative conductance. These measurements provide a quantitative analysis of the conductances induced by NMDA in central neurons of the lamprey spinal cord and directly establish the basis of the non-linear oscillatory behavior previously observed in the presence of NMDA. NMDA was shown specifically to activate a negative and a positive conductance, both of which were markedly affected by the membrane potential. It is shown that the net current in the presence of NMDA must be considered as the algebraic sum of currents in opposite directions.(ABSTRACT TRUNCATED AT 400 WORDS)

Active control of nonlinear pressure oscillations in combustion chambers

A theoretical analysis of active control of nonlinear acoustic instabilities in combustion chambers is developed in this article. The formulation starts with a generalized wave equation that describes the dynamic behavior of second-order nonlinear oscillations with distributed feedback actions. Control inputs are provided by the burning of the injected seconday fuel in the chamber, with its instantaneous mass flow rate modulated by a proportionalplus-integral (PI) controller located between the pressure sensor and the fuel injection mechanism. Various nonlinear stability characteristics, including the existence and stability of limit cycles, are studied analytically using the method of time averaging. In addition, an optimization procedure is developed for selecting controller gains. Nomenclature Anij = parameters associated with nonlinear acoustic coupling a = speed of sound in mixture Bnij = parameters associated with nonlinear acoustic coupling b = spatial distribution of burning of control fuel, Eq. (5b), Cv - constant-volume specific heat for two-phase mixture c = amplification factor of sensor output Dni = linear parameters, Eq. (11) Eni = linear parameters, Eq. (11) e - error signal between desired response and actual measurement

Controlling chaotic oscillators without feedback

Nonfeedback methods of controlling the chaotic behaviour of nonlinear oscillators are described. First an analytical method of controlling chaos in Duffing's oscillator which uses the classical approximate analysis of nonlinear oscillations is proposed. The second method allows us to control behaviour on the chaotic attractor in such a way as to obtain a strange nonchaotic trajectory. Finally the application of a ‘dynamical absorber’ to control chaos is shown. It is also shown that the presence of quasiperiodic noise in the system can limit the necessary changes of a control parameter. The methods presented are compared with the feedback Ott-Grebogi-Yorke method [E. Ott, C. Grebogi and Y.A. Yorke, Phys. Rev. Lett. 64, 1196 (1990)].

Heteroclinic orbit and subharmonic bifurcations and chaos of nonlinear oscillator

Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes van der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-for various resonant cases. Finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.

Demonstrations of nonlinear oscillators, parametric excitation, and solitons

The peculiar behavior of nonlinear oscillations will be demonstrated with a series of experiments selected from those shown previously in Austin [J. Acoust. Soc. Am. Suppl. 1 77, S35 (1985)]: (1) A stretched rubber band, driven by a loudspeaker, exhibits hysteresis due to its bent resonance curve. (2) A doubly bent resonance curve and hysteresis are exhibited by a parametrically driven pendulum bouncing against stops. (3) A hanging chain undergoes quasiperiodically modulated vibrations, where the modulations have a frequency independent of that of the drive. (4) A parametrically driven rigid pendulum will defy gravity by standing on end and oscillating upside down. (5) Standing waves on a trough of water will be parametrically driven, as seen by Michael Faraday. (6) A nonpropagating soliton [Phys. Rev. Lett. 52, 1421 (1984)] will be shown in a trough of water, and two of these solitons will oscillate about each other.