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One of the classic problems in the study of nonlinear dynamics has been the diode resonator. Previous work with the diode resonator sought to explain the causes of period doubling and chaos, and often used simplified models. This paper instead seeks to link the onset of nonlinear dynamical effects to measurable parameters by comparing experiments and numerical models.
We consider the interactions of two identical, orthogonally polarized vector solitons in a nonlinear optical fiber with two polarization directions, described by a coupled pair of nonlinear Schrödinger equations. We study a low-dimensional model system of Hamiltonian ordinary differential equations (ODEs) derived by Ueda and Kath and also studied by Tan and Yang. We derive a further simplified model which has similar dynamics but is more amenable to analysis. Sufficiently fast solitons move by each other without much interaction, but below a critical velocity the solitons may be captured. In certain bands of initial velocities the solitons are initially captured, but separate after passing each other twice, a phenomenon known as the two-bounce or two-pass resonance. We derive an analytic formula for the critical velocity. Using matched asymptotic expansions for separatrix crossing, we determine the location of these "resonance windows." Numerical simulations of the ODE models show they compare quite well with the asymptotic theory.
Resonance Raman profiles for 14 vibrational modes of betaine-30 in ethanol at room temperature were measured at wavelengths within the first charge-transfer absorption band. The absorption spectrum and resonance Raman profiles were analyzed using time-dependent theory and a Brownian oscillator model modified to account for nonlinear solvent response; i.e., dependence of the solvent reorganization energy on the electronic state of the solute. As in our previous study of betaine-30 in acetonitrile, the solvent reorganization energy for the excited electronic state, determined from resonance Raman spectroscopy, was found to be smaller than that for the ground electronic state, determined from the absorption spectrum. The mode-dependent internal reorganization energies of betaine-30 in ethanol were found to be slightly larger than those of betaine-30 in acetonitrile. Temperature-dependent solvent reorganization energies for the ground electronic state were determined from analysis of the absorption line shape from 279 to 332 K and were found to decrease with increasing temperature. The influence of hydrogen bonding on the solvent and internal reorganization energy of betaine-30 is considered, and the physical basis for nonlinear solvent response is discussed.
Earthquake faults occur in interacting networks having emergent space-time modes of behavior not displayed by isolated faults. Using simulations of the major faults in southern California, we find that the physics depends on the elastic interactions among the faults defined by network topology, as well as on the nonlinear physics of stress dissipation arising from friction on the faults. Our results have broad applications to other leaky threshold systems such as integrate-and-fire neural networks.
RNA virus evolution is generally considered to be highly unpredictable, but tests of determinism in the evolution of competing populations during viral infections have not been performed. Here we study the fate of two closely related evolving quasispecies of vesicular stomatitis virus, by determining the relative concentration of a wild-type clone and a surrogate marked virus subclone (MARM-C) upon extensive competitive replication in a constant cell culture environment. A highly predictable nonlinear behaviour of the two competing populations was found. In addition, the presence of critical points, which are defined as points from which viral competitions may follow different trajectories, has been documented. Critical points were reached after nearly constant periods of time. The dynamics of relative fitness values for both competing populations were calculated during the replication passages. Concomitant with expected fitness gain of both competing viral populations (which follow the Red Queen hypothesis) a tendency for the MARM-C to gain less fitness than the wild-type was observed. Although fitness variations were noisy, this tendency was seen in all evolutionary replicas. Thus, despite the stochastic process of mutation that leads to a continuous generation of mutant genomes during RNA virus replication, a nonlinear, nearly deterministic evolutionary behaviour has been observed. It is proposed that such a behaviour is mediated by a low-pass filter (averaging of mutational noise signals) due to competitive selection among variants.
We discuss some problems, concerning the application of nonlinear dynamics methods and ideas to vacuum microwave electronics. We consider such phenomena as solitons, deterministic chaos and pattern formation in different models of electron flows and devices. Our results reveal that microwave electronics is an interesting field of application of nonlinear dynamics. (c) 1996 American Institute of Physics.
The Gaussian surrogate-date procedure was applied to the measurement of the effect of cigarette smoking on the dimensional complexity of normal, resting EEG. Evidence of significant nonlinearity in the EEG was obtained, replicating previous results. However, unlike EEG dimensional complexity, EEG nonlinearity (difference between original and surrogate data) was not affected by smoking. This indicates that, under resting conditions, smoking/nicotine may have a modulating effect on input from the reticular activating system, with such input having a global, linearizing effect on cortical dynamics. Nonlinear dynamics resulting from intracortical processes appear not to be affected.
Photic stimulation of the human visual system with uniform but intermittent illumination has been recognized for well over a century to give rise to the perception of complex visual patterns and forms. A review of the literature pertaining to these "stroboscopic patterns" is presented, along with a review of theories addressing their origins. The theories range from attributing the patterns to inhomogeneities in the structure of the eye, to the formation of functional domains of neural activity. A novel account of the origins of the patterns is developed in this paper utilizing advances in the neurobiology of information processing and nonlinear dynamics. This view conceptualizes the patterns as examples of "dissipative structures." These dissipative structures are conceived of as self-organizing macrostates of spatio-temporal coherence, widespread among visual cortical neurons. This self-organization is described in terms of nonlinear dynamics, in which sets of attractors emerge within a neural phase space undergoing constant change as the energy contained within successive volleys of stimulation-induced action currents is recursively dissipated by the visual system through cooperative feedback. The trajectories of activity formed by these attractors may reflect the actual form of the perceived patterns themselves and are thought to correspond to some of the reported properties of "linking fields" in the visual cortex.
In this paper is shown how to interpret the nonlinear dynamics of a class of one-dimensional physical systems exhibiting soliton behavior in terms of Killing fields for the associated dynamical laws acting as generators of torus knots. Soliton equations are related to dynamical laws associated with the intrinsic kinematics of space curves and torus knots are obtained as traveling wave solutions to the soliton equations. For the sake of illustration a full calculation is carried out by considering the Killing field that is associated with the nonlinear Schrodinger equation. Torus knot solutions are obtained explicitly in cylindrical polar coordinates via perturbation techniques from the circular solution. Using the Hasimoto map, the soliton conserved quantities are interpreted in terms of global geometric quantities and it is shown how to express these quantities as polynomial invariants for torus knots. The techniques here employed are of general interest and lead us to make some conjectures on natural links between the nonlinear dynamics of one-dimensional extended objects and the topological classification of knots.
We investigate nonlinear dynamical behaviors of operational amplifiers. When the output terminal of an operational amplifier is connected to the inverting input terminal, the circuit exhibits period-doubling bifurcation, chaos, and periodic windows, depending on the voltages of the positive and the negative power supplies. We study these nonlinear dynamical characteristics of this electronic circuit experimentally.
A gas-sensing system based on a dynamic nonlinear response is reported to improve the selectivity in the sensor response toward sample gases. A cyclic temperature composed of fundamental and second harmonics was applied to a SnO(2) semiconductor gas sensor and the resulting conductance of the sensor was analyzed by fast Fourier transformation (FFT). The dynamic nonlinear responses to the gas species were further characterized depending on the scanning profile of the temperature. These characteristic sensor responses under the application of second-harmonic perturbation were theoretically considered based on a reaction-diffusion model for the semiconductor surface.
By using the femtosecond laser spectroscopic techniques, we have studied the ultrafast response and the nonlinear optical properties of three molecules with donor-acceptor structure (denoted as T01, T02, and T03). Two-photon absorption (2PA) cross sections measured by the open aperture Z-scan technique were determined to be 77, 90, and 410 GM for T01, T02, and T03, respectively. The relaxation dynamics of the excited states were measured by two-color femtosecond pump-probe and time-resolved photoluminescence (PL) experiments. By changing the solvent from chloroform (CHCl3) to dimethyl sulfoxide (DMSO), the transient dynamics was found changed significantly and the decay time of PL emission decreased dramatically because DMSO with large dipole moment accelerates the cross-transfer process and the nonradiative process in the molecules.
With the discovery of chaos came the hope of finding simple models that would be capable of explaining complex phenomena. Numerous papers claimed to find low-dimensional chaos in a number of areas ranging from the weather to the stock market. Years later, many of these claims have been disproved and the fantastic hopes pinned on chaos have been toned down as research with more realistic objectives follows. The difficulty in calculating reliable estimates of the correlation dimension and the maximal Lyapunov exponent, two of the hallmarks of chaos, are explored. Given that nonlinear dynamics is a relatively new and growing field of science, the need for statistical testing is greater than ever. Surrogate data provides one possible approach but great care is needed in generating relevant surrogates and in interpreting the results. Examples of misleading applications and challenges for the future of research in nonlinear dynamics are discussed.
STUDY OBJECTIVES: Sleep deprivation can affect the waking EEG that may reflect information processing of the brain. We examined the effect of total sleep deprivation (TSD) on nonlinear dynamics of the waking EEG. DESIGN: Paired-group design. SETTING: A sleep disorders laboratory in a hospital. PARTICIPANTS: Twenty healthy male volunteers. INTERVENTIONS: Waking EEG data were recorded from subjects with eyes closed after (a) an 8-hour night's sleep and (b) TSD for 24 hours. The dimensional complexity (D2), as a nonlinear measure of complexity, of the EEG after a full night sleep were compared with those of the EEG after TSD. MEASUREMENTS AND RESULTS: The sleep-deprived states had lower D2 values at three channels (P4, O2, and C3) than normal states. CONCLUSIONS: TSD results in the decrease of complexity in the brain, which may imply sub-optimal information processing of the cerebral cortex. We suggest that the investigation of the relation between nonlinear dynamics of the waking EEG induced by TSD and cognitive performance may offer fruitful clues for understanding the role of sleep and the effects of sleep deprivation on brain function.
In the lowest order of approximation quasi-two-dimensional dynamics of planetary atmospheres and of plasmas in a magnetic field can be described by a common convective vortex equation, the Charney and Hasegawa-Mima (CHM) equation. In contrast to the two-dimensional Navier-Stokes equation, the CHM equation admits "shielded vortex solutions" in a homogeneous limit and linear waves ("Rossby waves" in the planetary atmosphere and "drift waves" in plasmas) in the presence of inhomogeneity. Because of these properties, the nonlinear dynamics described by the CHM equation provide rich solutions which involve turbulent, coherent and wave behaviors. Bringing in nonideal effects such as resistivity makes the plasma equation significantly different from the atmospheric equation with such new effects as instability of the drift wave driven by the resistivity and density gradient. The model equation deviates from the CHM equation and becomes coupled with Maxwell equations. This article reviews the linear and nonlinear dynamics of the quasi-two-dimensional aspect of plasmas and planetary atmosphere starting from the introduction of the ideal model equation (CHM equation) and extending into the most recent progress in plasma turbulence.
In this paper, we investigated the acoustic characteristics of sustained and running vowels from normal subjects and patients with laryngeal pathologies. Perturbation methods (including jitter and shimmer), signal-to-noise ratio (SNR), and nonlinear dynamic methods (such as correlation dimension and second-order entropy) were used to analyze sustained and running vowels. We found that the sustained vowels and running voices from normal subjects and patients with laryngeal pathologies had low-dimensional dynamic characteristics. For sustained vowels, the analyses of jitter, shimmer, correlation dimension, and second-order entropy revealed significant differences between normal and pathological voices. For running voices, jitter and shimmer did not statistically discriminate between normal and pathological voices, but a significant difference was found for SNR, correlation dimension, and second-order entropy. The results suggest that nonlinear dynamic analysis and traditional SNR analysis may be valuable for the analysis of sustained and running vowels; perturbation analysis may be applicable for the analysis of sustained vowels but should be applied with caution for running voice analysis.
Quasicrystals are unique structures with long-range order but no periodicity. Their properties have intrigued scientists ever since their discovery and initial theoretical analysis. The lack of periodicity excludes the possibility of describing quasicrystal structures with well-established analytical tools, including common notions like Brillouin zones and Bloch's theorem. New and unique features such as fractal-like band structures and 'phason' degrees of freedom are introduced. In general, it is very difficult to directly observe the evolution of electronic waves in solid-state atomic quasicrystals, or the dynamics of the structure itself. Here we use optical induction to create two-dimensional photonic quasicrystals, whose macroscopic nature allows us to explore wave transport phenomena. We demonstrate that light launched at different quasicrystal sites travels through the lattice in a way equivalent to quantum tunnelling of electrons in a quasiperiodic potential. At high intensity, lattice solitons are formed. Finally, we directly observe dislocation dynamics when crystal sites are allowed to interact with each other. Our experimental results apply not only to photonics, but also to other quasiperiodic systems such as matter waves in quasiperiodic traps, generic pattern-forming systems as in parametrically excited surface waves, liquid quasicrystals, and the more familiar atomic quasicrystals.
The mammalian ventilatory behaviour exhibits nonlinear dynamics as reflected by certain nonlinearity or complexity indicators (e.g. correlation dimension, approximate entropy, Lyapunov exponents, etc.) but this is not sufficient to determine its possible chaotic nature. To address this, we applied the noise titration technique, previously shown to discern and quantify chaos in short and noisy time series, to ventilatory flow recordings obtained in quietly breathing normal humans. Nine subjects (8 men and 1 woman, 24-42 years) were studied during 15-min epochs of ventilatory steady-state (10.1+/-3.0 breaths/min, tidal volume 0.63+/-0.2 L). Noise titration applied to the unfiltered signals subsampled at 5 Hz detected nonlinearity in all cases (noise limit 20.2+/-12.5%). Noise limit values were weakly correlated to the correlation dimension and the largest Lyapunov exponent of the signals. This study shows that the noise titration approach evidences a chaotic dimension to the behavior of ventilatory flow over time in normal humans during tidal breathing.