Simple Dynamical System Research Articles

Time as a dynamical variable

Clocks are dynamical systems, notwithstanding our usual formalism in which time is treated as an external parameter. A dynamical variable which may be identified with time is explicitly constructed for a variety of simple dynamical systems. This variable is canonically conjugate to the Hamiltonian. The complications brought about by the semiboundedness of the Hamiltonian are taken into account and the spectral properties of the time operator are examined. For relativistic systems boosts affect the time variable in a familiar manner. It is pointed out that all physical clock times are cyclic, contrary to Newton's view of a ‘uniformly flowing time’.

Phase reset and dynamic stability during human gait

The human walking movement shows transient changes in response to single short-lived external perturbations, termed “stumbling reactions.” During the stumbling reactions, the walking phase is reset. It has been considered that the reactions contribute to stabilizing the motion, but less evidence bridging between the rhythm reset and the dynamic stability of the gait has been provided. The present study tries to establish the relationship between them. To this end, we construct a simple dynamical system model of the human musculo-skeletal system interacting with the ground, whose joint kinematics during walking is constrained by a given periodic joint-angles-profile. We show first that the model can exhibit a stable limit cycle corresponding to the steady walking with no perturbations. The responses of the limit cycle oscillation are examined by applying a type of perturbations at various timings with various intensities, elucidating the stability of the model’s walking when no phase reset is performed. We then observe that modifications of the periodic joint-angles-profile within a short time interval in response to the perturbation can alter the responses of the limit cycle oscillation and induce phase reset of the model’s walking. It is shown that appropriate amounts of the phase reset can prevent the model from falling, even for the perturbation that induces falling in the case without the phase reset. This suggests that those phase resets can improve the dynamic stability of the gait. Moreover, the appropriate phase resets predicted by the model are compared with the experimentally observed phase resets during human stumbling reaction to show they share similar characteristics.

Strange Attractors in Periodically-Kicked Limit Cycles and Hopf Bifurcations

We prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with SRB measures when certain simple dynamical systems are kicked at periodic time intervals. The settings considered include limit cycles and stationary points undergoing Hopf bifurcations.

Impact of the 11-yr solar activity on the QBO in the climate system

The QBO (quasi-biennial oscillation) in the climate system, with a mean cycle-length slightly above or below 2 years, is studied in a simple forced dynamical system. The fundamental cause of the quasi-biennial periodicity of the QBO is nonlinear resonance of the system to the seasonal forcing that is modulated by the 11-yr solar cycle. For a given nonlinearity, the cycle-length and the amplitude of the QBO depend on the intensity of both the unmodulated seasonal cycle and the 11-yr solar cycle, which may be one of the reasons why the QBO properties in climate vary with time and space.

Understanding and control of a simple dynamic system

AbstractThis study examined laypeople's understanding of a simple dynamic system, expressed in reasoning and strategies used by the subjects, and how it affected performance. Participants were 15 undergraduate psychology students, 4 male and 11 female; median age was 24 years, ranging from 21 to 31 years. The subjects' task was to establish equilibrium in a simple predator‐and‐prey system. A task analysis was performed to identify the problem structure, the vital aspects of the task, and the ideal strategies to perform the task. The subjects' actual performance was compared to these strategies. The results revealed that, even though the task was structurally simple, it was still difficult. Much of these difficulties seemed to stem from a low ability to apply indirect reasoning and thinking in terms of discrete time steps instead of in terms of continuous time. Copyright © 2003 John Wiley & Sons, Ltd.

Measurement-induced decoherence and Gaussian smoothing of the Wigner distribution function

We study the problem of measurement-induced decoherence using the phase-space approach employing the Gaussian-smoothed Wigner distribution function. Our investigation is based on the notion that measurement-induced decoherence is represented by the transition from the Wigner distribution to the Gaussian-smoothed Wigner distribution with the widths of the smoothing function identified as measurement errors. We also compare the smoothed Wigner distribution with the corresponding distribution resulting from the classical analysis. The distributions we computed are the phase–space distributions for simple one-dimensional dynamical systems such as a particle in a square-well potential and a particle moving under the influence of a step potential, and the time–frequency distributions for high-harmonic radiation emitted from an atom irradiated by short, intense laser pulses.

Tropical Atlantic seasonal predictability: The roles of El Niño remote influence and thermodynamic air‐sea feedback

Recent studies suggest that the thermodynamic feedback and the remote influence of El Niño Southern Oscillation (ENSO) are two dominant factors affecting climate variability in the tropical Atlantic sector. Given that both of these processes are included in an atmospheric general circulation model (AGCM) coupled to a mixed layer ocean (ML), such a simple coupled system is expected to possess useful skills in predicting regional climate variability on seasonal time scales. This letter reports some preliminary results from a set of prediction experiments using a coupled AGCM‐ML model. These results show promise for prospects of seasonal climate prediction in the tropical Atlantic sector using the simple dynamical seasonal prediction system.

Linear Stochastic Dynamical System under Uncertain Load: Inverse Reliability Analysis

The reliability of a linear dynamical system driven by a partially known Gaussian load process, specified only through its total average energy, is studied. A simple dynamic parallel and series system reliability network is investigated for the failure analysis using the crossing theory of stochastic processes. The critical input power spectral density of the load process which maximizes the mean crossing rate of a parallel or series system network emerges to be fairly narrow banded and hence fails to represent the erratic nature of the random input realistically. Consequently, a tradeoff curve between the maximum mean crossing rate of the reliability network and the disorder in the input, quantitatively measured through its entropy rate, is generated. Mathematically, the generation of the tradeoff curve of nondominated solutions, known as the Pareto optimal set, leads to a nonlinear, nonconvex, and multicriteria optimization problem with conflicting objectives. A recently developed Pareto optimization te...

DECOHERENCE INDUCED TRANSITION FROM QUANTUM TO CLASSICAL DYNAMICS

Framework for a general discussion of environmentally induced classical properties, like superselection rules, privileged basis and classical behavior, in quantum systems with both finite and infinite number of degrees of freedom is proposed. A number of examples showing that classical properties do not have to be postulated as an independent ingredient are given. In particular, it is shown that infinite open quantum systems in some cases may behave like simple classical dynamical systems.

The motion of a fluid ellipsoid in a general linear background flow

The study of the motion of a fluid ellipsoid has a long and fascinating history stretching back originally to Laplace in the late 18th century. Recently, this subject has been revived in the context of geophysical fluid dynamics, where it has been shown that an ellipsoid of uniform potential vorticity remains an ellipsoid in a background flow consisting of horizontal strain, vertical shear, and uniform rotation. The object of the present work is to present a simple, appealing, and practical way of investigating the motion of an ellipsoid not just in geophysical fluid dynamics but in general. The main result is that the motion of an ellipsoid may be reduced to the evolution of a symmetric, 3×3 matrix, under the action of an arbitrary 3×3 ‘flow’ matrix. The latter involves both the background flow, which must be linear in the Cartesian coordinates at the surface of the ellipsoid, and the self-induced flow, which was given by Laplace.The resulting simple dynamical system lends itself ideally to both numerical and analytical study. We illustrate a few examples and then present a theory for the evolution of a vortex within a slowly varying background flow. We show that a vortex may evolve quasi-adiabatically, that is, it stays close to an equilibrium form associated with the instantaneous background flow. The departure from equilibrium, on the other hand, is proportional to the rate of change of the background flow.

INTERACTION OF A SOLITARY WAVE WITH AN EXTERNAL FORCE IN THE EXTENDED KORTEWEG–DE VRIES EQUATION

The interaction of a strongly nonlinear solitary wave with an external force is studied using the extended Korteweg–de Vries equation as a model. This equation has several different families of nonlinear wave solutions: solitons, the so-called "thick" solitons, algebraic solitons and breathers, depending upon the sign of the cubic nonlinear term. A simple nonlinear dynamical system of the second order for the amplitude and position of the solitary wave is derived, and used to study the interaction. Its solutions are investigated in the phase plane. The conditions for the capture or reflection of a solitary wave by a single localized external force are obtained, with an emphasis on the role of the cubic nonlinear term.

Experiments on Lagrangian transport in steady vortex-breakdown bubbles in a confined swirling flow

In a recent study, Sotiropoulos et al. (2001) studied numerically the chaotic particle paths in the interior of stationary vortex-breakdown bubbles that form in a closed cylindrical container with a rotating lid. Here we report the first experimental verification of these numerical findings along with new insights into the dynamics of vortex-breakdown bubbles. We visualize the Lagrangian transport within the bubbles using planar laser-induced fluorescence (LIF) and show that even though the flow fields are steady – from the Eulerian standpoint – the spatial distribution of the dye tracer varies continuously, and in a seemingly random manner, over very long observation intervals. This finding is consistent with the arbitrarily long šil'nikov transients of upstream-originating orbits documented numerically by Sotiropoulos et al. (2001). Sequences of instantaneous LIF images also show that the steady bubbles exchange fluid with the outer flow via random bursting events during which blobs of dye exit the bubble through the spiral-in saddle. We construct experimental Poincaré maps by time-averaging a sufficiently long sequence of instantaneous LIF images. Ergodic theory concepts (Mezić & Sotiropoulos 2002) can be used to formally show that the level sets of the resulting time-averaged light intensity field reveal the invariant sets (unmixed islands) of the flow. The experimental Poincaré maps are in good agreement with the numerical computations. We apply this method to visualize the dynamics in the interior of the vortex-breakdown bubble that forms in the wake of the first bubble for governing parameters in the steady, two-bubble regime. In striking contrast with the asymmetric image obtained for the first bubble, the time-averaged light intensity field for the second bubble is remarkably axisymmetric. Numerical computations confirm this finding and further reveal that the apparent axisymmetry of this bubble is due to the fact that orbits in its interior exhibit quasi-periodic dynamics. We argue that this stark contrast in dynamics should be attributed to differences in the swirl-to-axial velocity ratio in the vicinity of each bubble. By studying the bifurcations of a simple dynamical system, with manifold topology resembling that of a vortex-breakdown bubble, we show that sufficiently high swirl intensities can stabilize the chaotic orbits, leading to quasi-periodic dynamics.

Models of the predictability of a simple nonlinear dynamical system

Chaotic nonlinear low order systems are often regarded as paradigms of the atmospheric behaviour, not least because of their limited predictability. In this paper, the predictability of a forced nonlinear system proposed by Lorenz is examined. The system is a compelling heuristic model of the mid-latitude global circulation; it exhibits limited predictability. While weak error growth takes place throughout most of the phase space of the system, strong error growth is confined to very limited regions of phase space. A “discrete scattering” model of error growth is proposed.

Measuring Dynamical Prediction Utility Using Relative Entropy

A new parameter of dynamical system predictability is introduced that measures the potential utility of predictions. It is shown that this parameter satisfies a generalized second law of thermodynamics in that for Markov processes utility declines monotonically to zero at very long forecast times. Expressions for the new parameter in the case of Gaussian prediction ensembles are derived and a useful decomposition of utility into dispersion (roughly equivalent to ensemble spread) and signal components is introduced. Earlier measures of predictability have usually considered only the dispersion component of utility. A variety of simple dynamical systems with relevance to climate and weather prediction is introduced, and the behavior of their potential utility is analyzed in detail. For the climate systems examined here, the signal component is at least as important as the dispersion in determining the utility of a particular set of initial conditions. The simple ‘‘weather’’ system examined (the Lorenz system) exhibited different behavior with the dispersion being more important than the signal at short prediction lags. For longer lags there appeared no relation between utility and either signal or dispersion. On the other hand, there was a very strong relation at all lags between utility and the location of the initial conditions on the attractor.

Chaotic diffusion of particles with finite mass in oscillating convection flows.

Deterministic diffusion in temporally oscillating convection is studied for particles with finite mass. The particles are assumed to obey a simple dissipative dynamical system and the particle diffusion is induced by the strange attractor. The diffusion constants are numerically calculated for convection models with free and rigid boundary conditions.

Spurious Numerical Solutions in Coupled Natural Ventilation and Thermal Analyses

It is well known that iterative solution processes can lead to divergence when dealing with coupled airflow and thermal analyses for buildings ventilated either naturally or by a mixed-mode system. The Newton- Raphson method or its variants are used in almost all existing multi-zone airflow models. This paper discusses the qualitative features of the iterative solution processes of the Newton-Raphson method when used for coupled thermal and ventilation analyses of a simple one-zone building with two openings. Chaotic solutions are obtained for three different coupling strategies in combining the airflow and indoor air temperature calculations. The spurious solutions obtained in this paper are due to the inherent characteristics of the Newton-Raphson method. Our analyses on these spurious numerical solutions are based on a simple dynamical systems approach. Unfortunately, there seems to be no general approach to distinguish the true steady-state solution from the spurious numerical solutions in practical simulations. The fact that chaotic solutions were obtained for a very simple problem in this paper may indicate the need for a further understanding of its implications to more complex multi-zone problems. The findings in this paper may serve as a tool for further development of effective numerical methods for multi-zone airflows and may also provide useful knowledge for interpreting the divergence behaviours of some of the numerical simulations in building ventilation.

Relation between Kinematic Boundaries, Stirring, and Barriers for the Antarctic Polar Vortex

Maximum stretching lines in the lower stratosphere around the Antarctic polar vortex are diagnosed using a method based on finite-size Lyapunov exponents. By analogy with the mathematical results known for simple dynamical systems, these curves are identified as stable and unstable manifolds of the underlying hyperbolic structure of the flow. For the first time, the exchange mechanism associated with lobe dynamics is characterized using atmospheric analyzed winds. The tangling manifolds form a stochastic layer around the vortex. It is found that fluid is not only expelled from this layer toward the surf zone but also is injected inward from the surf zone, through a process similar to the turnstile mechanism in lobe dynamics. The vortex edge, defined as the location of the maximum gradient in potential vorticity or tracer, is found to be the southward (poleward) envelope of this stochastic layer. Exchanges with the inside of the vortex are therefore largely decoupled from those, possibly intense, exchanges between the stochastic layer and the surf zone. It is stressed that using the kinematic boundary defined by the hyperbolic points and the manifolds as an operational definition of vortex boundary is not only unpractical but also leads to spurious estimates of exchanges. The authors anticipate that more accurate dynamical systems tools are needed to analyze stratospheric transport in terms of lobe dynamics.

Oscillatory finite-time singularities in finance, population and rupture

We present a simple two-dimensional dynamical system where two nonlinear terms, exerting respectively positive feedback and reversal, compete to create a singularity in finite time decorated by accelerating oscillations. The power law singularity results from the increasing growth rate. The oscillations result from the restoring mechanism. As a function of the order of the nonlinearity of the growth rate and of the restoring term, a rich variety of behavior is documented analytically and numerically. The dynamical behavior is traced back fundamentally to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin. The interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log-periodic oscillations with remarkable scaling properties. Three domains of applications are discussed: (1) the stock market with a competition between nonlinear trend-followers and nonlinear value investors; (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity; (3) the failure of a material subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing.

Molecular amplification in a dynamic system by ammonium cations

Using hydrazone chemistry, a simple dynamic system of macrocyclic pseudo-peptides was prepared which contains a cyclic structure that resembles a known receptor for ammonium cations. A series of ammonium salts were tested as templates and the expected receptor was amplified in different degrees according to the template used. Competitive non-covalent interactions with a crown ether were used to demonstrate in situ that the templated system is still dynamic. Formation of host–guest complexes was proven by ESI-MS and 1H ROESY experiments. Relative affinities of the various templates for the isolated receptor correlate well with the effect they produce on the library composition.

Non-stationary characteristics of instability in a single-mode Nd:YVO 4 laser with fiber feedback

Random chaotic burst generation was experimentally observed in a single-mode microchip Nd:YVO4 laser with ber feedback. As the feedback strength was increased, a transition from stable relax- ation oscillation state to unstable random chaotic burst state appeared. Furthermore, the non-stationary characteristic of probability association was experimentally identied at the transition of the two states while similar characteristics were reported only by numerical simulations of simple dynamical systems. This implies the general feature of non-stationary property of the dynamic switching between two states at transition. The observed chaotic burst generation and non-stationary nature were reproduced numerically based on the Lang-Kobayashi model. PACS. 42.55.Rz Doped-insulator lasers and other solid state lasers