Crossing bifurcations and unstable dimension variability

A crisis is a global bifurcation in which a chaotic attractor has a discontinuous change in size or suddenly disappears as a scalar parameter of the system is varied. In this Letter, we describe a global bifurcation in three dimensions which can result in a crisis. This bifurcation does not involve a tangency and cannot occur in maps of dimension smaller than 3. We present evidence of unstable dimension variability as a result of the crisis. We then derive a new scaling law describing the density of the new portion of the attractor formed in the crisis. We illustrate this new type of bifurcation with a specific example of a three-dimensional chaotic attractor undergoing a crisis.     

Crisis and unstable dimension variability in the bailout embedding map

The dynamics of inertial particles in 2-d incompressible flows can be modeled by 4-d bailout embedding maps. The density of the inertial particles, relative to the density of the fluid, is a crucial parameter which controls the dynamical behaviour of the particles. We study here the dynamical behaviour of aerosols, i.e. particles heavier than the flow. An attractor widening and merging crisis is seen in the phase space in the aerosol case. Crisis-induced intermittency is seen in the time series and the laminar length distribution of times before bursts give rise to a power law with the exponent β = −1/3. The maximum Lyapunov exponent near the crisis fluctuates around zero indicating unstable dimension variability (UDV) in the system. The presence of unstable dimension variability is confirmed by the behaviour of the probability distributions of the finite time Lyapunov exponents.      

Crisis-induced unstable dimension variability in a dynamical system

Unstable dimension variability is an extreme form of non-hyperbolic behavior in chaotic systems whose attractors have periodic orbits with a different number of unstable directions. We propose a new mechanism for the onset of unstable dimension variability based on an interior crisis, or a collision between a chaotic attractor and an unstable periodic orbit. We give a physical example by considering a high-dimensional dissipative physical system driven by impulsive periodic forcing.     

Periodic orbit analysis of the Lorenz attractor

We present an application of the periodic orbit formalism to the Lorenz attractor at the standard parameter values. The symbolic encoding of trajectories, the effects of symmetries and scaling properties of trajectories are discussed. Good results for the Hausdorff dimension and the Lyapunov exponent are obtained. The classical spectral density is computed and positions and widths of resonances are compared with those found in correlation functions.     

Periodic orbit analysis demonstrates genetic constraints, variability, and switching in Drosophila courtship behavior

We use symbolic dynamics to describe Drosophila courtship communication. We posit that behavior should be defined in terms of irreducible periodic orbits of fundamental acts. This leads to a first operational definition of behavior, which allows for a fine grained quantitative analysis of behavior. We obtain evidence that during Drosophila courtship, individual characteristics of the protagonists are exchanged (predominantly from the male to the female) and that males in the presence of fruitless males perform a behavioral switch from male to female behavior.     

Universality of fractal dimension at the onset of period-doubling Chaos

We present numerical evidence for the hypothesis that, at the threshold of period-doubling chaos in a dynamical system, the fractal dimension of the associated strange attractor assumes a universal value.     

Mechanisms for the development of unstable dimension variability and the breakdown of shadowing in coupled chaotic systems

A chaotic attractor containing unstable periodic orbits with different numbers of unstable directions is said to exhibit unstable dimension variability (UDV). We present general mechanisms for the progressive development of UDV in uni- and bidirectionally coupled systems of chaotic elements. Our results are applicable to systems of dissimilar elements without invariant manifolds. We also quantify the severity of UDV to identify coupling ranges where the shadowability and modelability of such systems are significantly compromised.     

Periodic orbit expansions for the Lorentz gas

We apply the periodic orbit expansion to the calculation of transport, thermodynamic, and chaotic properties of the finite-horizon triangular Lorentz gas. We show numerically that the inverse of the normalized Lyapunov number is a good estimate of the probability of an individual periodic orbit. We investigate the convergence of the periodic orbit expansion and compare it with the convergence of the cycle expansions obtained from the Ruelle dynamical -function. For this system with severe pruning we find that applying standard convergence acceleration schemes to the periodic orbit expansion is superior to the dynamical -function approach. The averages obtained from the periodic orbit expansion are within 8% of the values obtained from direct numerical time and ensemble averaging. None of the periodic orbit expansions used here is computationally competitive with the standard simulation approaches for calculating averages. However, we believe that these expansion methods are of fundamental importance, because they give a direct route to the phase space distribution function.     

Periodic orbit quantization of the anisotropic Kepler problem

The periodic orbit quantization on the anisotropic Kepler problem is tested. By computing the stability and action of some 2000 of the shortest periodic orbits, the eigenvalue spectrum of the anisotropic Kepler problem is calculated. The aim is to test the following claims for calculating the quantum spectrum of classically chaotic systems: (1) Curvature expansions of quantum mechanical zeta functions offer the best semiclassical estimates; (2) the real part of the cycle expansions of quantum mechanical zeta functions cut at appropriate cycle length offer the best estimates; (3) cycle expansions are superfluous; and (4) only a small subset of cycles (irreducible cycles) suffices for good estimates for the eigenvalues. No evidence is found to support any of the four claims.     

Periodic solutions and flip bifurcation in a linear impulsive system

In this paper, the dynamical behaviour of a linear impulsive system is discussed both theoretically and numerically. The existence and the stability of period-one solution are discussed by using a discrete map. The conditions of existence for flip bifurcation are derived by using the centre manifold theorem and bifurcation theorem. The bifurcation analysis shows that chaotic solutions appear via a cascade of period-doubling in some interval of parameters. Moreover, the periodic solutions, the bifurcation diagram, and the chaotic attractor, which show their consistence with the theoretical analyses, are given in an example.     

Periodic orbit quantization of chaotic maps by harmonic inversion

A method for the semiclassical quantization of chaotic maps is proposed, which is based on harmonic inversion. The power of the technique is demonstrated for the baker's map as a prototype example of a chaotic map.     

Accuracy and variability of acoustic measures of voicing onset

Five commonly used methods for determining the onset of voicing of syllable-initial stop consonants were compared. The speech and glottal activity of 16 native speakers of Cantonese with normal voice quality were investigated during the production of consonant vowel (CV) syllables in Cantonese. Syllables consisted of the initial consonants /ph/, /th/, /kh/, /p/, /t/, and /k/ followed by the vowel /a/. All syllables had a high level tone, and were all real words in Cantonese. Measurements of voicing onset were made based on the onset of periodicity in the acoustic waveform, and on spectrographic measures of the onset of a voicing bar (f0), the onset of the first formant (F1), second formant (F2), and third formant (F3). These measurements were then compared against the onset of glottal opening as determined by electroglottography. Both accuracy and variability of each measure were calculated. Results suggest that the presence of aspiration in a syllable decreased the accuracy and increased the variability of spectrogram-based measurements, but did not strongly affect measurements made from the acoustic waveform. Overall, the acoustic waveform provided the most accurate estimate of voicing onset; measurements made from the amplitude waveform were also the least variable of the five measures. These results can be explained as a consequence of differences in spectral tilt of the voicing source in breathy versus modal phonation.     

Periodic orbit theory in acoustics: spectral fluctuations in circular and annular waveguides

Formulas based on the theory of Weyl are widely used to obtain the average number of modes at or below a given frequency in acoustic and vibrational waveguides. These formulas are valid at asymptotically high frequencies; at finite frequencies they are subject to some error, due to fluctuations in the mode count, which depend on the shape of the waveguide. The periodic orbit theory of semiclassical physics is used to give estimates of the variance of these fluctuations and these results are compared with numerical estimates based on eigenvalues obtained by root-finding. The comparison is good but shows errors that can be related to the nature of the periodic orbit theory. Engineering formulas are provided that give an accurate approximation without significant computational cost. The results are valid for membranes, ducts, and thin plates with clamped and/or simply supported boundary conditions.     

Stability analysis of coupled map lattices at locally unstable fixed points

Numerical simulations of coupled map lattices (CMLs) and other complex model systems show an enormous phenomenological variety that is difficult to classify and understand. It is therefore desirable to establish analytical tools for exploring fundamental features of CMLs, such as their stability properties. Since CMLs can be considered as graphs, we apply methods of spectral graph theory to analyze their stability at locally unstable fixed points for different updating rules, different coupling scenarios, and different types of neighborhoods. Numerical studies are found to be in excellent agreement with our theoretical results.