Accelerating cycle expansions by dynamical conjugacy

Periodic orbit theory provides two important functions---the dynamical zeta function and the spectral determinant for the calculation of dynamical averages in a nonlinear system. Their cycle expansions converge rapidly when the system is uniformly hyperbolic but greatly slowed down in the presence of non-hyperbolicity. We find that the slow convergence can be associated with singularities in the natural measure. A properly designed coordinate transformation may remove these singularities and results in a dynamically conjugate system where fast convergence is restored. The technique is successfully demonstrated on several examples of one-dimensional maps and some remaining challenges are discussed

Time-Frequency Analysis of Cardiovascular Signals and Their Dynamic Interactions

Cardiovascular signals are intrinsically non-stationary and interact through dynamic mechanisms to maintain blood pressure homeostasis in response to internal and external perturbations. The assessment of changes in cardiovascular signals and in their interactions provides valuable information regarding the cardiovascular function. Time-frequency analysis is a useful tool to study the time-varying nature of the cardiovascular system because it provides a joint representation of a signal in the temporal and spectral domain that allows to track the instantaneous frequency, amplitude and phase of non-stationary processes. The time-frequency distributions described in this chapter belong to the Cohen’s class, and can be derived from the Wigner-Ville distribution, which represents the fundamental basis of this unified framework. Time-frequency analysis can be extended to the study of the dynamic interactions between two or more non-stationary processes. Time-frequency coherence, phase-delay, phase-locking and partial-spectra are estimators that assess changes in the coupling and phase shift of signals generated by a complex system. This chapter introduces the reader to multivariate time-frequency analysis and covers both theoretical and practical aspects. The application of these methodologies in the study of the dynamic interactions between the most important variables of the cardiovascular function is discussed. In the introduction, classical spectral analysis of cardiovascular signals is reviewed along with its physiological interpretation. The limitations of this framework provides a motivation for implementing non-stationary tools. In the first section, time-frequency representations based on the Wigner-Ville distribution are introduced and important aspects, such as the interference cross-terms and their elimination, the time and frequency resolution and the estimation of time-frequency spectra, are described. The second section describes algorithms to assess the dynamic interactions between non-stationary signals, including time-frequency coherence, phase delay and partial spectra, while the third section provides examples of multivariate time-frequency analysis of cardiovascular data recorded during a standard test to induce an autonomic response