Compactification for asymptotically autonomous dynamical systems: theory, applications and invariant manifolds

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.

This paper reviews modern geometrical dynamics and control of humanoid robots. This general Lagrangian and Hamiltonian formalism starts with a proper definition of humanoid's configuration manifold, which is a set of all robot's active joint angles. Based on the ‘covariant force law’, the general humanoid's dynamics and control are developed. Autonomous Lagrangian dynamics is formulated on the associated ‘humanoid velocity phase space’, while autonomous Hamiltonian dynamics is formulated on the associated ‘humanoid momentum phase space’. Neural-like hierarchical humanoid control naturally follows this geometrical prescription. This purely rotational and autonomous dynamics and control is then generalized into the framework of modern non-autonomous biomechanics, defining the Hamiltonian fitness function. The paper concludes with several simulation examples.

The phase transitions of charged Anti-de Sitter (AdS) black holes are characterized by studying null geodesics in the vicinity of the critical curve of photon trajectories around black holes as well as their optical appearance as the black hole images. In the present work, the critical parameters including the orbital half-period $$\tau ,$$ τ , the angular Lyapunov exponent $$\lambda _L,$$ λ L , and the temporal Lyapunov exponent $$\gamma _L$$ γ L are employed to characterize black hole phase transitions within both the extended phase space and holographic thermodynamics frameworks. Under certain conditions, we observe multi-valued function behaviors of these parameters as functions of bulk pressure and temperature in the respective approaches. We propose that $$\tau ,$$ τ , $$\lambda _L,$$ λ L , and $$\gamma _L$$ γ L can serve as order parameters due to their discontinuous changes at first-order phase transitions. To validate this, we provide detailed analytical calculations demonstrating that these optical critical parameters follow scaling behavior near the critical phase transition point. Notably, the critical exponents for these parameters are found to be 1/2, consistent with those of the van der Waals fluid. Our findings suggest that static and distant observers can study black hole thermodynamics by analyzing the images of regions around the black holes.

We consider the gauged model of Siegel type chiral boson with a Lorentz non-covariant mass-like term for the gauge fields which is found to be equivalent to a model of boson where the kinetic term is like the kinetic term of Floreanini-Jackiw type chiral boson. We carry out the quantization of gauge non-invariant version this model in both the Lagrangian and Hamiltonian formulation. The quantization of the gauge-invariant version of this model in the extended phase space also has been carried out in the Lagrangian formulation. The gauge-invariant version of this model in the extended phase space is found to map onto the physical phase space with the appropriate gauge fixing condition. BRST symmetry associated with this model has been studied with different gauge fixing terms. It has been shown that the same model shows off-shell as well as on-shell BRST invariance depending on the choice of gauge fixing term.

Geometrical thermodynamics and P-V criticality of charged accelerating AdS black holes

For charged black holes in Horava-Lifshitz gravity, a second order phase transition takes place in extended phase space where the cosmological constant is taken as thermodynamic pressure. We relate the second order nature of phase transition to the fact that the phase transition occurs at a sharp temperature and not over a temperature interval. Once we know the continuity of the first derivatives of the Gibbs free energy, we show that all the Ehrenfest equations are readily satisfied. We study the effect of the perturbation of the cosmological constant as well as the perturbation of the electric charge on thermodynamic stability of Horava-Lifshitz black hole. We also use thermodynamic geometry to study phase transition in extended phase space. We investigate the behavior of scalar curvature of Weinhold, Ruppeiner, and Quevedo metric in extended phase space of charged Horava-Lifshitz black holes. It is checked if these curvatures could reproduce the result of specific heat for the phase transition.

The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated by Nipp and Stoffer, Numer. Math. 70 (1995), 245–257, for Runge-Kutta methods and in K. Nipp and D. Stoffer, Numer. Math. 74 (1996), 305–323, for linear multistep methods. As main result of this paper it is shown that also for Runge-Kutta methods and linear multistep methods applied to a index-2 problem of Hessenberg form there is a (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation.

Chaos and Predictable Time-Scale of the Brown Planthopper Nilaparvata Lugens (Stål) Occurrence System

In general relativity, it is difficult to localise observables such as energy, angular momentum, or centre of mass in a bounded region. The difficulty is that there is dissipation. A self-gravitating system, confined by its own gravity to a bounded region, radiates some of the charges away into the environment. At a formal level, dissipation implies that some diffeomorphisms are not Hamiltonian. In fact, there is no Hamiltonian on phase space that would move the region relative to the fields. Recently, an extension of the covariant phase space has been introduced to resolve the issue. On the extended phase space, the Komar charges are Hamiltonian. They are generators of dressed diffeomorphisms. While the construction is sound, the physical significance is unclear. We provide a critical review before developing a geometric approach that takes into account dissipation in a novel way. Our approach is based on metriplectic geometry, a framework used in the description of dissipative systems. Instead of the Poisson bracket, we introduce a Leibniz bracket - a sum of a skew-symmetric and a symmetric bracket. The symmetric term accounts for the loss of charge due to radiation. On the metriplectic space, the charges are Hamiltonian, yet they are not conserved under their own flow.

In this paper, by proposing a generalized $specific~volume$, we restudy the $P-V$ criticality of charged AdS black holes in the extended phase space. The results show that most of the previous conclusions can be generalized without change, but the ratio $\tilde{\rho}_c$ should be $3 \tilde{\alpha}/16$ in general case. Further research on the thermodynamical phase transition of black hole leads us to a natural interpretation of our assumption, and more black hole properties can be generalized. Finally, we study the number density for charged AdS black hole in higher dimensions, the results show the necessity of our assumption.

In this paper, we use Conley index theory to develop necessary and sufficient conditions for stability of equilibrium and periodic solutions of nonlinear continuous-time, discrete-time, and impulsive dynamical systems. The Conley index is a topological generalization of Morse theory which has been developed to analyze dynamical systems using topological methods. In particular, the Conley index of an invariant set with respect to a dynamical system is defined as the relative homology of an index pair for the invariant set. The Conley index can then be used to examine the structure of the system invariant set as well as the system dynamics within the invariant set, including system stability properties. Efficient numerical algorithms using homology theory have been developed in the literature to compute the Conley index and can be used to deduce the stability properties of nonlinear dynamical systems

Most dynamical systems in real-life applications are usually nonlinear and nonstationary, such as biomechanical systems, neuroscience, healthcare, economics, and engineering. Nowadays, the availability of data makes it possible to investigate and discover the hidden dynamics of target systems. However, numerous concerns arise in modeling and analysis of such systems, including complex system structure, data-inherent uncertainty in observations, poor interpretability, etc. To this end, we will investigate dynamical behaviors by proposing data-driven machine learning and dynamical system techniques. Specifically, we present various approaches for extracting characteristics or patterns that have good interpretability in various complex dynamical system applications.In the first study, we consider a pattern recognition problem in biomechanical system using multivariate human kinematic data and propose data-driven analytical methods to model human movement. We first perform recurrence quantification analysis on the phase space of the kinematic system to visualize and evaluate the recurrence behavior of the folded phase space trajectory. However, this method could miss out the relationship between variables. We therefore introduce a new modeling approach based on the self-expressive assumption to discover the interactions between variables and characterize the patterns of human motion. Our model measures correlations between variables by underlying linear subspaces, which can be further extended to nonlinear subspaces. Based on these correlations, our model improves the classification accuracy by 20% - 30% over other standard state-of-art correlation measurements and clustering methods. In the second study, we consider a specific neuroscience problem, seizure detection, to recognize abnormal brain states by learning from highly nonlinear electroencephalography. Based on dynamical systems theory, Koopman operator theory, and dynamic mode decomposition, we project the complex brain dynamics into an infinite-dimensional linear space, where the intrinsic linear dynamic shares the same dynamical behavior as the original nonlinear dynamical system. We propose a new system measurement to unify the projections in different Koopman spaces. Our measurement indicates the stability of the dynamical system, which represents the system state trajectory with respect to the corresponding attractor. We demonstrate the effectiveness of our approach with a seizure onset detection task. In this case study, our model gives high accuracy with a low false-positive rate. In the last study, we consider dynamical systems with multiple modalities. We design a Koopman-based deep learning network to identify and simplify the intrinsic nonlinear dynamics from different sources/observations of a multi-modal dynamical system. Since these different modalities describe the same system, they jointly share the same intrinsic dynamics. We develop an autoencoder framework with custom loss functions to find proper project functions/operators and to obtain the intrinsic Koopman coordinates. Our framework can reconstruct the original system states based on the joint dynamics. Once the intrinsic linear dynamics is learned, future system states can be predicted by the evolution in the Koopman space. This approach allows us to uncover and linearize the hidden nonlinear dynamics, which enables linear dynamical tools, for example, system control, spectrum analysis, to analyze complex dynamical systems.--Author's abstract

325 Food does not affect the pharmacokinetics of CS-7017 in healthy subjects: results from an open label, phase I, two-treatment, three-period, crossover study

Parallel Markov Chain Monte Carlo (pMCMC) algorithms generate clouds of proposals at each step to efficiently resolve a target probability distribution μ. We build a rigorous foundational framework for pMCMC algorithms that situates these methods within a unified ‘extended phase space’ measure-theoretic formalism. Drawing on our recent work that provides a comprehensive theory for reversible single-proposal methods, we herein derive general criteria for multiproposal acceptance mechanisms that yield ergodic chains on general state spaces. Our formulation encompasses a variety of methodologies, including proposal cloud resampling and Hamiltonian methods, while providing a basis for the derivation of novel algorithms. In particular, we obtain a top-down picture for a class of methods arising from ‘conditionally independent’ proposal structures. As an immediate application of this formalism, we identify several new algorithms including a multiproposal version of the popular preconditioned Crank–Nicolson (pCN) sampler suitable for high- and infinite-dimensional target measures that are absolutely continuous with respect to a Gaussian base measure. To supplement the aforementioned theoretical results, we carry out a selection of numerical case studies that evaluate the efficacy of these novel algorithms. First, noting that the true potential of pMCMC algorithms arises from their natural parallelizability and the ease with which they map to modern high-performance computing architectures, we provide a limited parallelization study using TensorFlow and a graphics processing unit to scale pMCMC algorithms that leverage as many as 100k proposals at each step. Second, we use our multiproposal pCN algorithm (mpCN) to resolve a selection of problems in Bayesian statistical inversion for partial differential equations motivated by fluid measurement. These examples provide preliminary evidence of the efficacy of mpCN for high-dimensional target distributions featuring complex geometries and multimodal structures.

A semiquantal analysis of condensed phase chemical dynamics, outlined recently for a double-well linearly coupled to dissipative harmonic bath, is formulated in detail to clarify its general features as well as the specifics of the linear and quadratic coupling cases. The theory may be called a "semiquantal time-dependent Hartree (SQTDH)" approach, as it assumes a factorized product of the squeezed coherent state wave packets for the variational subspace of the many-dimensional time-dependent wave function. Due to this assumption, it straightforwardly satisfies the canonicity condition introduced by Marumori et al. is described by a set of Hamilton equations of motion in an extended phase space that includes auxiliary coordinates representing the wave packet widths. The potential in the extended phase space provides a pictorial understanding of the quantum effects affected due to the bath coupling, e.g., suppression of the wave packet spreading in terms of the potential wall developing along the auxiliary coordinates. The idea is illustrated by prototypical models of quartic double-well and cubic metastable potentials linearly and quadratically coupled to the bath. Further applications and extensions, where the SQTDH method will offer a practical approach for introducing quantum effects into realistic molecular dynamics simulations, are also discussed.