Asymptotic Behavior and Stability for Solutions of a Biochemical Reactor Distributed Parameter Model

This paper deals with the dynamical analysis of a tubular biochemical reactor. The existence of nonnegative state trajectories and the invariance of the set of all physically feasible state values under the dynamical equation as well as the convergence of the state trajectories to equilibrium profiles are proved. In addition, the existence of multiple equilibrium profiles is analyzed. It is proved that, under physically meaningful conditions, the system has two stable and one unstable equilibrium profiles.      

Lyapunov and converse Lyapunov theorems for stochastic semistability

This paper develops Lyapunov and converse Lyapunov theorems for stochastic semistable nonlinear dynamical systems. Semistability is the property whereby the solutions of a stochastic dynamical system almost surely converge to (not necessarily isolated) Lyapunov stable in probability equilibrium points determined by the system initial conditions. Specifically, we provide necessary and sufficient Lyapunov conditions for stochastic semistability and show that stochastic semistability implies the existence of a continuous Lyapunov function whose infinitesimal generator decreases along the dynamical system trajectories and is such that the Lyapunov function satisfies inequalities involving the average distance to the set of equilibria.     

Constructing subsystems for nonlinear controlled systems

We consider the problem of constructing subsystems representing simplified models of nonlinear controlled dynamical systems. Subsystems define a part of phase trajectories of the original controlled system.     

Analysis of the Caratheodory's Theorem on Dynamical System Trajectories Under Numerical Uncertainty

The current work proposes a new and constructive proof for the Caratheodory's theorem on existence and uniqueness of trajectories of dynamical systems. The key concern is the numerical uncertainty, i.e., the discrepancy between mathematical proofs, algorithms, and their implementations, which may affect the correct functioning of a control system. Due to growing demands on security and compliance with specifications, correctness of the control system functioning is becoming ever more important. Since in both dynamical systems and many control design approaches, one of the central notions is the system trajectory, it is important to address existence and uniqueness of system trajectories in a way which incorporates numerical uncertainty. Constructive analysis is a particular approach to formalizing numerical uncertainty and is used as the basis of the current work. The major difficulties of guaranteeing existence and uniqueness of system trajectories arise in the case of systems and controllers which possess discontinuities in time, since classical solutions to initial value problems do not exist. This issue is addressed in Caratheodory's theorem. A particular constructive variant of the theorem is proven which covers a large class of problems found in practice.      

Computing combinatorial types of trajectories in Pfaffian Dynamics

Suppose that the state space of a dynamical system has a finite partition, and each element of the partition is labelled by a letter of some alphabet. Then every trajectory of the system is naturally labelled by a word in this alphabet. This word is called the combinatorial type of the trajectory. In applications it is important to decide whether among a certain family of trajectories there is at least one trajectory of a given type, or whether all the trajectories in this family have the same type. In this paper we construct algorithms for solving this sort of questions for a wide class of Pfaffian dynamical systems, which have elementary (doubly-exponential) upper complexity bounds.     

System identification for chaotic integrate-and-fire dynamics

We discuss applications of the fact that dynamical state information can be reconstructed from a series of interspike interval (ISI) measurements. This system analysis allows system identification and prediction from spike train history. Secondly, using this reconstruction, unstable periodic trajectories of the underlying system can be controlled by small changes in a system parameter. The underlying assumption is an integrate-and-fire model coupling the dynamical system to the observable spike train. © 1997 John Wiley & Sons, Inc.     

Fractals in neurodynamics: From hybrid dynamical systems point of view

Recently, hybrid dynamical systems have attracted considerable attention in the automatic control domain. In this article, a theory for recurrent neural networks is presented from a hybrid dynamical systems point of view. The hybrid dynamical system is defined by a continuous dynamical system discretely switched by external temporal inputs. The theory suggests that the dynamics of continuous-time recurrent neural networks, which are stochastically excited by external temporal inputs, are generally characterized by a set of continuous trajectories with a fractal-like structure in hyper-cylindrical phase space. This work was presented, in part, at the 7th International Symposium on Artificial Life and Robotics, Oita, Japan, January 16–18, 2002     

Quantitative stability of dynamical systems

The stability of general dynamical systems is studied from a quantitative viewpoint (involving specific bounds on the system trajectories). Theorems yielding necessary and sufficient conditions for quantitative stability and asymptotic quantitative stability are proven.     

Reversibility and PoincarÉ Recurrence in Linear Dynamical Systems

In this paper, we study the Poincare recurrence phenomenon for linear dynamical systems, that is, linear systems whose trajectories return infinitely often to neighborhoods of their initial condition. Specifically, we provide several equivalent notions of Poincare recurrence and review sufficient conditions for nonlinear dynamical systems that ensure that the system exhibits Poincare recurrence. Furthermore, we establish necessary and sufficient conditions for Poincare recurrence in linear dynamical systems. In addition, we show that in the case of linear systems the absence of volume-preservation is equivalent to the absence of Poincare recurrence implying irreversibility of a dynamical system. Finally, we introduce the notion of output reversibility and show that in the case of linear systems, Poincare recurrence is a sufficient condition for output reversibility.     

A fuzzy information space approach to speech signal non‐linear analysis

A new approach for analyzing the similarity of dynamical systems is presented, with applications to speech analysis. This approach is based on a temporal fuzzy set representation of the trajectories of the dynamical system. The similarity between segments of the speech signal is determined via similarity measures of the corresponding temporal fuzzy sets. We present an application of the method to vowel recognition in the samples (amplitude–time) space. © 2000 John Wiley & Sons, Inc.     

On Characterizations of Exponential Stability of Nonlinear Discrete Dynamical Systems on Bounded Regions

In this paper, we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space. Through introducing two new concepts-the Lip constant of a nonlinear operator and strongly equivalent metrics of a norm-we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics, or if and only if the Lip constant of the nonlinear operator is less than one. In the latter case, we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the Lip constant. Based on the obtained results, we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the Lip constant and strongly equivalent metrics. The obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new, useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems.     

Global behavior of dynamical agents in directed network

This paper investigates the global behavior of controlled dynamical agents in directed networks. The agents are Lyapunov stable, are distributed in a line, and communicate through a directed network. The communication topology of the network is characterized by a directed graph and the control protocol is designed in simple linear decentralized feedback law. We study the different conditions under which agents will achieve aggregation, and critical and divergent trajectories, respectively. Our investigation on the dynamical agent system under network is extended to the time-delay network case. Furthermore, we study the case with two pre-specified virtual leaders in the system. Numerical simulations are given and demonstrate that our theoretical results are effective.     

Fast diffeomorphic matching to learn globally asymptotically stable nonlinear dynamical systems

We propose a new diffeomorphic matching algorithm and use it to learn nonlinear dynamical systems with the guarantee that the learned systems have global asymptotic stability. For a given set of demonstration trajectories, and a reference globally asymptotically stable time-invariant system, we compute a diffeomorphism that maps forward orbits of the reference system onto the demonstrations. The same diffeomorphism deforms the whole reference system into one that reproduces the demonstrations, and is still globally asymptotically stable.     

A Contraction Theory Approach to Stochastic Incremental Stability

We investigate the incremental stability properties of Ito stochastic dynamical systems. Specifically, we derive a stochastic version of nonlinear contraction theory that provides a bound on the mean square distance between any two trajectories of a stochastically contracting system. This bound can be expressed as a function of the noise intensity and the contraction rate of the noise-free system. We illustrate these results in the contexts of nonlinear observers design and stochastic synchronization.     

Phase Space Projection of Dynamical Systems

Dynamical systems are commonly used to describe the state of time-dependent systems. In many engineering and control problems, the state space is high-dimensional making it difficult to analyze and visualize the behavior of the system for varying input conditions. We present a novel dimensionality reduction technique that is tailored to high-dimensional dynamical systems. In contrast to standard general purpose dimensionality reduction algorithms, we use energy minimization to preserve properties of the flow in the high-dimensional space. Once the projection operator is optimized, further high-dimensional trajectories are projected easily. Our 3D projection maintains a number of useful flow properties, such as critical points and flow maps, and is optimized to match geometric characteristics of the high-dimensional input, as well as optional user constraints. We apply our method to trajectories traced in the phase spaces of second-order dynamical systems, including finite-sized objects in fluids, the circular restricted three-body problem and a damped double pendulum. We compare the projections with standard visualization techniques, such as PCA, t-SNE and UMAP, and visualize the dynamical systems with multiple coordinated views interactively, featuring a spatial embedding, projection to subspaces, our dimensionality reduction and a seed point exploration tool.     

Equilibrium Trajectories in Dynamical Bimatrix Games with Average Integral Payoff Functionals

This paper considers models of evolutionary non-zero-sum games on the infinite time interval. Methods of differential game theory are used for the analysis of game interactions between two groups of participants. We assume that participants in these groups are controlled by signals for the behavior change. The payoffs of coalitions are defined as average integral functionals on the infinite horizon. We pose the design problem of a dynamical Nash equilibrium for the evolutionary game under consideration. The ideas and approaches of non-zero-sum differential games are employed for the determination of the Nash equilibrium solutions. The results derived in this paper involve the dynamic constructions and methods of evolutionary games. Much attention is focused on the formation of the dynamical Nash equilibrium with players strategies that maximize the corresponding payoff functions and have the guaranteed properties according to the minimax approach. An application of the minimax approach for constructing optimal control strategies generates dynamical Nash equilibrium trajectories yielding better results in comparison to static solutions and evolutionary models with the replicator dynamics. Finally, we make a comparison of the dynamical Nash equilibrium trajectories for evolutionary games with the average integral payoff functionals and the trajectories for evolutionary games with the global terminal payoff functionals on the infinite horizon.     

Reduced Modeling of Unknown Trajectories

This paper deals with model order reduction of parametrical dynamical systems. We consider the specific setup where the distribution of the system’s trajectories is unknown but the following two sources of information are available: (i) some “rough” prior knowledge on the system’s realisations; (ii) a set of “incomplete” observations of the system’s trajectories. We propose a Bayesian methodological framework to build reduced-order models (ROMs) by exploiting these two sources of information. We emphasise that complementing the prior knowledge with the collected data provably enhances the knowledge of the distribution of the system’s trajectories. We then propose an implementation of the proposed methodology based on Monte-Carlo methods. In this context, we show that standard ROM learning techniques, such e.g., proper orthogonal decomposition or dynamic mode decomposition, can be revisited and recast within the probabilistic framework considered in this paper. We illustrate the performance of the proposed approach by numerical results obtained for a standard geophysical model.     

Coupling geometric analysis and viability theory for system exploration: Application to a living food system

This paper addresses the issue of studying a food complex system in a reverse engineering manner with the aim of identifying the set of all possible actions that makes it reach a quality target with respect to manufacturing constraints. Once the set of actions is identified, several criteria can be considered to identify interesting trajectories and control policies. A viability approach, coupling the viability theory and a geometric approach of robustness, is proposed to study complex dynamical systems. It can be implemented for several types of systems, from linear to non linear or hybrid systems. The proposed framework was adapted to a living food system: a ripening model of Camembert cheese to identify the set of states and actions (capture basin) from which it is possible to reach a predefined quality target. Within the set of viable trajectories, particular trajectories that improve the Camembert cheese ripening process are identified using the proposed approach. The results are applied at a pilot scale and are discussed in this paper.     

An adaptive neural control scheme for articulatory synthesis of CV sequences

Reproducing the smooth vocal tract trajectories is critical for high quality articulatory speech synthesis. This paper presents an adaptive neural control scheme for such a task using fuzzy logic and neural networks. The control scheme estimates motor commands from trajectories of flesh-points on selected articulators. These motor commands are then used to reproduce the trajectories of the underlying articulators in a 2nd order dynamical system. Initial experiments show that the control scheme is able to manipulate the mass-spring based elastic tract walls in a 2-dimensional articulatory synthesizer and to realize efficient speech motor control. The proposed controller achieves high accuracy during on-line tracking of the lips, the tongue, and the jaw in the simulation of consonant–vowel sequences. It also offers salient features such as generality and adaptability for future developments of control models in articulatory synthesis.     

五平衡点非线性系统的多核混沌运动分析

研究了外力矩作用下刚体转动动力学系统存在的混沌吸引子类型、数量和特性,分析了5个平衡点的属性。系统状态在5个平衡点邻域沿某方向的收敛和沿某方向的发散,可以形成各种各样的周期吸引子,单核、双核、三核、连体四核、单体四核混沌吸引子。该系统中包含的各类混沌运动形式复杂,吸引子种类繁多。