Cybernetic causality: A unitary theory of causal recursion in natural and social systems

The phenomena connected with non-linear difference and differential equations are approached from a unifying point of view, offered by the general concept of causal recursion (Sections 1 and 2). After a short discussion of nilpotent causal recursions (Section 3), the properties of full causal recursions near or steadily developing states are considered in detail (Section 4). In the general case of causal recursion, a theorem based on a differential-geometric argument is proved concerning the convergence of trajectories, and an exhaustive classification of systems having causal recursion is obtained (Section 5). It gives, among some other things, a mathematical foundation to the concept of a self-steering system (as distinguished from that of a self-regulating system), applied in recent sociocybernetic analyses.     

Dissipativity theory for discontinuous dynamical systems: Basic input, state, and output properties, and finite-time stability of feedback interconnections

In this paper, we develop dissipativity theory for discontinuous dynamical systems. Specifically, using set-valued supply rate maps and set-valued connective supply rate maps consisting of locally Lebesgue integrable supply rates and connective supply rates, respectively, and set-valued storage maps consisting of piecewise continuous storage functions, dissipativity properties for discontinuous dynamical systems are presented. Furthermore, extended Kalman–Yakubovich–Popov set-valued conditions, in terms of the discontinuous system dynamics, characterizing dissipativity via generalized Clarke gradients and locally Lipschitz continuous storage functions are derived. Finally, these results are used to develop feedback interconnection stability results for discontinuous dynamical systems by appropriately combining the set-valued storage maps for the forward and feedback systems.     

Unifying theory for stability of continuous,discontinuous, and discrete-time dynamical systems

Continuous-time dynamical systems whose motions are continuous with respect to time (called continuous dynamical systems), may be viewed as special cases of continuous-time dynamical systems whose motions are not necessarily continuous with respect to time (called discontinuous dynamical systems, or DDS). We show that the classical Lyapunov stability results for continuous dynamical systems are embedded in the authors’ stability results for DDS (given in [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474]), in the following sense: if the hypotheses for a given Lyapunov stability result for continuous dynamical systems are satisfied, then the hypotheses of the corresponding stability result for DDS are also satisfied. This shows that the stability results for DDS in [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] are much more general than was previously known, and that the quality of the DDS results therein is consistent with that of the classical Lyapunov stability results for continuous dynamical systems.By embedding discrete-time dynamical systems into a class of DDS with equivalent stability properties, we also show that when the hypotheses of the classical Lyapunov stability results for discrete-time dynamical systems are satisfied, then the hypotheses of the corresponding DDS stability results are also satisfied. This shows that the results for DDS in [H. Ye, A.N. Michel, L. Hou Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] are much more general than previously known, having connections even with discrete-time dynamical systems!Finally, we demonstrate by the means of a specific example that the stability results for DDS are less conservative than corresponding classical Lyapunov stability results for continuous dynamical systems.     

Cybernetic causality III: The qualitative theory of self-steering and social development

A mathematical connection, based on a representation postulate, is established between system dynamics and actor theory, as a result of which a general theorem concerning the double representation of causal recursion in action-systems can be proved (Section 1). For self-steering actors the theorem permits an indeterministic interpretation in terms of the existence of free will in such actors (Section 1.4). The laws of requisite variety and requisite hierarchy are connected with a stoachastic process of self-organization in complex self-regulating actors and actor-hierarchies, respectively (Sections 2 and 3). The mathematical apparatus created in Sections 1–3 is applied to a foundational study, in terms of dynamical systems, of the reasons for social development and underdevelopment (Section 4), and to an analysis of the governability of human society (Section 5).     

A theory for synchronization of dynamical systems

In this paper, a theory for synchronization of multiple dynamical systems under specific constraints is developed from a theory of discontinuous dynamical systems. The concepts on synchronization of two or more dynamical systems to specific constraints are presented. The synchronization, desynchronization and penetration of multiple dynamical systems to multiple specified constraints are discussed, and the necessary and sufficient conditions for such synchronicity are developed. The synchronicity of two dynamical systems to a single specific constraint and to multiple specific constraints is investigated. Finally, the synchronization and the corresponding complexity for multiple slave systems with multiple master systems are discussed briefly. The meaning of synchronization for dynamical systems with constraints is extended as a generalized, universal concept. The theory presented in this paper may be as a universal theory for dynamical systems. The paper provides a theoretic frame work in order to control slave systems which can be synchronized with master systems through specific constraints in a general sense.     

Multistability,bifurcations, and biological neural networks: A synaptic drive firing model for cerebral cortex transition in the induction of general anesthesia

This paper focuses on multistability theory for discontinuous dynamical systems having a set of multiple isolated equilibria and/or a continuum of equilibria. Multistability is the property whereby the solutions of a dynamical system can alternate between two or more mutually exclusive Lyapunov stable and convergent equilibrium states under asymptotically slowly changing inputs or system parameters. In this paper, we extend the definition and theory of multistability to discontinuous autonomous dynamical systems. In particular, nontangency Lyapunov-based tests for multistability of discontinuous systems with Filippov and Carathéodory solutions are established. The results are then applied to excitatory and inhibitory biological neuronal networks to explain the underlying mechanism of action for anesthesia and consciousness from a multistable dynamical system perspective, thereby providing a theoretical foundation for general anesthesia using the network properties of the brain.     

Stability results involving time-averaged Lyapunov function derivatives

The stability results which comprise the Direct Method of Lyapunov involve the existence of auxiliary functions (Lyapunov functions) endowed with certain definiteness properties. Although the Direct Method is very general and powerful, it has some limitations: there are dynamical systems with known stability properties for which there do not exist Lyapunov functions which satisfy the hypotheses of a Lyapunov stability theorem.In the present paper we identify a scalar switched dynamical system whose equilibrium (at the origin) has known stability properties (e.g., uniform asymptotic stability) and we prove that there does not exist a Lyapunov function which satisfies any one of the Lyapunov stability theorems (e.g., the Lyapunov theorem for uniform asymptotic stability). Using this example as motivation, we establish stability results which eliminated some of the limitations of the Direct Method alluded to. These results involve time-averaged Lyapunov function derivatives (TALFD’s). We show that these results are amenable to the analysis of the same dynamical systems for which the Direct Method fails. Furthermore, and more importantly, we prove that the stability results involving TALFD’s are less conservative than the results which comprise the Direct Method (which henceforth, we refer to as the classical Lyapunov stability results).While we confine our presentation to continuous finite-dimensional dynamical systems, the results presented herein can readily be extended to arbitrary continuous dynamical systems defined on metric spaces. Furthermore, with appropriate modifications, stability results involving TALFD’s can be generalized to discontinuous dynamical systems (DDS).     

Decomposition and simulation of sequential dynamical systems

Sequential dynamical systems have been developed as a basis for a theory of computer simulation. This paper contains a generalization of this concept. The notion of morphism of sequential dynamical systems is introduced, formalizing the concept of simulating one system by another. Several examples of morphisms are given. Using the morphism concept, it is shown that every sequential dynamical system decomposes uniquely into a product of indecomposable systems.     

New approach in dynamics of regenerative chatter research of turning

In this paper, regenerative chatter phenomena in a turning process is discussed from impulsive dynamical point of view. By introducing the instantaneous pulse when vibration occurs and the vibratory condition set, we optimize the models and present a certain kind of second-order impulsive differential systems, which is a specific discontinuous dynamical system. Then we search for the general results of the nonoccurrence of chatter phenomena by discussing the number of the vibration pulse times, utilizing the method of flow theory in discontinuous systems and transversal property at the boundary. Our results give a convenient way to estimate the available parameters to keep the turning process stable.     

Financial systems: a few theoretical and algebraic considerations for their modeling

In this paper I suggest that observable entities, usually named ‘financial systems’, may be related to the general conceptual framework of systems theory. Starting from the requisite properties of a (general) system, I derive a strong and operational concept for specific financial systems (Section 1). Then a general modeling procedure is proposed, mainly based upon graph theory (with an additional and complementary use of linear systems analysis), through which it is possible to establish the general static and almost dynamic properties of these specific systems and their implications for financial analysis itself (Section 2). A numerical example (Section 3) illustrates most of the concepts and ideas introduced throughout the paper.     

Periodic and chaotic synchronizations of two distinct dynamical systems under sinusoidal constraints

In this paper, periodic and chaotic synchronizations between two distinct dynamical systems under specific constraints are investigated from the theory of discontinuous dynamical systems. The analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator were obtained, and the invariant domain of sinusoidal synchronization is achieved. From analytical conditions, the control parameter map is developed. Numerical illustrations for partial and full sinusoidal synchronizations of chaotic and periodic motions of the controlled pendulum with the Duffing oscillator are carried out. This paper presents how to apply the theory of discontinuous dynamical systems to dynamical system synchronization with specific constraints. The function synchronization of two distinct dynamical systems with specific constraints should be identified only by G-functions. The significance of function synchronization of distinct dynamical systems is to make the synchronicity behaviors hidden, which is very useful for telecommunication synchronization and network security.     

Impulsive non‐autonomous dynamical systems and impulsive cocycle attractors

In this work, we define the notions of ‘impulsive non‐autonomous dynamical systems’ and ‘impulsive cocycle attractors’. Such notions generalize (we will see that not in the most direct way) the notions of autonomous dynamical systems and impulsive global attractors in the current published literature. We also establish conditions to ensure the existence of an impulsive cocycle attractor for a given impulsive non‐autonomous dynamical system, which are analogous to the continuous case. Moreover, we prove the existence of such attractor for a non‐autonomous 2D Navier–Stokes equation with impulses, using energy estimates. Copyright © 2016 John Wiley & Sons, Ltd.     

Dissipative differential inclusions,set-valued energy storage and supply rate maps,and stability of discontinuous feedback systems

In this paper, we develop dissipativity notions for dynamical systems with discontinuous vector fields. Specifically, we consider dynamical systems with Lebesgue measurable and locally essentially bounded vector fields characterized by differential inclusions involving Filippov set-valued maps specifying a set of directions for the system velocity and admitting Filippov solutions with absolutely continuous curves. In particular, we introduce a generalized definition of dissipativity for discontinuous dynamical systems in terms of set-valued supply rate maps and set-valued storage maps consisting of locally Lebesgue integrable supply rates and Lipschitz continuous storage functions, respectively. In addition, we introduce the notion of a set-valued available storage map and a set-valued required supply map, and show that if these maps have closed convex images they specialize to single-valued maps corresponding to the smallest available storage and the largest required supply of the differential inclusion, respectively. Furthermore, we show that all system storage functions are bounded from above by the largest required supply and bounded from below by the smallest available storage, and hence, a dissipative differential inclusion can deliver to its surroundings only a fraction of its generalized stored energy and can store only a fraction of the generalized work done to it. Moreover, extended Kalman–Yakubovich–Popov conditions, in terms of the discontinuous system dynamics, characterizing dissipativity via generalized Clarke gradients and locally Lipschitz continuous storage functions are derived. Finally, these results are then used to develop feedback interconnection stability results for discontinuous systems thereby providing a generalization of the small gain and positivity theorems to systems with discontinuous vector fields.     

Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication

This paper investigates drive-response synchronization of chaotic systems with discontinuous right-hand side. Firstly, a general model is proposed to describe most of known discontinuous chaotic system with or without time-varying delay. An uniform impulsive controller with multiple unknown time-varying delays is designed such that the response system can be globally exponentially synchronized with the drive system. By utilizing a new lemma on impulsive differential inequality and the Lyapunov functional method, several synchronization criteria are obtained through rigorous mathematical proofs. Results of this paper are universal and can be applied to continuous chaotic systems. Moreover, numerical examples including discontinuous chaotic Chen system, memristor-based Chua’s circuit, and neural networks with discontinuous activations are given to verify the effectiveness of the theoretical results. Application of the obtained results to secure communication is also demonstrated in this paper.     

On the connections between symmetries and conservation rules of dynamical systems

The strict connection between Lie point‐symmetries of a dynamical system and its constants of motion is discussed and emphasized through old and new results. It is shown in particular how the knowledge of the symmetry of a dynamical system can allow us to obtain conserved quantities that are invariant under the symmetry. In the case of Hamiltonian dynamical systems, it is shown that if the system admits a symmetry of a ‘weaker’ type (specifically, a λ or a Λ‐symmetry), then the generating function of the symmetry is not a conserved quantity, but the deviation from the exact conservation is ‘controlled’ in a well‐defined way. Several examples illustrate the various aspects. Copyright © 2012 John Wiley & Sons, Ltd.     

Discontinuous Dynamical Systems in Optimal Control Theory

We develop foundations of the theory of discontinuous Hamiltonian systems appearing in the problems of optimal control. We consider analogs of the classical Poisson and Liouville theorems for discontinuous Hamiltonian systems. We study the local geometry of discontinuous dynamical systems and describe singularities in general position and the behavior of integral trajectories near an elliptical submanifold (sliding mode).     

A theory for flow switchability in discontinuous dynamical systems

The G-functions for discontinuous dynamical systems are introduced to investigate singularity in discontinuous dynamical systems. Based on the new G-function, the switchability of a flow from a domain to an adjacent one is discussed. Further, the full and half sink and source, non-passable flows to the separation boundary in discontinuous dynamical systems are discussed. A flow to the separation boundary in a discontinuous dynamical system can be passable or non-passable. Therefore, the switching bifurcations between the passable and non-passable flows are presented. Finally, the first integral quantity increment for discontinuous dynamical systems is given instead of the Melnikov function to develop the iterative mapping relations.     

Dynamical behavior for a class of predator–prey system with general functional response and discontinuous harvesting policy

In this paper, we introduce a class of predator–prey system with general functional response, whose harvesting policy is modeled by a discontinuous function. Based on the differential inclusions theory, topological degree theory in set‐valued analysis and generalized Lyapunov approach, we analyze the existence, uniqueness and global asymptotic stability of positive periodic solution. In particular, a series of useful criteria on existence, uniqueness and global asymptotic stability of the positive equilibrium point are established for the autonomous system corresponding to the non‐autonomous biological and mathematical model with a discontinuous right‐hand side. Moreover, some new sufficient conditions are provided to guarantee the global convergence in measure of harvesting solution and convergence in finite time of any positive solution for the autonomous discontinuous biological system. The obtained results, which improve and generalize previous works on dynamical behavior in the literature, are of interest for understanding and designing biological system with not only continuous or even Lipschitz continuous but also discontinuous harvesting function. Finally, we give three examples with numerical simulations to show the applicability and effectiveness of our main results. Copyright © 2015 John Wiley & Sons, Ltd.     

The Brouwer Fixed Point Theorem applied to rumour transmission

The results of the Brouwer Fixed Point Theorem are extended to continuous dynamical systems. It is shown that if there exists a compact convex positive invariant set for the dynamical system, then this convex positive invariant set contains an equilibrium point. The existence of an interior equilibrium is shown for a general model of rumour transmission.