Autonomous Dynamical Systems Research Articles

Complex dynamical behavior in memristor–capacitor systems

The parallel and series circuits of a Hewlett–Packard memristor and a capacitor are foundational building blocks for realistic memristive circuits. Due to the nonlinearity of the memristor, traditional studies with a single sinusoidal stimulus are limited in their ability to reveal the complex characteristics of memristors. By converting the circuits to an autonomous dimensionless dynamical system, we show that both memristors and capacitors can generate complex dynamical behaviors such as high periodic limit cycles and chaos under a combined periodic stimulus of two sinusoidal signals. To verify the existence of chaos, we present a computer-assisted rigorous proof by a topological horseshoe as well as a circuit implementation. In this way, we uncover a new property of the memristor–capacitor systems: for a typical memristor, e.g., $$R_\mathrm{OFF}/R_\mathrm{ON}=100$$ , no matter what values other parameters take, there often exists a periodic stimulus to make the circuits chaotic. Furthermore, under combined excitation of multiple periodic stimulus, the chaos in the systems still exists irrespective of a certain pattern that the frequencies of these stimulus have.

Individual Functions Method for Power System Transient Stability Assessment

This paper introduces an alternative description of autonomous dynamical systems as a sum of individual vector fields. This structure of a dynamical system gives us the opportunity to construct individual Lyapunov-like functions to certify stability of an equilibrium point. A theorem on individual functions is developed in a quadratic form to maintain computational efficiency. As a corollary to the developed theory, we conclude that regions of attraction can be estimated by sub-level sets of individual quadratic functions, and through geometric programs they can easily be maximized efficiently. Theoretical findings in this paper are examined on transient stability problem in power systems for both single machine to infinite bus and multi-machine systems with promising results for critical clearing time estimation compared to the values obtained from time domain simulations.

An approach to stability analyses in general relativity via symplectic geometry

We begin with a review of the statements of non-linear, linear and mode stability of autonomous dynamical systems in classical mechanics, using symplectic geometry. We then discuss what the Arnowitt–Deser–Misner (ADM) phase space and the ADM Hamiltonian of general relativity are, what constitutes a dynamical system, and subsequently present a nascent attempt to draw a formal analogy between the notions of stability in these two theories. We wish to note that we have not discussed here the construction of the reduced phase space by forming the quotient space of the constrained phase space with the gauge orbits. Our approach here is pedagogical and geometric, and the motivation is to unify and simplify a formal understanding of the statements regarding the stability of stationary solutions of general relativity. That is, typically the governing equations of motion of a Hamiltonian dynamical system are simply the flow equations of the associated symplectic Hamiltonian vector field, defined on phase space, and the non-linear stability analysis of its critical points have simply to do with the divergence of its flow there. Further, the linear stability of a critical point is related to the properties of the tangent flow of the Hamiltonian vector field. In the second half of this work, we posit that a study of the genericity of a particular black hole or naked singularity spacetime forming as an endstate of gravitational collapse is equivalent to an inquiry of how sensitive the orbits of the symplectic Hamiltonian vector field of general relativity are to changes in initial data. We demonstrate this by conducting a restricted non-linear stability analysis of the formation of a Schwarzschild black hole, working in the usual initial value formulation of general relativity.

DYNAMICS OF A SIMPLE ENDOGENOUS GROWTH MODEL WITH FINANCIAL INTERMEDIATION

We study an impact of the financial intermediation on economic growth. We assume the simple model of the economic growth in the form of an autonomous dynamical system with a financial sector represented by banks and real sector represented by households and firms. We assume that financial intermediation services are described by financial intermediation technology which is a function depending on the share of labor employed by banks. Investments realized by firms depend not only on savings accumulated by banks but also on financial intermediation technology. We obtain a three-dimensional dynamical system and analyze the existence of a saddle equilibrium in the growth process associated with financial intermediation. Using mathematical methods of dynamical systems, we analyze growth paths, and we study the stationary states of the system and their stability. We found that equilibrium is reached only by trajectories located on two submanifolds. The resulting analysis provides an insight into the saddle solution with a stable incoming separatrix lying on one of the invariant manifolds.

Generalized logarithmic equation of state in classical and loop quantum cosmology dark energy–dark matter coupled systems

In this paper we shall study the phase space of a coupled dark energy–dark matter fluids system, in which the dark energy has a generalized logarithmic corrected equation of state. Particularly, the equation of state for the dark energy will contain a logarithmic function of the dark energy density ρd and will also have quadratic and Chaplygin gas-like terms, expressed in terms of ρd. We shall use the dynamical system approach in order to study the cosmological dynamics, and by appropriately choosing the dynamical system variables, we shall construct an autonomous dynamical system. The study will be performed in the context of classical and loop quantum cosmology, and the focus is on finding stable de Sitter attractors. As we demonstrate, in both the classical and loop quantum cosmology cases, there exist stable de Sitter attractors in the phase space, with the loop quantum cosmology case though having a wider range of the free parameter values for which the stable de Sitter attractors may occur. It is emphasized that the use of a generalized dark energy equation of state makes possible the existence of de Sitter attractors, which were absent in the case that a simple logarithmic term constitutes the dark energy equation of state.

Analysis and spline approximation of surface charge and potential at planar electrochemical interfaces

In this paper, we consider the Poisson‐Boltzmann theory to model the electrostatic potential of a bulk electrolyte containing a single planar charged surface. In the case of a constant surface charge density, we address the problem as a ODE system by using the stability theory for autonomous dynamical systems. By stating that the surface charge density and the surface potential are non‐linearly related through the Grahame equation, we give a description of the stable manifold of the system. To solve the Grahame equation and obtain an approximation for the stable manifold, we propose a smoothing collocation method based on cubic splines, including implementation details and numerical results.

Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems

<p style="text-indent:20px;">In this paper, we first prove that the property of being a gradient-like general dynamical system and the existence of a Morse decomposition are equivalent. Next, the stability of gradient-like general dynamical systems is analyzed. In particular, we show that a gradient-like general dynamical system is stable under perturbations, and that Morse sets are upper semi-continuous with respect to perturbations. Moreover, we prove that any solution of perturbed general dynamical systems should be close to some Morse set of the unperturbed gradient-like general dynamical system. We do not assume local compactness for the metric phase space <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula>, unlike previous results in the literature. Finally, we extend the Morse decomposition theory of single-valued nonautonomous dynamical systems to the multi-valued case, without imposing any compactness of the parameter spaces.

Chaotic system with bondorbital attractors

Two kinds of chaotic attractors with nontrivial topologies are found in a 4D autonomous continuous dynamical system. Since the equilibria of the system are located on both sides of the basic structures of these attractors, and the basic structures of these attractors are bond orbits, we call them bondorbital attractors. They have fractional dimensions and their Kaplan–Yorke dimensions are greater than 3. The generation mechanisms of the two types of attractors are explored and analyzed based on the Shilnikov’s theorems. The type I attractors generated by system with parameter $$P_{1}$$ possess coexistence features, and the type II attractors generated by system with parameter $$P_{2}$$ have the ability to realize consecutive bond orbits. Furthermore, the type I attractors have continuous attracting basins with diagonal distribution and can be caught by means of a method of shorting capacitors in hardware experiments, whereas the type II attractors possess discrete basins of attraction and are difficult to be captured in hardware experiments. The difference between the two types of bondorbital attractors and traditional self-excited attractors in generation method is analyzed, and we also verify that they are not hidden attractors. Based on the step function sequence f(x, M, N), the type II attractors with at most ( $$N+M+1$$ )-fold structures can be generated in the system with parameter $$P_{2}$$ . Two sets of symmetric specific initial conditions are used to verify that the system with parameter $$P_{2}$$ can generate bondorbital attractors with fourfold and fivefold basic structures based on f(x, 2, 2). Some characteristics of the two classes of bondorbital attractors are listed in tabular form.

Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an [Formula: see text]-asymptotically [Formula: see text]-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

Method to Construct Periodic Solutions of Controlled Second-Order Dynamical Systems

Nonconservative mechanical systems with one degree of freedom are considered. The goal is to provide the existence of steady-state oscillations with the prescribed properties. The system’s behavior is modeled by a second-order autonomous dynamical system with one variable parameter describing the amplifying coefficient of the control action. A numerical-analytic method to find the amplifying coefficient is proposed. Conditions of the orbital stability are obtained for the steady-state oscillations. An example of the application of the method is provided. The proposed approach can be applied to solve control problems and to find periodic solutions of second-order autonomous dynamical systems.

The Structure of Global Attractors for Non-autonomous Perturbations of Gradient-Like Dynamical Systems

In this paper we give the complete description of the structure of compact global (forward) attractors for non-autonomous perturbations of autonomous gradient-like dynamical systems under the assumption that the original autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed non-autonomous (in particular $$\tau $$ -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular the perturbed systems has the same number of $$\tau $$ -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff solutions). It is shown the compact global (forward) attractor of non-autonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections.

Inhibitory autapse mediates anticipated synchronization between coupled neurons.

Two identical autonomous dynamical systems unidirectionally coupled in a sender-receiver configuration can exhibit anticipated synchronization (AS) if the receiver neuron also receives a delayed negative self-feedback. Recently, AS was shown to occur in a three-neuron motif with standard chemical synapses where the delayed inhibition was provided by an interneuron. Here, we show that a two-neuron model in the presence of an inhibitory autapse, which is a massive self-innervation present in the cortical architecture, may present AS. The GABAergic autapse regulates the internal dynamics of the receiver neuron and acts as the negative delayed self-feedback required by dynamical systems in order to exhibit AS. In this biologically plausible scenario, a smooth transition from the usual delayed synchronization (DS) to AS typically occurs when the inhibitory conductance is increased. The phenomenon is shown to be robust when model parameters are varied within a physiological range. For extremely large values of the inhibitory autapse the system undergoes to a phase-drift regime in which the receiver is faster than the sender. Furthermore, we show that the inhibitory autapse promotes a faster internal dynamics of the free-running Receiver when the two neurons are uncoupled, which could be the mechanism underlying anticipated synchronization and the DS-AS transition.

Mapping of non-autonomous dynamical systems to autonomous ones

Using a proper choice of the dynamical variables, we show that a non-autonomous dynamical system transforming to an autonomous dynamical system with a certain coordinate transformations can be obtained by solving a general nonlinear first-order partial differential equations. We examine some special cases and provide particular physical examples. Our framework constitutes general machineries in investigating other non-autonomous dynamical systems.

Koopman operator-based model reduction for switched-system control of PDEs

We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. Using a recent convergence result for the numerical approximation via Extended Dynamic Mode Decomposition (EDMD), we show that the value of the K-ROM based objective function converges in measure to the value of the full objective function. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier–Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.

Finite-time singularities in swampland-related dark-energy models

In this work we shall investigate the singularity structure of the phase space corresponding to an exponential quintessence dark-energy model recently related to swampland models. The dynamical system corresponding to the cosmological system is an autonomous polynomial dynamical system, and by using a mathematical theorem we shall investigate whether finite-time singularities can occur in the dynamical system variables. As we demonstrate, the solutions of the dynamical system are non-singular for all cosmic times and this result is general, meaning that the initial conditions corresponding to the regular solutions, belong to a general set of initial conditions and not to a limited set of initial conditions. As we explain, a dynamical system singularity is not directly related to a physical finite-time singularity. Then, by assuming that the Hubble rate with functional form , is a solution of the dynamical system, we investigate the implications of the absence of finite-time singularities in the dynamical system variables. As we demonstrate, Big Rip and a Type-IV singularities can always occur if and , respectively. However, Type-II and Type-III singularities cannot occur in the cosmological system, if the Hubble rate we quoted is considered a solution of the cosmological system.

Dynamical system analysis of a three fluid cosmological model: an invariant manifold approach

The present paper considers a three-fluid cosmological model consisting of noninteracting dark matter, dark energy and baryonic matter in the background of the Friedman–Robertson–Walker–Lemaître flat spacetime. It has been assumed that the dark matter takes the form of dust whereas the dark energy is a quintessence (real) scalar field with exponential potential. It has been further assumed that the baryonic matter is a perfect fluid with barotropic equation of states. The field equations for this model takes the form of an autonomous dynamical system after some suitable changes of variables. Then a complete stability analysis is done considering all possible parameter (the adiabatic index of the baryonic matter and the parameter arising from the dark energy potential) values and for both the cases of hyperbolic and non-hyperbolic critical points. For non-hyperbolic critical points, the invariant manifold theory (center manifold approach) is applied. Finally various topologically different phase planes and vector field diagrams are produced and the cosmological interpretation of this model is presented.

Hopf bifurcations in dynamical systems

The onset of instability in autonomous dynamical systems (ADS) of ordinary differential equations is investigated. Binary, ternary and quaternary ADS are taken into account. The stability frontier of the spectrum is analyzed. Conditions necessary and sufficient for the occurring of Hopf, Hopf–Steady, Double-Hopf and unsteady aperiodic bifurcations—in closed form—and conditions guaranteeing the absence of unsteady bifurcations via symmetrizability, are obtained. The continuous triopoly Cournot game of mathematical economy is taken into account and it is shown that the ternary ADS governing the Nash equilibrium stability, is symmetrizable. The onset of Hopf bifurcations in rotatory thermal hydrodynamics is studied and the Hopf bifurcation number (threshold that the Taylor number crosses at the onset of Hopf bifurcations) is obtained.

Identifiability from a Few Species for a Class of Biochemical Reaction Networks.

Under mass-action kinetics, biochemical reaction networks give rise to polynomial autonomous dynamical systems whose parameters are often difficult to estimate. We deal in this paper with the problem of identifying the kinetic parameters of a class of biochemical networks which are abundant, such as multisite phosphorylation systems and phosphorylation cascades (for example, MAPK cascades). For any system of this class, we explicitly exhibit a single species for each connected component of the associated digraph such that the successive total derivatives of its concentration allow us to identify all the parameters occurring in the component. The number of derivatives needed is bounded essentially by the length of the corresponding connected component of the digraph. Moreover, in the particular case of the cascades, we show that the parameters can be identified from a bounded number of successive derivatives of the last product of the last layer. This theoretical result induces also a heuristic interpolation-based identifiability procedure to recover the values of the rate constants from exact measurements.

Integrable cosmological model with van der Waals gas and matter creation

A cosmological model with van der Waals gas and dust has been studied in the context of a three-component autonomous non-linear dynamical system involving the time evolution of the particle number density, the Hubble parameter and the temperature. Due to the presence of a symmetry of the model, the temperature evolution law is determined (in terms of the particle number density) and with this the dynamical system reduces to a two-component one which is fully integrable. The globally conserved Hamiltonian is identified and, in addition to it, some special (second) integrals, defined and conserved on a lower-dimensional manifold, are found. The parameter choices and their implication for the global dynamics in terms of cosmological relevance are comprehensively studied and the physically meaningful parameter values are identified.

Van der Waals universe with adiabatic matter creation

A FRWL cosmological model with perfect fluid comprising of van der Waals gas and dust has been studied in the context of dynamical analysis of a three-component autonomous non-linear dynamical system for the particle number density n, the Hubble parameter H, and the temperature T. Perfect fluid isentropic particle creation at rate proportional to an integer power alpha of H has been incorporated. The existence of a global first integral allows the determination of the temperature evolution law and hence the reduction of the dynamical system to a two-component one. Special attention is paid to the cases of alpha = 2 and alpha = 4 and these are illustrated with numerical examples. The global dynamics is comprehensively studied for different choices of the values of the physical parameters of the model. Trajectories in the (n, H) phase space are identified for which temporary inflationary regime exists.