Finite-dimensional Dynamical Systems Research Articles

Strong versions of impulsive controllability and sampled observability

We give a simple proof of the (perhaps not so) well known fact that exponential polynomials of order k with real exponents have at most k−1 real zeros. We deduce several results that relate to impulsive controllability and sampled observability of finite-dimensional linear time-invariant dynamical systems. We prove that the initial state of a continuous-time linear time-invariant dynamical system of dimension n can be uniquely reconstructed from the sampled output regardless of its sampling time sequence of length n if and only if the system is observable and all the eigenvalues of the system matrix are real. This result thus characterizes sampled observability with arbitrary sampling times. Likewise, we prove that the system is controllable by means of n impulses regardless of when they occur if and only if the system is controllable and all the eigenvalues of the system matrix are real. We also show that, if the system is observable and all the eigenvalues of the system matrix are real, then there exists an initial state such that the times where the output crosses a prescribed threshold are prescribed m times with m<n.

Equivariant embedding of finite-dimensional dynamical systems

AbstractWe prove an equivariant version of the classical Menger-Nöbeling theorem regarding topological embeddings: Whenever a group G acts on a finite-dimensional compact metric space X, a generic continuous equivariant function from X into $$([0,1]^r)^G$$ ( [ 0 , 1 ] r ) G is a topological embedding, provided that for every positive integer N the space of points in X with orbit size at most N has topological dimension strictly less than $$\frac{rN}{2}$$ rN 2 . We emphasize that the result imposes no restrictions whatsoever on the acting group G (beyond the existence of an action on a finite-dimensional space). Moreover if G is finitely generated then there exists a finite subset $$F\subset G$$ F ⊂ G so that for a generic continuous map $$h:X\rightarrow [0,1]^{r}$$ h : X → [ 0 , 1 ] r , the map $$ h^{F}:X\rightarrow ([0,1]^{r})^{F}$$ h F : X → ( [ 0 , 1 ] r ) F given by $$x\mapsto (f(gx))_{g\in F}$$ x ↦ ( f ( g x ) ) g ∈ F is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.

Total and partial observation–detection in linear dynamical systems with characterized sources: finite-dimensional cases

In this work, we address the partial observation–detection problem for finite-dimensional dynamical linear systems that may not be fully observable or detectable. We introduce the concepts of `observation–detection' and `partial observation–detection,' which involve reconstructing either the entirety or a portion of the system state and the source reacting on the system, even when the system is not fully observable or detectable. We provide characterizations of `observable–detectable systems' and `observable–detectable spaces.' The reconstruction of the state and source on the observable–detectable subspace is achieved through orthogonal projection, leveraging the algebraic structure of the given finite-dimensional system. Additionally, we present examples to illustrate our approach.

Direct Lyapunov Method in the Problem of Ensuring the Stability of a Compact Minimum Set of a Dynamic System and the Formation of a Halo-Orbit Near the Lagrange Point L2

The efficiency of using the direct Lyapunov method for ensuring the stability of motion in compact invariant sets of finite-dimensional dynamical systems is shown. In the linear model of the restricted three-body problem, the possibility of ensuring the asymptotic stability of the periodic motion of a spacecraft (SC) in the vicinity of the collinear Lagrange point L2 using light pressure forces without the consumption of the working fluid is considered. The required area of control surfaces is estimated depending on the mass of the SC.

Crises and chaotic scattering in hydrodynamic pilot-wave experiments.

Theoretical foundations of chaos have been predominantly laid out for finite-dimensional dynamical systems, such as the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g., weather, arise in systems with many (formally infinite) degrees of freedom, which limits direct quantitative analysis of such systems using chaos theory. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results, which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions.

Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

&lt;p style='text-indent:20px;'&gt;In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.&lt;/p&gt;

An RNA-based theory of natural universal computation

Life is confronted with computation problems in a variety of domains including animal behavior, single-cell behavior, and embryonic development. Yet we currently do not know of a naturally existing biological system that is capable of universal computation, i.e., Turing-equivalent in scope. Generic finite-dimensional dynamical systems (which encompass most models of neural networks, intracellular signaling cascades, and gene regulatory networks) fall short of universal computation, but are assumed to be capable of explaining cognition and development. I present a class of models that bridge two concepts from distant fields: combinatory logic (or, equivalently, lambda calculus) and RNA molecular biology. A set of basic RNA editing rules can make it possible to compute any computable function with identical algorithmic complexity to that of Turing machines. The models do not assume extraordinarily complex molecular machinery or any processes that radically differ from what we already know to occur in cells. Distinct independent enzymes can mediate each of the rules and RNA molecules solve the problem of parenthesis matching through their secondary structure. In the most plausible of these models all of the editing rules can be implemented with merely cleavage and ligation operations at fixed positions relative to predefined motifs. This demonstrates that universal computation is well within the reach of molecular biology. It is therefore reasonable to assume that life has evolved – or possibly began with – a universal computer that yet remains to be discovered. The variety of seemingly unrelated computational problems across many scales can potentially be solved using the same RNA-based computation system. Experimental validation of this theory may immensely impact our understanding of memory, cognition, development, disease, evolution, and the early stages of life.

Estimate the spectrum of affine dynamical systems from partial observations of a single trajectory data

In this paper, we study the nonlinear inverse problem of estimating the spectrum of a system matrix, that drives a finite-dimensional affine dynamical system, from partial observations of a single trajectory data. In the noiseless case, we prove an annihilating polynomial of the system matrix, whose roots are a subset of the spectrum, can be uniquely determined from data. We then study which eigenvalues of the system matrix can be recovered and derive various sufficient and necessary conditions to characterize the relationship between the recoverability of each eigenvalue and the observation locations. We propose various reconstruction algorithms with theoretical guarantees, generalizing the classical Prony method, ESPRIT, and matrix pencil method. We test the algorithms over a variety of examples with applications to graph signal processing, disease modeling and a real-human motion dataset. The numerical results validate our theoretical results and demonstrate the effectiveness of the proposed algorithms.

Connection among Stochastic Hamilton–Jacobi–Bellman Equation, Path-Integral, and Koopman Operator on Nonlinear Stochastic Optimal Control

The path-integral control, which stems from the stochastic Hamilton-Jacobi-Bellman equation, is one of the methods to control stochastic nonlinear systems. This paper gives a new insight into nonlinear stochastic optimal control problems from the perspective of Koopman operators. When a finite-dimensional dynamical system is nonlinear, the corresponding Koopman operator is linear. Although the Koopman operator is infinite-dimensional, adequate approximation makes it tractable and useful in some discussions and applications. Employing the Koopman operator perspective, it is clarified that only a specific type of observable is enough to be focused on in the control problem. This fact becomes easier to understand via path-integral control. Furthermore, the focus on the specific observable leads to a natural power-series expansion; coupled ordinary differential equations for discrete-state space systems are derived. A demonstration for nonlinear stochastic optimal control shows that the derived equations work well.

A Lipschitz Version of the $$\lambda $$-Lemma and a Characterization of Homoclinic and Heteroclinic Orbits

In this paper we consider finite dimensional dynamical systems generated by a Lipschitz function. We prove a version of the Whitney’s Extension Theorem on compact manifolds to obtain a version of the well-known \(\lambda \)-lemma for Lipschitz functions. The notions of Lipschitz transversality and hyperbolicity are investigated in the finite dimensional framework with a norm between \(C^1\)-norm and \(C^0\)-norm. As an application, we study homoclinic and heteroclinic orbits obtaining, as a consequence, a stability result for Lipschitz Morse–Smale functions.

A Degasperis–Procesi equation II with multi-peakon solutions

Based on a 3 × 3 matrix spectral problem, a new hierarchy of nonlinear evolution equations is obtained by means of the zero-curvature equation and the Lenard recursion equation. With the help of the trace identity, the Hamiltonian structures of this hierarchy are established. Then a new nonlinear wave equation, called the Degasperis–Procesi equation II, is derived from a negative flow, which admits exact solutions with N-peakons. Finite-dimensional dynamical systems related to N-peakon solutions and an infinite sequence of conserved quantities of the Degasperis–Procesi equation II are obtained.

Partial observability of finite dimensional linear systems

Abstract In this work, we consider the partial observability problem for finite dimensional dynamical linear systems that are not necessarily observable. For that purpose we introduce the so called “observable subspaces” and “partial observability” to find a way to reconstruct the observable part of the system state. Some characterizations of “observable subspaces” have been provided. The reconstruction of the orthogonal projection of the state on the observable subspace is obtained. We give some examples to illustrate our theoretical approach.

Bifurcation of the neuronal population dynamics of the modified theta model: Transition to macroscopic gamma oscillation

Interactions of inhibitory neurons produce gamma oscillations (30–80 Hz) in the local field potential, which is known to be involved in functions such as cognition and attention. In this study, the modified theta model is considered to investigate the theoretical relationship between the microscopic structure of inhibitory neurons and their gamma oscillations under a wide class of distribution functions of tonic currents on individual neurons. The stability and bifurcation of gamma oscillations for the Vlasov equation of the model is investigated by the generalized spectral theory. It is shown that as a connection probability of neurons increases, a pair of generalized eigenvalues crosses the imaginary axis twice, which implies that a stable gamma oscillation exists only when the connection probability has a value within a suitable range. On the other hand, when the distribution of tonic currents on individual neurons is the Lorentzian distribution, the Vlasov equation is reduced to a finite dimensional dynamical system. The bifurcation analyses of the reduced equation exhibit equivalent results with the generalized spectral theory. It is also demonstrated that the numerical computations of neuronal population follow the analyses of the generalized spectral theory as well as the bifurcation analysis of the reduced equation.

Approximating dynamics of a number-conserving cellular automaton by a finite-dimensional dynamical system

The local structure theory for cellular automata (CA) can be viewed as an finite-dimensional approximation of infinitely dimensional system. While it is well known that this approximation works surprisingly well for some CA, it is still not clear why it is the case, and which CA rules have this property. In order to shed some light on this problem, we present an example of a four input CA for which probabilities of occurrence of short blocks of symbols can be computed exactly. This rule is number conserving and possesses a blocking word. Its local structure approximation correctly predicts steady-state probabilities of small length blocks, and we present a rigorous proof of this fact, without resorting to numerical simulations. We conjecture that the number-conserving property together with the existence of the blocking word are responsible for the observed perfect agreement between the finite-dimensional approximation and the actual infinite-dimensional dynamical system.

Static Self-gravitating Newtonian Elastic Balls

The existence of static self-gravitating Newtonian elastic balls is proved under general assumptions on the constitutive equations of the elastic material. The proof uses methods from the theory of finite-dimensional dynamical systems and the Euler formulation of elasticity theory for spherically symmetric bodies introduced recently by the authors. Examples of elastic materials covered by the results of this paper are Saint Venant-Kirchhoff, John and Hadamard materials.

Nonlinear Autoregressive Moving Average-L2 Model Based Adaptive Control of Nonlinear Arm Nerve Simulator System

This paper considers the trouble of the usage of approximate strategies for realizing the neural controllers for nonlinear SISO systems. In this paper, we introduce the nonlinear autoregressive moving average (NARMA-L2) model which might be approximations to the NARMA model. The nonlinear autoregressive moving average (NARMA-L2) model is an precise illustration of the input–output behavior of finite-dimensional nonlinear discrete time dynamical systems in a neighborhood of the equilibrium state. However, it isn't always handy for purposes of neural networks due to its nonlinear dependence on the manipulate input. In this paper, nerves system based arm position sensor device is used to degree the precise arm function for nerve patients the use of the proposed systems. In this paper, neural network controller is designed with NARMA-L2 model, neural network controller is designed with NARMA-L2 model system identification based predictive controller and neural network controller is designed with NARMA-L2 model based model reference adaptive control system. Hence, quite regularly, approximate techniques are used for figuring out the neural controllers to conquer computational complexity. Comparison were made among the neural network controller with NARMA-L2 model, neural network controller with NARMA-L2 model system identification based predictive controller and neural network controller with NARMA-L2 model reference based adaptive control for the preferred input arm function (step, sine wave and random signals). The comparative simulation result shows the effectiveness of the system with a neural network controller with NARMA-L2 model based model reference adaptive control system. Index Terms--- Nonlinear autoregressive moving average, neural network, Model reference adaptive control, Predictive controller DOI: 10.7176/JIEA/10-3-03 Publication date: April 30 th 2020

A priori analysis of reduced description of dynamical systems using approximate inertial manifolds

The treatment of turbulent flows as finite-dimensional dynamical systems opens new paths for modeling and development of reduced-order descriptions of such systems. For certain types of dynamical systems, a property known as the inertial manifold (IM) exists, allowing for the dynamics to be represented in a sub-space smaller than the entire state-space. While the existence of an IM has not been shown for the three-dimensional Navier-Stokes equations, it has been investigated for variations of the two-dimensional version and for similar canonical systems such as the Kuramoto-Sivashinsky equation (KSE). Based on this concept, a computational analysis of the use of IMs for modeling turbulent flows is conducted. In particular, an approximate IM (AIM) is used where the flow is decomposed into resolved and unresolved dynamics, similar to conventional large eddy simulation (LES). Instead of the traditional approach to subfilter modeling, a dynamical systems approach is used to obtain the closure terms. In the a priori estimation of the AIM approach for the Kuramoto-Sivashinsky equation, it is shown that the small-scale dynamics are accurately reconstructed even when using only a small number of resolved modes. Further, it is demonstrated that the number of resolved variables needed for this reconstruction is dependent on the dimension of the attractor.

Time dependent center manifold in PDEs

<p style='text-indent:20px;'>We consider externally forced equations in an evolution form. Mathematically, these are skew systems driven by a finite dimensional dynamical system. Two very common cases included in our treatment are quasi-periodic forcing and forcing by a stochastic process. We allow that the evolution is a PDE and even that it is not well-posed and that it does not define a flow (not all initial conditions lead to a solution). <p style='text-indent:20px;'>We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, smoothness, etc) assumptions, establishes the existence of a "time-dependent invariant manifold" (TDIM). These manifolds evolve with the forcing. They are such that the original equation is always tangent to a vector field in the manifold. Hence, for initial data in the TDIM, the original equation is equivalent to an ordinary differential equation. This allows us to define families of solutions of the full equation by studying the solutions of a finite dimensional system. Note that this strategy may apply even if the original equation is ill posed and does not admit solutions for arbitrary initial conditions (the TDIM selects initial conditions for which solutions exist). It also allows that the TDIM is infinite dimensional. <p style='text-indent:20px;'>Secondly, we construct the center manifold for skew systems driven by the external forcing. <p style='text-indent:20px;'>Thirdly, we present concrete applications of the abstract result to the differential equations whose linear operators are exponential trichotomy subject to quasi-periodic perturbations. The use of TDIM allows us to establish the existence of quasi-periodic solutions and to study the effect of resonances.

Nonlinear dynamics of dewetting thin films

Fluid films spreading on hydrophobic solid surfaces exhibit complicated dynamics that describe transitions leading the films to break up into droplets. For viscous fluids coating hydrophobic solids this process is called "dewetting". These dynamics can be represented by a lubrication model consisting of a fourth-order nonlinear degenerate parabolic partial differential equation (PDE) for the evolution of the film height. Analysis of the PDE model and its regimes of dynamics have yielded rich and interesting research bringing together a wide array of different mathematical approaches. The early stages of dewetting involve stability analysis and pattern formation from small perturbations and self-similar dynamics for finite-time rupture from larger amplitude perturbations. The intermediate dynamics describes further instabilities yielding topological transitions in the solutions producing sets of slowly-evolving near-equilibrium droplets. The long-time behavior can be reduced to a finite-dimensional dynamical system for the evolution of the droplets as interacting quasi-steady localized structures. This system yields <i>coarsening</i>, the successive re-arrangement and merging of smaller drops into fewer larger drops. To describe macro-scale applications, mean-field models can be constructed for the evolution of the number of droplets and the distribution of droplet sizes. We present an overview of the mathematical challenges and open questions that arise from the stages of dewetting and how they relate to issues in multi-scale modeling and singularity formation that could be applied to other problems in PDEs and materials science.

Local Theory for Spatio-Temporal Canards and Delayed Bifurcations

We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical system for slow variables. We prove the existence of center-manifolds for generic models of this type, and study the reduced, finite-dimensional dynamics near bifurcations of (possibly) patterned steady states in the layer problem. Theoretical results are complemented with detailed examples and numerical simulations covering systems of local and nonlocal reaction-diffusion equations, neural field models, and delay-differential equations. We provide analytical foundations for numerical observations recently reported in the literature, such as spatio-temporal canards and slow passages through Hopf bifurcations in spatially extended systems subject to slow parameter variations. We also provide a theoretical analysis of slow passage through a Turing bifurcation in local and nonlocal models.