Flow Curvature Method Research Articles

Slow Invariant Manifolds of Memristor-Based Chaotic Circuits

This work presents an efficient approach for computing the slow invariant manifold of the fourth-order canonical memristor-based Chua circuits using the flow curvature method. First, the magnetic-flux and charge characteristic curve is generated from the classical circuit with a piecewise-linear function. Then, the characteristic curve is generated from the circuit with the piecewise-linear function replaced by a cubic function. Further, the duality principle is applied to studying such memristor-based circuits in the three-dimensional flux-linkage and charge phase space and then in the four-dimensional current–voltage phase space. It is demonstrated that the slow invariant manifolds of these fourth-order memristor-based chaotic circuits can be more directly computed for the first case than the second.

Flow curvature manifold and energy of generalized Liénard systems

In his famous book entitled Theory of Oscillations, Nicolas Minorsky wrote: “each time the system absorbs energy the curvature of its trajectory decreases and vice versa”. By using the Flow Curvature Method, we establish that, in the ε-vicinity of the slow invariant manifold of generalized Liénard systems, the curvature of trajectory curve increases while the energy of such systems decreases. Hence, we prove Minorsky's statement for the generalized Liénard systems. These results are then illustrated with the classical Van der Pol and generalized Liénard singularly perturbed systems.

On Differential Geometric Formulations of Slow Invariant Manifold Computation: Geodesic Stretching and Flow Curvature

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.

Slow Invariant Manifold of Laser with Feedback

Previous studies have demonstrated, experimentally and theoretically, the existence of slow–fast evolutions, i.e., slow chaotic spiking sequences in the dynamics of a semiconductor laser with AC-coupled optoelectronic feedback. In this work, the so-called Flow Curvature Method was used, which provides the slow invariant manifold analytical equation of such a laser model and also highlights its symmetries if any exist. This equation and its graphical representation in the phase space enable, on the one hand, discriminating the slow evolution of the trajectory curves from the fast one and, on the other hand, improving our understanding of this slow–fast regime.

Slow Invariant Manifolds of Slow–Fast Dynamical Systems

Slow–fast dynamical systems, i.e. singularly or nonsingularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating their equations. This paper aims, on the one hand, to propose a classification of the most important of them into two great categories: singular perturbation-based methods and curvature-based methods, and on the other hand, to prove the equivalence between any methods belonging to the same category and between the two categories. Then, a deep analysis and comparison between each of these methods enable to state the efficiency of the Flow Curvature Method which is exemplified with paradigmatic Van der Pol singularly perturbed dynamical system and Lorenz slow–fast dynamical system.

Modeling critical phenomena in a generalized van der Pol model

The van der Pol model has a long history of being used in both the physical and biological sciences. This paper considers modeling critical phenomena in one of three-time scales fast-slow autonomous dynamical systems. We investigate a model that is a generalization of the classical van der Pol model. In the framework of this paper, we construct an invariant manifold with variable stability for the generalized van der Pol model. Expansions of the manifold and corresponding gluing function were constructed using the flow curvature method. We show the existence of a sufficiently smooth invariant surface (a black swan surface) for the model. Computer algebra methods are used for the quantitative analysis of the model.

Invariant manifold of variable stability in the Koper model

The Koper model is a vector field that was developed to study electrochemical oscillations arising in diffusion processes. In the framework of this paper, we consider the Koper model of chemical reactors. It is a three-dimensional autonomous slow-fast system with a folded node and a supercritical singular Hopf bifurcation. The aim is to construct the expansion of invariant manifold with variable stability and corresponding gluing function using the flow curvature method. We show that Koper model has sufficiently smooth invariant surface, called black swan. Computer simulation and computer algebra methods are used for the quantitative analysis of the model.

Flow curvature in a self-coupled FitzHugh-Nagumo model

In this paper we consider a self-coupled FitzHugh-Nagumo model describes a prototype of an excitable system (e.g., a neuron). The aim is to construct the expansion of invariant manifold with variable stability and corresponding gluing function using flow curvature method. We show that a self-coupled FitzHugh-Nagumo model has sufficiently smooth invariant surface, called “black swan”. Computer simulation and computer algebra methods are used for the quantitative analysis of the model.

Black swan and curvature in an autocatalator model

The aim of this paper is to extend canard constructing method given by J. Ginoux, J. Llibre, and L. Chua [1,2]. In this paper, we consider a black swan construction for three-dimensional autocatalator model using modified flow curvature method.

The Slow Invariant Manifold of the Lorenz–Krishnamurthy Model

During this last decades, several attempts to construct slow invariant manifold of the Lorenz-Krishnamurthy five-mode model of slow-fast interactions in the atmosphere have been made by various authors. Unfortunately, as in the case of many two-time scales singularly perturbed dynamical systems the various asymptotic procedures involved for such a construction diverge. So, it seems that till now only the first-order and third-order approximations of this slow manifold have been analytically obtained. While using the Flow Curvature Method we show in this work that one can provide the eighteenth-order approximation of the slow manifold of the generalized Lorenz-Krishnamurthy model and the thirteenth-order approximation of the "conservative" Lorenz-Krishnamurthy model. The invariance of each slow manifold is then established according to Darboux invariance theorem.

CANARDS FROM CHUA'S CIRCUIT

The aim of this work is to extend Benoît's theorem for the generic existence of "canards" solutions in singularly perturbed dynamical systems of dimension three with one fast variable to those of dimension four. Then, it is established that this result can be found according to the Flow Curvature Method. Applications to Chua's cubic model of dimension three and four enable to state the existence of "canards" solutions in such systems.

Investigation on Cubic Chua's Circuit via Differential Geometry

In this work, we analyze Chua's model with differential geometry, within the framework of the application of which the slow manifold and connecting curve are discussed using the flow curvature method and the curvature of the model, respectively. Consequently, we can study the metric properties of the model and also find the interesting and available properties intuitively by way of the numerical computations.

The flow curvature method applied to canard explosion

The aim of this work is to establish that the bifurcation parameter value leading to a canard explosion in dimension 2 obtained by the so-called geometric singular perturbation method can be found according to the flow curvature method. This result will be then exemplified with the classical Van der Pol oscillator.

Flow curvature method

Flow curvature method

SLOW INVARIANT MANIFOLDS AS CURVATURE OF THE FLOW OF DYNAMICAL SYSTEMS

Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the curvature of the trajectory curves of any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of high-dimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the so-called Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chua's piecewise linear and cubic models of dimensions three, four and five will be provided as tutorial examples exemplifying this method as well as those of high-dimensional dynamical systems.