Modifications of the method of variation of parameters

AbstractIn spite of being a classical method for solving differential equations, the method of variation of parameters continues having a great interest in theoretical and practical applications, as in astrodynamics. In this paper we analyse this method providing some modifications and generalised theoretical results. Finally, we present an application to the determination of the ephemeris of an artificial satellite, showing the benefits of the method of variation of parameters for this kind of problems

Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh-Rose model

Background: Development of effective and plausible numerical tools is an imperative task for thorough studies of nonlinear dynamics in life science applications. Results: We have developed a complementary suite of computational tools for twoparameter screening of dynamics in neuronal models. We test a ‘brute-force’ effectiveness of neuroscience plausible techniques specifically tailored for the examination of temporal characteristics, such duty cycle of bursting, interspike interval, spike number deviation in the phenomenological Hindmarsh-Rose model of a bursting neuron and compare the results obtained by calculus-based tools for evaluations of an entire spectrum of Lyapunov exponents broadly employed in studies of nonlinear systems. Conclusions: We have found that the results obtained either way agree exceptionally well, and can identify and differentiate between various fine structures of complex dynamics and underlying global bifurcations in this exemplary model. Our future planes are to enhance the applicability of this computational suite for understanding of polyrhythmic bursting patterns and their functional transformations in small networks

Bregman Proximal Gradient Algorithm with Extrapolation for a class of Nonconvex Nonsmooth Minimization Problems

In this paper, we consider an accelerated method for solving nonconvex and nonsmooth minimization problems. We propose a Bregman Proximal Gradient algorithm with extrapolation(BPGe). This algorithm extends and accelerates the Bregman Proximal Gradient algorithm (BPG), which circumvents the restrictive global Lipschitz gradient continuity assumption needed in Proximal Gradient algorithms (PG). The BPGe algorithm has higher generality than the recently introduced Proximal Gradient algorithm with extrapolation(PGe), and besides, due to the extrapolation step, BPGe converges faster than BPG algorithm. Analyzing the convergence, we prove that any limit point of the sequence generated by BPGe is a stationary point of the problem by choosing parameters properly. Besides, assuming Kurdyka-{\'L}ojasiewicz property, we prove the whole sequences generated by BPGe converges to a stationary point. Finally, to illustrate the potential of the new method BPGe, we apply it to two important practical problems that arise in many fundamental applications (and that not satisfy global Lipschitz gradient continuity assumption): Poisson linear inverse problems and quadratic inverse problems. In the tests the accelerated BPGe algorithm shows faster convergence results, giving an interesting new algorithm.Comment: Preprint submitted for publication, February 14, 201

A General Condition Number for Polynomials

This paper presents a generic condition number for polynomials that is useful for polynomial evaluation of a finite series of polynomial basis defined by means of a linear recurrence. This expression extends the classical one for the power and Bernstein bases, but it also provides us a general framework for all the families of orthogonal polynomials like Chebyshev, Legendre, Gegenbauer, Jacobi, and Sobolev orthogonal polynomial bases. The standard algorithm for the evaluation of finite series in any of these polynomial bases is the extended Clenshaw algorithm. The use of this new condition number permits us to give a general theorem about the forward error for that evaluation algorithm. A running-error bound of the extended algorithm is also presented and all the bounds are compared in several numerical examples

Homoclinic organization in the Hindmarsh-Rose model: A three parameter study

Bursting phenomena are found in a wide variety of fast-slow systems. In this article, we consider the Hindmarsh-Rose neuron model, where, as it is known in the literature, there are homoclinic bifurcations involved in the bursting dynamics. However, the global homoclinic structure is far from being fully understood. Working in a three-parameter space, the results of our numerical analysis show a complex atlas of bifurcations, which extends from the singular limit to regions where a fast-slow perspective no longer applies. Based on this information, we propose a global theoretical description. Surfaces of codimension-one homoclinic bifurcations are exponentially close to each other in the fast-slow regime. Remarkably, explained by the specific properties of these surfaces, we show how the Hindmarsh-Rose model exhibits isolas of homoclinic bifurcations when appropriate two-dimensional slices are considered in the three-parameter space. On the other hand, these homoclinic bifurcation surfaces contain curves corresponding to parameter values where additional degeneracies are exhibited. These codimension-two bifurcation curves organize the bifurcations associated with the spike-adding process and they behave like the "spines-of-a-book, " gathering "pages" of bifurcations of periodic orbits. Depending on how the parameter space is explored, homoclinic phenomena may be absent or far away, but their organizing role in the bursting dynamics is beyond doubt, since the involved bifurcations are generated in them. This is shown in the global analysis and in the proposed theoretical scheme

Order in chaos: Structure of chaotic invariant sets of square-wave neuron models

Bursting phenomena and, in particular, square-wave or fold/hom bursting, are found in a wide variety of mathematical neuron models. These systems have different behavior regimes depending on the parameters, whether spiking, bursting, or chaotic. We study the topological structure of chaotic invariant sets present in square-wave bursting neuron models, first detailed using the Hindmarsh–Rose neuron model and later exemplary in the more realistic model of a leech heart neuron. We show that the unstable periodic orbits that form the skeleton of the chaotic invariant sets are deeply related to the spike-adding phenomena, typical from these models, and how there are specific symbolic sequences and a symbolic grammar that organize how and where the periodic orbits appear. Linking this information with the topological template analysis permits us to understand how the internal structure of the chaotic invariants is modified and how more symbolic sequences are allowed. Furthermore, the results allow us to conjecture that, for these systems, the limit template when the small parameter ¿, which controls the slow gating variable, tends to zero is the complete Smale topological template

Unbounded dynamics in dissipative flows: Rössler model

Transient chaos and unbounded dynamics are two outstanding phenomena that dominate in chaotic systems with large regions of positive and negative divergences. Here, we investigate the mechanism that leads the unbounded dynamics to be the dominant behavior in a dissipative flow. We describe in detail the particular case of boundary crisis related to the generation of unbounded dynamics. The mechanism of the creation of this crisis in flows is related to the existence of an unstable focus-node (or a saddle-focus) equilibrium point and the crossing of a chaotic invariant set of the system with the weak-(un)stable manifold of the equilibrium point. This behavior is illustrated in the well-known Rössler model. The numerical analysis of the system combines different techniques as chaos indicators, the numerical computation of the bounded regions, and bifurcation analysis. For large values of the parameters, the system is studied by means of Fenichel's theory, providing formulas for computing the slow manifold which influences the evolution of the first stages of the orbit

Inertial Nonconvex Alternating Minimizations for the Image Deblurring

In image processing, total variation (TV) regularization models are commonly used to recover the blurred images. One of the most efficient and popular methods to solve the convex TV problem is the alternating direction method of multipliers (ADMM) algorithm, recently extended using the inertial proximal point method. Although all the classical studies focus on only a convex formulation, recent articles are paying increasing attention to the nonconvex methodology due to its good numerical performance and properties. In this paper, we propose to extend the classical formulation with a novel nonconvex alternating direction method of multipliers with the inertial technique (IADMM). Under certain assumptions on the parameters, we prove the convergence of the algorithm with the help of the Kurdyka-Lojasiewicz property. We also present numerical simulations on the classical TV image reconstruction problems to illustrate the efficiency of the new algorithm and its behavior compared with the well-established ADMM method

Sensorless torque estimation and control in a 1 DoF geared robotic arm

Torque control techniques have been proven to control robotic arms more adaptive and safely, adding accuracy to the robot and thus allowing a wide range of applications in which uncertainties are existent such as the presence of humans in the workspace or working with materials with different stiffness. In order to implement torque control, an awareness of the aggregate of torques intervening in the system to control is needed, separating the internal torques caused by elements in the robot itself and the external torques caused by external elements. Therefore, general solution uses torque sensors to measure the external torques. Torque sensors give information with much noise, and also increase the budget, making it more expensive their use in industrial applications, together with the collocation problem present in their correct positioning on the design. Furthermore, sensors add weight, increasing the inertia of the dynamic system which makes the control of the robot more complex and computationally expensive. Moreover, stable contact with the environment is difficult when a torque sensor is used because of its soft mechanical structure and narrow bandwidth of torque sensing [1]. In this thesis, a disturbance observer (DOb) and a reaction torque observer (RTOb) are proposed, implemented and validated in a 1 DoF geared robotic arm system for torque estimation purposes, and their performance has been compared with torque sensor information. The correct functioning of disturbance observers on a geared system also add difficulty to the system due to non-linearities on the friction on the reduction gear. Moreover, simulations have been carried out and experimental data has been collected in order to validate the observers and the control algorithm diagram. With the real system mounted, tests have been performed to find the parameters that define the system dynamics accordingly. With the parameters found, the implementation in the real system of the control loop has been done and validate