Filippov Type Research Articles

High Energetic Deuteron Ion Irradiation of Al Samples by Dense Plasma Focus Device

In this paper energetic ion beams of a 90 kJ filippov type plasma focus were utilized to irradiation aluminum samples. The working gases were pure D2, and D2+Kr2%. The phenomena of melting, micro cracks, evaporation, and sputtering of the surface layer have been noticeable on the samples. Surface smoothing was remarkable when a heavy rare element added to D2. Adding Kr to the D2 filling gas lead to collapsing bubbles and greater surface damage. Micro hardness of surface layer tends to decrease particularly in the central region of the sample where destruction is more intense. The decrease in micro hardness for D2+2%Kr is more than that compared to deuteron ion treated sample.

On a boundary value problem for a Sturm-Liouville type differential inclusion

This paper proves a Filippov type existence theorem for solutions of a boundary value problem for a Sturm-Liouville type differential inclusion defined by a nonconvex set-valued map. The method consists in application of the contraction principle in the space of selections of the set-valued map instead of the space of solutions.

Micro-chaotic dynamics due to digital sampling in hybrid systems of Filippov type

This paper describes a novel mechanism leading to the sudden onset of chaos that may occur in Filippov type systems when the switching decision is made at discrete times. Such systems can be thought of as models of switched digital control systems. Our findings are explained in detail using a representative planar example. Results are then extended to more general planar switched systems with digital sampling. It is conjectured that this mechanism is also typical of n -dimensional switched systems with digital sampling, and numerical investigations of a third order relay feedback system are presented to support this; the onset of chaos is triggered in the way which is equivalent to that revealed for the planar case.

MPEC strategies for optimization of a class of hybrid dynamic systems

With the development and widespread use of large-scale nonlinear programming (NLP) tools for process optimization, there has been an associated application of NLP formulations with complementarity constraints in order to represent discrete decisions. In particular, these constraints arise frequently in equation-based formulations for real-time optimization. Also known as mathematical programs with equilibrium constraints (MPECs), these formulations can be used to model certain classes of discrete events and can be more efficient than a mixed integer formulation, particularly for large systems with many discrete decisions, such as dynamic systems with switches at any point in time. In this study, we consider and extend MPEC formulations for the optimization of a class of hybrid dynamic models, where the differential states remain continuous over time. These include differential inclusions of the Filippov type. Here, particular care is required in the formulation in order to preserve smoothness properties of the dynamic system. Results on three case studies, including process control examples, illustrate the effectiveness and accuracy of the proposed MPEC optimization methodology for a class of hybrid dynamic systems.

Current Sheath Dynamics and its Evolution Studies in Sahand Filippov Type Plasma Focus

One of the most important factors for optimizing the plasma focus device operation is the dynamics of the plasma. In this paper, we investigated the profile and dynamics of the current sheath by measuring the velocity and distribution of current sheath in Sahand as a Filippov type plasma focus device. For this purpose, the discharge is produced in pure neon gas with capacitor bank stored energies in the range of 14–50 kJ. The current sheath is monitored using two sets of magnetic probes, one with four and other with three equi-distant probe coils. These probes, installed in both radial and axial directions near the edge of the interior electrode (anode), are used for monitoring the distributions and dynamics of the current sheath. The maximum current sheath velocities at radial and axial phase are 4 ± 0.13 and 3.51 ± 0.22 (cm/μs) respectively for 0.25 Torr. The decreasing of CS velocities in going move away from anode surface is one of the our results in this paper. In this paper we conclude that the current sheath velocity at radial phase in Sahand is greater than axial phase. The effect of the neon working gas pressure and working voltage on the current sheath dynamics and its spatial evolution is investigated and presented.

Sliding Motion in Filippov Differential Systems: Theoretical Results and a Computational Approach

In this work, we discuss some theoretical and numerical aspects of solving differential equations with discontinuous right-hand sides of Filippov type. In particular, (i) we propose second order corrections to the theory of Filippov, (ii) we provide a systematic and nonambiguous way to define the vector field on the intersection of several surfaces of discontinuity, and (iii) we propose, and implement, a numerical method to approximate a trajectory of systems with discontinuous right-hand sides and illustrate its performance on a few examples.

Robustness of Grazing-sliding Bifurcations in Hybrid Systems

In the paper we consider how orbits characterised by the presence of so-called sliding motion, which orbits typically occur in hybrid systems of Filippov type, are affected by stable singular perturbations. To be able to pursuit our analysis we consider a planar minimal system and we tune a system parameter such that a stable periodic orbit of the system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. We then check the effect of stable singular perturbation on the grazing cycle. In the unperturbed system the periodic orbit remains stable, and its local return map becomes piecewise linear. The effect of an arbitrarily small stable singular perturbation is that the local return map changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos.

Study of dense plasma-surface interaction by a Filippov type plasma focus device

In this paper we have tried to investigate of dense plasma-material interaction in a Filippov type plasma focus (90KJ, 25KV, 288µf) facility. Dense plasma focus discharges are known as a remarkable source of hot plasma bunches and fast streams, fast neutrons, hard and soft X-rays, and energetic particles such as ions and electrons. The Aluminum made targets with 20cm in diameter and thickness 5mm, were placed at the central part of plasma focus cathode. These targets were exposed to perpendicular dense high temperature plasma stream incidence. The working gases for these experiments were Argon, Deuterium +Krypton, Helium, and pure Deuterium. We took advantage of changeable targets surface and used SEM technique for analyzing irradiated samples by various working gases. As it shown from SEM pictures and surface profiles, melt layer erosion by melt motion, surface smoothing, and bubble formation were some of different effects caused by diverse working conditions.

An investigation of the plasma behaviour in a Filippov type plasma focus device

Plasma behaviour in the 90 kJ Filippov type plasma focus (PF) device Dena, is studied both experimentally and theoretically. The latest experimental data obtained by the use of the Dena facility are presented. Then the experimental data are compared with the simulated data obtained through the ML model. This study shows that the ML model, to a good extent, is capable of predicting the plasma behaviour in the Filippov type PF. The experimental and the theoretical results show that the increment of the discharge voltage leads to an almost linear decrement of the pinch time. It is also shown that the increment of the pressure leads to a decrement of the current sheath expansion velocity. Finally, a semiempirical method for determination of the permitted values of the current efficiency factor and the mass shedding factor is presented.

On the existence of solutions for a higher order differential inclusion without convexity

We prove a Filippov type existence theorem for solutions of a higher order differential inclusion in Banach spaces with nonconvex valued right hand side by applying the contraction principle in the space of the derivatives of solutions instead of the space of solutions.

Preliminary measurements in Sahand plasma-focus emphasizing on the temporal characteristics of hard and soft X-rays

Plasma Focus has been known as an intense source of soft and hard X-rays (SXR and HXR). In this system a hot and dense plasma column is formed as a result of fast discharging a high current (∼MA) in a given gas. Various radiations are emitted from the pinched plasma. The aim of this paper is to give some information about the temporal behavior of the emitted X-rays from Sahand Filippov type plasma focus device. The optimum conditions for the X-ray production are also investigated. The results show that at each discharge voltage, there is an optimum pressure in which the pinch current is maximum as well as the HXR and SXR yields in the case of argon as the working gas. The optimum pressure increases linearly with the operation voltage. Furthermore the pinch current along with the yield of X-rays increases with the operating voltage at the optimum pressure. The study of the time resolved and time integrated signals of hard and soft X-rays in neon, argon and deuterium as working gases, indicate that the small peaks in the time resolved signals of HXR can be attributed to “micro pinches” which are observed in all above mentioned three gases. Attributing the main peak to the onset of the instability in the plasma column, one can declare that the life-time of the pinch decreases with the atomic number of the gas. The full width at half maximum (FWHM) of the pulses which actually represents the radiation time is maximum for Ne and minimum for Ar.

Studies of the Hard X-ray Emission from the Filippov Type Plasma Focus Device, Dena

This article is about the characteristics of the hard X-ray (HXR) emission from the Filippov type plasma focus (PF) device, Dena. The article begins with a brief presentation of Dena, and the mechanism of the HXR production in PF devices. Then using the differential absorption spectrometry, the energy resolved spectrum of the HXR emission from a 37 kJ discharge in Dena, is estimated. The energy flux density and the energy fluence of this emission have also been calculated to be 1.9 kJ cm−2 s−1 and 9.4 × 10−5 J cm−2. In the end, after presentation of radiography of sheep bones and calf ribs, the medical application of the PF devices has been discussed.

Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin

In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.

Adaptation of Sing Lee's model to the Filippov type plasma focus geometry

A new model for plasma behaviour in Filippov type plasma focus (PF) systems has been described and used. This model is based on the so-called slug model and Sing Lee's model for Mather type PF devices. Using the model, the discharge current and its derivative as a function of time, and the pinch time and the maximum discharge current as a function of pressure, have been predicted. At the end, the predicted data are compared with the experimental data obtained through a Filippov type PF facility with a nominal maximum energy of 90 kJ.

X-ray and neutron emission studies in a new Filippov type plasma focus

We have performed experimental comparative studies of the x-ray and neutron emission generated by the new Filippov-type plasma focus Dena, (90 kJ, 25 kV, 288mF) in the pressure range of 0.6-1 torr. Time-integrated and time-resolved detectors, together with an x-ray pin-hole camera, along with a Be filter of 10mm thickness have been used. For a working gas of neon and a at insert anode, the maximum soft and hard x-rays (SXR-HXR) yield obtained was 16 V and 1.5 V/shot over a 4p solid angle, respectively, for a charging voltage range of 16-20 kV. As for the argon gas, the similar results such as 3.5 and 2 V/shot have been found, leading to a total conversion eAciency of x-ray emission of 0.09 % (for neon) and 0.03 % (for argon) of the stored energy. These efficiencies have been improved by the employment of a conic insert anode up to 0.4% and 0.1%. With deuterium puffing gas and a at insert anode, the maximum emission yield has been found to be 2.5 V for SXR and 1 V for HXR/shot which produce an ultimate emission profile width (FWHM) of 70-90 ns for x-rays and neutrons, giving rise to a maximum neutron yield of 1.2 × 109. Nevertheless, the maximum yield has been increased up to 5.5 times with the conic insert anode. In order to increase the neutron yield, we have introduced a krypton admixture to the deuterium filling gas and found that, for a krypton pressure of about 0.1 torr, the neutron yield increases by a factor of 3.5 for the flat insert and 1.5 for the conic insert anodes.

Bifurcations in Nonlinear Discontinuous Systems

This paper treats bifurcations of periodic solutions indiscontinuous systems of the Filippov type. Furthermore, bifurcations offixed points in non-smooth continuous systems are addressed. Filippov'stheory for the definition of solutions of discontinuous systems issurveyed and jumps in fundamental solution matrices are discussed. It isshown how jumps in the fundamental solution matrix lead to jumps of theFloquet multipliers of periodic solutions. The Floquet multipliers canjump through the unit circle causing discontinuous bifurcations.Numerical examples are treated which show various discontinuousbifurcations. Also infinitely unstable periodic solutions are addressed.

Stability and Euler Approximation of One-sided Lipschitz Differential Inclusions

Ordinary differential and functional-differential inclusions with compact right-hand sides are considered. Stability theorems of Filippov's type in the convex and nonconvex case are proved under a one-sided Lipschitz condition, which extends the notions of Lipschitz continuity, dissipativity, and the uniform one-sided Lipschitz condition for set-valued mappings. The accuracy of approximation of the solution sets by means of the Euler discretization scheme for both types of inclusions is estimated.

Filippov Implicit Function Theorem for Quasi-Carathéodory Functions

Several Filippov type implicit function theorems are known for Carathéodory functionsf(t,x), i.e., allf(·,x) are measurable andf(t,·) are continuous. We prove some generalisations of this theorem supposing only each functionf(t,·) to be quasicontinuous with closed values.

Megajoule scale plasma focus as efficient X-ray source

Our experiments have established that a plasma focus (PF) with Filippov type electrodes can convert 10% of the capacitor bank energy (for W = 0.9 MJ) into K-shell lines of Ne radiation. This result when compared to the data obtained for a PF operated at W ≤ 5 kJ indicates a Y x ∼ I p 3.5–4 radiation scaling ( I p is the pinch current) and is in conformity to the resistive heating mechanism of Ne-plasma. Measurements of the X-ray energy flux were carried out with specially developed sensors, based on the explosive evaporation of the filtered targets.

Filippov type plasma focus as intense source of hard X-rays (E/sub x/≆50 keV)

This paper evaluates the use of a plasma focus machine (Filippov type) as a pulsed intense source of hard X-rays. It is shown experimentally that discharging a capacitor bank of W=50 kJ in the new arrangement of the discharge parameters, one can deliver into the test cavity (volume of 6*10/sup 3/ cm/sup 3/) an average of /spl Phi//sub x/=8*10/sup -4/ J/cm/sup 2/. The delivered energy fluence varies in the range of 2*10/sup -4/ J/cm/sup 2//spl les//spl Phi//sub x//spl les/2.9*10/sup -3/ J/cm/sup 2/, depending on the probe position. The X-ray energy spectra were established by the differential absorption method using thermo-luminescent detectors (time-integrated spectra) or plastic scintillators (time-resolved spectra). The average energy of X-ray photons is 40 keV<E/sub x//sup av/<50 keV (20 keV/spl les/E/sub x//spl les/170 keV) and remains quasi-constant during the main part of the pulse (/spl sim/40 ns). The electron energy that can produce such bremsstrahlung radiation is in the range of 95 keV/spl les/E/sub e//spl les/170 keV. However, time-resolved experiments reveal that at each instant of time only one energy of electron beam occurs. The aforementioned radiation characteristics reveal new potentials for the plasma-focus machine for a variety of applications. Some insight into the physics behind production mechanism of quasi-monoenergetic relativistic-electron beams will be concurrently reviewed.