Higher order zero time discontinuity mapping for analysis of degenerate grazing bifurcations of impacting oscillators

2018 ◽  
Vol 437 ◽  
pp. 209-222 ◽  
Author(s):  
Shan Yin ◽  
Guilin Wen ◽  
Huidong Xu ◽  
Xin Wu
Keyword(s):  
Higher Order ◽  
Order Zero ◽  
Discontinuity Mapping ◽  
Zero Time ◽  
Time Discontinuity ◽  
Grazing Bifurcations

scholarly journals NONSMOOTH BIFURCATIONS, TRANSIENT HYPERCHAOS AND HYPERCHAOTIC BEATS IN A MEMRISTIVE MURALI–LAKSHMANAN–CHUA CIRCUIT

2013 ◽  
Vol 23 (06) ◽  
pp. 1350098 ◽  
Author(s):  
A. ISHAQ AHAMED ◽  
M. LAKSHMANAN
Keyword(s):  
Time Series ◽  
Piecewise Linear ◽  
Power Spectra ◽  
Phase Portraits ◽  
Chua Circuit ◽  
Smooth System ◽  
Zero Time ◽  
Active Flux ◽  
Time Discontinuity ◽  
Grazing Bifurcations

In this paper, a memristive Murali–Lakshmanan–Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three-segment piecewise-linear active flux controlled memristor. The bistability nature of the memristor introduces two discontinuity boundaries or switching manifolds in the circuit topology. As a result, the circuit becomes a piecewise-smooth system of second order. Grazing bifurcations, which are essentially a form of discontinuity-induced nonsmooth bifurcations, occur at these boundaries and govern the dynamics of the circuit. While the interaction of the memristor-aided self oscillations of the circuit and the external sinusoidal forcing result in the phenomenon of beats occurring in the circuit, grazing bifurcations endow them with chaotic and hyperchaotic nature. In addition, the circuit admits a codimension-5 bifurcation and transient hyperchaos. Grazing bifurcations as well as other behaviors have been analyzed numerically using time series plots, phase portraits, bifurcation diagram, power spectra and Lyapunov spectrum, as well as the recent 0–1 K test for chaos, obtained after constructing a proper Zero Time Discontinuity Map (ZDM) and Poincaré Discontinuity Map (PDM) analytically. Multisim simulations using a model of piecewise linear memristor have also been used to confirm some of the behaviors.

scholarly journals Sliding Bifurcations in the Memristive Murali–Lakshmanan–Chua Circuit and the Memristive Driven Chua Oscillator

2020 ◽  
Vol 30 (14) ◽  
pp. 2050214
Author(s):  
A. Ishaq Ahamed ◽  
M. Lakshmanan
Keyword(s):  
Numerical Simulations ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Filippov Systems ◽  
Chua Circuit ◽  
Linear Characteristic ◽  
Discontinuity Mapping ◽  
Chua Oscillator ◽  
Zero Time ◽  
Time Discontinuity

In this paper, we report the occurrence of sliding bifurcations admitted by the memristive Murali–Lakshmanan–Chua circuit [Ishaq & Lakshmanan, 2013] and the memristive driven Chua oscillator [Ishaq et al., 2011]. Both of these circuits have a flux-controlled active memristor designed by the authors in 2011, as their nonlinear element. The three-segment piecewise-linear characteristic of this memristor bestows on the circuits two discontinuity boundaries, dividing their phase spaces into three subregions. For proper choice of parameters, these circuits take on a degree of smoothness equal to one at each of their two discontinuities, thereby causing them to behave as Filippov systems. Sliding bifurcations, which are characteristic of Filippov systems, arise when the periodic orbits in each of the subregions, interact with the discontinuity boundaries, giving rise to many interesting dynamical phenomena. The numerical simulations are carried out after incorporating proper zero time discontinuity mapping (ZDM) corrections. These are found to agree well with the experimental observations which we report here appropriately.

scholarly journals Discontinuity Induced Hopf and Neimark–Sacker Bifurcations in a Memristive Murali–Lakshmanan–Chua Circuit

2017 ◽  
Vol 27 (06) ◽  
pp. 1730021 ◽  
Author(s):  
A. Ishaq Ahamed ◽  
M. Lakshmanan
Keyword(s):  
Floquet Theory ◽  
Equilibrium Points ◽  
Multiple Equilibrium ◽  
Hopf Bifurcations ◽  
Chua Circuit ◽  
Local Bifurcations ◽  
Generalized Differential ◽  
Multiple Equilibrium Points ◽  
Zero Time ◽  
Time Discontinuity

We report using Clarke’s concept of generalized differential and a modification of Floquet theory to nonsmooth oscillations, the occurrence of discontinuity induced Hopf bifurcations and Neimark–Sacker bifurcations leading to quasiperiodic attractors in a memristive Murali–Lakshmanan–Chua (memristive MLC) circuit. The above bifurcations arise because of the fact that a memristive MLC circuit is basically a nonsmooth system by virtue of having a memristive element as its nonlinearity. The switching and modulating properties of the memristor which we have considered endow the circuit with two discontinuity boundaries and multiple equilibrium points as well. As the Jacobian matrices about these equilibrium points are noninvertible, they are nonhyperbolic, some of these admit local bifurcations as well. Consequently when these equilibrium points are perturbed, they lose their stability giving rise to quasiperiodic orbits. The numerical simulations carried out by incorporating proper discontinuity mappings (DMs), such as the Poincaré discontinuity map (PDM) and zero time discontinuity map (ZDM), are found to agree well with experimental observations.

Higher-order zero crossings for HSI ATD

2003 ◽  
Author(s):  
Dalton S. Rosario ◽  
Nasser M. Nasrabadi
Keyword(s):  
Higher Order ◽  
Order Zero ◽  
Zero Crossings

scholarly journals Higher-order functional languages and intensional logic

1999 ◽  
Vol 9 (5) ◽  
pp. 527-564 ◽  
Author(s):  
P. RONDOGIANNIS ◽  
W. W. WADGE
Keyword(s):  
Broad Class ◽  
Higher Order ◽  
Final Outcome ◽  
Intensional Logic ◽  
New Technique ◽  
Functional Languages ◽  
Order Zero ◽  
Functional Program ◽  
Source Program ◽  
Context Manipulation

In this paper we demonstrate that a broad class of higher-order functional programs can be transformed into semantically equivalent multidimensional intensional programs that contain only nullary variable definitions. The proposed algorithm systematically eliminates user-defined functions from the source program, by appropriately introducing context manipulation (i.e. intensional) operators. The transformation takes place in M steps, where M is the order of the initial functional program. During each step the order of the program is reduced by one, and the final outcome of the algorithm is an M-dimensional intensional program of order zero. As the resulting intensional code can be executed in a purely tagged-dataflow way, the proposed approach offers a promising new technique for the implementation of higher-order functional languages.

scholarly journals Double grazing bifurcations of the non-smooth railway wheelset systems

2021 ◽  
Author(s):  
Pengcheng Miao ◽  
Denghui Li ◽  
Shan Yin ◽  
Jianhua Xie ◽  
Celso Grebogi ◽  
...  
Keyword(s):  
Dynamical Behavior ◽  
Autonomous Systems ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Railway Vehicle ◽  
Impact Model ◽  
Railway Systems ◽  
Discontinuity Mapping ◽  
Stable Periodic ◽  
Grazing Bifurcations

Abstract There are numerous non-smooth factors in railway vehicle systems, such as flange impact, dry friction, creep force, and so on. Such non-smooth factors heavily affect the dynamical behavior of the railway systems. In this paper, we investigate and mathematically analyze the double grazing bifurcations of the railway wheelset systems with flange contact. Two types of models of flange impact are considered, one is a rigid impact model and the other is a soft impact model. First, we derive Poincaré maps near the grazing trajectory by the Poincaré-section discontinuity mapping (PDM) approach for the two impact models. Then, we analyze and compare the near grazing dynamics of the two models. It is shown that in the rigid impact model the stable periodic motion of the railway wheelset system translates into a chaotic motion after the gazing bifurcation, while in the soft impact model a pitchfork bifurcation occurs and the system tends to the chaotic state through a period doubling bifurcation. Our results also extend the applicability of the PDM of one constraint surface to that of two constraint surfaces for autonomous systems.

scholarly journals Transferring Instantly the State of Higher-Order Linear Descriptor (Regular) Differential Systems Using Impulsive Inputs

2009 ◽  
Vol 2009 ◽  
pp. 1-29 ◽  
Author(s):  
Athanasios D. Karageorgos ◽  
Athanasios A. Pantelous ◽  
Grigoris I. Kalogeropoulos
Keyword(s):  
Higher Order ◽  
Initial State ◽  
Approximation Process ◽  
Differential Systems ◽  
Order Linear ◽  
High Magnitude ◽  
Analytical Formulae ◽  
Dirac Function ◽  
Zero Time ◽  
Very High

In many applications, and generally speaking in many dynamical differential systems, the problem of transferring the initial state of the system to a desired state in (almost) zero-time time is desirable but difficult to achieve. Theoretically, this can be achieved by using a linear combination of Dirac -function and its derivatives. Obviously, such an input is physically unrealizable. However, we can think of it approximately as a combination of small pulses of very high magnitude and infinitely small duration. In this paper, the approximation process of the distributional behaviour of higher-order linear descriptor (regular) differential systems is presented. Thus, new analytical formulae based on linear algebra methods and generalized inverses theory are provided. Our approach is quite general and some significant conditions are derived. Finally, a numerical example is presented and discussed.

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