Attractor and Vector Structure Analyses of Bursting Oscillation with Sliding Bifurcation in Filippov Systems

The main purpose of this paper is to investigate the mechanism of sliding phenomenon in Filippov (nonsmooth) dynamical systems by attractor analysis and vector analysis. A corresponding simple model based on Chua’s circuit with periodic excitation was introduced as an example. The attractor analysis proposed in our previous work is used to discuss the complicated oscillations of the Filippov system. However, it failed to perfectly explain the sliding phenomena and establish an analytical method of constant voltage control. Therefore, the geometric structure and analytic conditions of sliding bifurcations in the general n-dimensional piecewise smooth system are discussed in detail by vector structure analysis. The prospects of practical application of this method are also discussed in the end.

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Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417300427 ◽

2017 ◽

Vol 27 (12) ◽

pp. 1730042 ◽

Cited By ~ 4

Author(s):

David J. W. Simpson

Keyword(s):

Periodic Orbit ◽

Periodic Orbits ◽

Piecewise Linear ◽

Simple Type ◽

Asymptotically Stable ◽

Sliding Bifurcation ◽

Continuous Maps ◽

Stable Periodic ◽

Smooth System ◽

Structurally Stable

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

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Control of a class of nonsmooth dynamical systems

Journal of Vibration and Control ◽

10.1177/1077546313505634 ◽

2013 ◽

Vol 21 (11) ◽

pp. 2212-2222 ◽

Cited By ~ 5

Author(s):

Hamid Reza Erfanian ◽

Mohammad Hadi Noori Skandari ◽

Ali Vahidian Kamyad

Keyword(s):

Dynamical Systems ◽

Nonsmooth Dynamical Systems

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The dynamics of language

Behavioral and Brain Sciences ◽

10.1017/s0140525x99251825 ◽

1999 ◽

Vol 22 (2) ◽

pp. 284-285

Author(s):

Peter W. Culicover ◽

Andrzej Nowak

Keyword(s):

Phase Space ◽

Dynamical Systems ◽

Simple Model ◽

Syntactic Structure ◽

Neural Assemblies ◽

Systems Perspective ◽

Model Based ◽

Associative Structures

To deal with syntactic structure, one needs to go beyond a simple model based on associative structures, and to adopt a dynamical systems perspective, where each phrase and sentence of a language is represented as a trajectory in a syntactic phase space. Neural assemblies could possibly be used to produce dynamics that in principle could handle syntax along these lines.

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New Discontinuity-Induced Bifurcations in Chua's Circuit

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741550090x ◽

2015 ◽

Vol 25 (06) ◽

pp. 1550090 ◽

Cited By ~ 4

Author(s):

Shihui Fu ◽

Qishao Lu ◽

Xiangying Meng

Keyword(s):

Dynamical Systems ◽

Equilibrium Points ◽

Dynamical Behavior ◽

Chua’S Circuit ◽

Jacobian Matrices ◽

Chua's Circuit ◽

Before And After ◽

Boundary Equilibrium ◽

Discontinuous Bifurcations ◽

Nonsmooth Dynamical Systems

Chua's circuit, an archetypal example of nonsmooth dynamical systems, exhibits mostly discontinuous bifurcations. More complex dynamical phenomena of Chua's circuit are presented here due to discontinuity-induced bifurcations. Some new kinds of classical bifurcations are revealed and analyzed, including the coexistence of two classical bifurcations and bifurcations of equilibrium manifolds. The local dynamical behavior of the boundary equilibrium points located on switch boundaries is found to be determined jointly by the Jacobian matrices evaluated before and after switching. Some new discontinuous bifurcations are also observed, such as the coexistence of two discontinuous and one classical bifurcation.

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Marks: A New Interval Tool for Uncertainty, Vagueness and Indiscernibility

Mathematics ◽

10.3390/math9172116 ◽

2021 ◽

Vol 9 (17) ◽

pp. 2116

Author(s):

Miguel A. Sainz ◽

Remei Calm ◽

Lambert Jorba ◽

Ivan Contreras ◽

Josep Vehi

Keyword(s):

Dynamical Systems ◽

Measurement Errors ◽

Practical Application ◽

Modal Interval ◽

The Difference ◽

Systems Simulation ◽

And Control ◽

Physical Quantities ◽

Successive Calculation

The system of marks created by Dr. Ernest Gardenyes and Dr. Lambert Jorba was first published as a doctoral thesis in 2003 and then as a chapter in the book Modal Interval Analysis in 2014. Marks are presented as a tool to deal with uncertainties in physical quantities from measurements or calculations. When working with iterative processes, the slow convergence or the high number of simulation steps means that measurement errors and successive calculation errors can lead to a lack of significance in the results. In the system of marks, the validity of any computation results is explicit in their calculation. Thus, the mark acts as a safeguard, warning of such situations. Despite the obvious contribution that marks can make in the simulation, identification, and control of dynamical systems, some improvements are necessary for their practical application. This paper aims to present these improvements. In particular, a new, more efficient characterization of the difference operator and a new implementation of the marks library is presented. Examples in dynamical systems simulation, fault detection and control are also included to exemplify the practical use of the marks.

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The Conceptual Design of a Perfusion Reactor Using a Complexity Measure

Volume 4: 20th International Conference on Design Theory and Methodology; Second International Conference on Micro- and Nanosystems ◽

10.1115/detc2008-49266 ◽

2008 ◽

Author(s):

Andreas Bischof ◽

Jorge Angeles ◽

Lucienne Blessing

Keyword(s):

Mathematical Model ◽

Porous Medium ◽

Dynamical Systems ◽

Simple Model ◽

Conceptual Design ◽

Living Cells ◽

Complexity Measure ◽

Ceramic Foam ◽

Cell Life ◽

The Subject

The conceptual design of a perfusion reactor is the subject of this paper. The main objective of the reactor is the provision of nutrients to living cells grown in a porous medium fabricated of a given ceramic foam. In order to increase reactor throughput, the nutrients should be provided in a minimum time, without affecting the cell life. Various layouts of identical ceramic-foam pieces hosting the cells are proposed, the purpose being to select the variant with the highest likelihood of optimum performance, in the absence of a detailed mathematical model. A simple model is proposed, drawn from the discipline of hydraulic dynamical systems, which leads to a flow-complexity measure. The variant with the lowest complexity is then selected, for which a possible embodiment is proposed.

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Periodicity and Nonlinearity of Nonlinear Dynamical Systems Under Periodic Excitation

Volume 10: Mechanical Systems and Control, Parts A and B ◽

10.1115/imece2009-10619 ◽

2009 ◽

Cited By ~ 2

Author(s):

Lu Han ◽

Liming Dai ◽

Huayong Zhang

Keyword(s):

Nonlinear Dynamics ◽

Lyapunov Exponent ◽

Dynamical Systems ◽

Dynamic Systems ◽

Nonlinear Dynamic ◽

Nonlinear Dynamical Systems ◽

Research Work ◽

Nonlinear Dynamic Systems ◽

Periodic Excitation ◽

Nonlinear Dynamical

Periodicity and nonlinearity of nonlinear dynamic systems subjected to regular external excitations are studied in this research work. Diagnoses of regular and chaotic responses of nonlinear dynamic systems are performed with the implementation of a newly developed Periodicity Ratio in combining with the application of Lyapunov Exponent. The properties of the nonlinear dynamics systems are classified into four categories: periodic, irregular-nonchaotic, quasiperiodic and chaotic, each corresponding to their Periodicity Ratios. Detailed descriptions about diagnosing the responses of the four categories are presented with utilization of the Periodicity Ratio.

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Application of Reduced Order Modeling for Design of Interacting Maritime Structures

Volume 6: Ocean Engineering ◽

10.1115/omae2011-49260 ◽

2011 ◽

Author(s):

Jan Holterman ◽

Jan Peters ◽

Erik-Jan de Ridder

Keyword(s):

Dynamical Systems ◽

Deep Sea ◽

Reduced Order Modeling ◽

Design Approach ◽

Practical Application ◽

Reduced Order ◽

Model Based

A model based design approach for maritime structures is presented. This approach, involving reduced order modeling, is especially intended for floating maritime structures with one or more separate dynamical systems. It can be used for qualitative ranking of engineering solutions, but also for quantitative optimization of engineering activities. The approach is illustrated for a practical application, i.e., dredging at deep sea by means of a draghead suspended by wires.

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