scholarly journals The Effect of Slow Invariant Manifold and Slow Flow Dynamics on the Energy Transfer and Dissipation of a Singular Damped System with an Essential Nonlinear Attachment

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jamal-Odysseas Maaita ◽  
Efthymia Meletlidou
Keyword(s):  
Energy Transfer ◽  
Total Energy ◽  
Invariant Manifold ◽  
Nonlinear Oscillator ◽  
Invariant Manifolds ◽  
Dissipative System ◽  
Flow Dynamics ◽  
Slow Flow ◽  
Damped System ◽  
Slow Invariant Manifold

We study the effect of slow flow dynamics and slow invariant manifolds on the energy transfer and dissipation of a dissipative system of two linear oscillators coupled with an essential nonlinear oscillator with a mass much smaller than the masses of the linear oscillators. We calculate the slow flow of the system, the slow invariant manifold, the total energy of the system, and the energy that is stored in the nonlinear oscillator for different sets of the parameters and show that the bifurcations of the SIM and the dynamics of the slow flow play an important role in the energy transfer from the linear to the nonlinear oscillator and the rate of dissipation of the total energy of the initial system.

scholarly journals Energy transfer and dissipation in nonlinear oscillators

2014 ◽  
Author(s):  
Τζαμάλ-Οδυσσέας Μαάϊτα
Keyword(s):  
Energy Transfer ◽  
Invariant Manifold ◽  
Nonlinear Oscillators ◽  
Singularity Analysis ◽  
Slow Flow ◽  
Relaxation Oscillations ◽  
Slow Invariant Manifold

Στην παρούσα διδακτορική διατριβή μελετάμε ένα σύστημα τριών συζευγμένων ταλαντωτών με τριβή, δύο γραμμικών με έναν μη γραμμικό. Τέτοια συστήματα ταλαντωτών έχουν μεγάλο ενδιαφέρον ιδιαίτερα όταν η μάζα του μη γραμμικού ταλαντωτή είναι πολύ μικρότερη από τους γραμμικούς με συνέπεια ο μη γραμμικός ταλαντωτής να λειτουργεί ως καταβόθρα ενέργειας.Αυτού του είδους τα συστήματα, στα οποία συνυπάρχουν ένας αργός και ένας γρήγορος χρόνος μπορούν να μελετηθούν με τη βοήθεια της singularity analysis, των αναλλοίωτων πολλαπλοτήτων, ενώ σημαντική πληροφορία για τη δυναμική του συστήματος δίνεται και από την δυναμική της αργής ροής (Slow Flow) του συστήματος. Στην παρούσα διατριβή μελετάμε το σύστημα μέσω της μελέτης της αργής αναλλοίωτης πολλαπλότητας (Slow Invariant Manifold- SIM-). Με τη βοήθεια του θεωρήματος του Tikhonov κατηγοριοποιούμε τις διάφορες περιπτώσεις της αργής αναλλοίωτης πολλαπλότητας και ορίζουμε αναλυτικά τις συνθήκες με τις οποίες μπορούμε να οδηγηθούμε στην κάθε περίπτωση. Σε επόμενο βήμα μελετάμε την δυναμική της αργής ροής και παρατηρούμε ότι η δυναμική της είναι πλούσια αφού οι τροχιές της μπορούν να είναι κανονικές, να κάνουν ταλαντώσεις ηρεμίας (relaxation oscillations), ή να είναι χαοτικές. Από την μελέτη της ενέργειας που αποθηκεύεται στον μη γραμμικό ταλαντωτή και από τον ρυθμό απόσβεσης της συνολικής ενέργειας του συστήματος παρατηρούμε ότι τόσο η ύπαρξη διακλαδώσεων της αργής αναλλοίωτης πολλαπλότητας, όσο και η δυναμική της αργής ροής παίζουν καθοριστικό ρόλο στην μεταφορά ενέργειας από τον γραμμικό στον μη γραμμικό ταλαντωτή. Επίσης, στις περιπτώσεις που βλέπουμε μεταφορά ενέργειας παρατηρούμε ότι ο ρυθμός απόσβεσης της συνολικής ενέργειας του συστήματος είναι μεγαλύτερος από τον ρυθμό απόσβεσης όταν δεν μεταφέρεται ενέργεια στον μη γραμμικό ταλαντωτή. Η μελέτη του συστήματος των τριών συζευγμένων ταλαντωτών κλείνει με την πρόταση ενός μη γραμμικού ηλεκτρικου κυκλώματος το οποίο υλοποιεί την μη γραμμική διαφορική εξίσωση δεύτερης τάξης με την οποία προσεγγίσαμε το αρχικό σύστημα. Το συγκεκριμένο κύκλωμα έχει ενδιαφέρον γιατί μας δίνει τη δυνατότητα να μελετήσουμε και πειραματικά διάφορα από τα φαινόμενα που είδαμε στην θεωρητική μας ανάλυση.

Targeted energy transfer in stochastically excited system with nonlinear energy sink

2018 ◽  
Vol 30 (5) ◽  
pp. 869-886
Author(s):  
P. KUMAR ◽  
S. NARAYANAN ◽  
S. GUPTA
Keyword(s):  
Energy Transfer ◽  
Equations Of Motion ◽  
Nonlinear Energy Sink ◽  
Flow Dynamics ◽  
Slow Flow ◽  
Targeted Energy Transfer ◽  
Optimal Regime ◽  
Excited System ◽  
Energy Sink ◽  
Nonlinear Energy

This study investigates the phenomenon of targeted energy transfer (TET) from a linear oscillator to a nonlinear attachment behaving as a nonlinear energy sink for both transient and stochastic excitations. First, the dynamics of the underlying Hamiltonian system under deterministic transient loading is studied. Assuming that the transient dynamics can be partitioned into slow and fast components, the governing equations of motion corresponding to the slow flow dynamics are derived and the behaviour of the system is analysed. Subsequently, the effect of noise on the slow flow dynamics of the system is investigated. The Itô stochastic differential equations for the noisy system are derived and the corresponding Fokker–Planck equations are numerically solved to gain insights into the behaviour of the system on TET. The effects of the system parameters as well as noise intensity on the optimal regime of TET are studied. The analysis reveals that the interaction of nonlinearities and noise enhances the optimal TET regime as predicted in deterministic analysis.

THE SLOW INVARIANT MANIFOLD OF A CONSERVATIVE PENDULUM-OSCILLATOR SYSTEM

1996 ◽  
Vol 06 (04) ◽  
pp. 673-692 ◽  
Author(s):  
IOANNIS T. GEORGIOU ◽  
IRA B. SCHWARTZ
Keyword(s):  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Slow Manifold ◽  
Equilibrium States ◽  
Periodic Motions ◽  
Nonlinear Normal Mode ◽  
Oscillator System ◽  
Slow Invariant Manifold ◽  
Nonlinear Normal ◽  
Global Invariant

We analyze the motions of a conservative pendulum-oscillator system in the context of invariant manifolds of motion. Using the singular perturbation methodology, we show that whenever the natural frequency of the oscillator is sufficiently larger than that of the pendulum, there exists a global invariant manifold passing through all static equilibrium states and tangent to the linear eigenspaces at these equilibrium states. The invariant manifold, called slow, carries a continuum of slow periodic motions, both oscillatory and rotational. Computations to various orders of approximation to the slow invariant manifold allow analysis of motions on the slow manifold, which are verified with numerical experiments. Motion on the slow invariant manifold is identified with a slow nonlinear normal mode.

Slow invariant manifold assessments in multi-route reaction mechanism

2019 ◽  
Vol 284 ◽  
pp. 265-270 ◽  
Author(s):  
M. Shahzad ◽  
F. Sultan ◽  
M. Ali ◽  
W.A. Khan ◽  
M. Irfan
Keyword(s):  
Reaction Mechanism ◽  
Invariant Manifold ◽  
Slow Invariant Manifold

scholarly journals Dynamical Tangles in Third-Order Oscillator with Single Jump Function

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Jiří Petržela ◽  
Tomas Gotthans ◽  
Milan Guzan
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Nonlinear Oscillator ◽  
Invariant Manifolds ◽  
Order Differential Equation ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Third Order ◽  
Newtonian Dynamics ◽  
Jump Function

This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity. The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems. The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

Invariant Manifold Detection From Phase Space Trajectories

2008 ◽  
Author(s):  
Joseph Kuehl ◽  
David Chelidze
Keyword(s):  
Phase Space ◽  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Basins Of Attraction ◽  
Detection Methods ◽  
Trajectory Data ◽  
Space Trajectories ◽  
Data Set ◽  
Phase Space Warping ◽  
Phase Space Trajectories

Invariant manifolds provide important information about the structure of flows. When basins of attraction are present, the stable invariant manifold serves as the boundary between these basins. Thus, in experimental applications such as vibrations problems, knowledge of these manifolds is essential to understanding the evolution of phase space trajectories. Most existing methods for identifying invariant manifolds of a flow rely on knowledge of the flow field. However, in experimental applications only knowledge of phase space trajectories is available. We provide modifications to several existing invariant manifold detection methods which enables them to deal with trajectory only data, as well as introduce a new method based on the concept of phase space warping. The method of Stochastic Interrogation applied to the damped, driven Duffing equation is used to generate our data set. The result is a set of trajectory data which randomly populates a phase space. Manifolds are detected from this data set using several different methods. First is a variation on manifold “growing,” and is based on distance of closest approach to a hyperbolic trajectory with “saddle like behavior.” Second, three stretching based schemes are considered. One considers the divergence of trajectory pairs, another quantifies the deformation of a nearest neighbor cloud, and the last uses flow fields calculated from the trajectory data. Finally, the new phase space warping method is introduced. This method takes advantage of the shifting (warping) experienced by a phase space as the parameters of the system are slightly varied. This results in a shift of the invariant manifolds. The region spanned by this shift, provides a means to identify the invariant manifolds. Results show that this method gives superior detection and is robust with respect to the amount of data.

scholarly journals Peripheral Vascular Anomalies – Essentials in Periinterventional Imaging

2019 ◽  
Vol 192 (02) ◽  
pp. 150-162 ◽  
Author(s):  
Maliha Sadick ◽  
Daniel Overhoff ◽  
Bettina Baessler ◽  
Naema von Spangenberg ◽  
Lena Krebs ◽  
...  
Keyword(s):  
Therapeutic Approach ◽  
Vascular Malformations ◽  
Vascular Anomaly ◽  
Flow Dynamics ◽  
Vascular Anomalies ◽  
Vascular Tumors ◽  
Slow Flow ◽  
Peripheral Vascular ◽  
Fast Flow ◽  
Selection Of

Background Peripheral vascular anomalies represent a rare disease with an underlying congenital mesenchymal and angiogenetic disorder. Vascular anomalies are subdivided into vascular tumors and vascular malformations. Both entities include characteristic features and flow dynamics. Symptoms can occur in infancy and adulthood. Vascular anomalies may be accompanied by characteristic clinical findings which facilitate disease classification. The role of periinterventional imaging is to confirm the clinically suspected diagnosis, taking into account the extent and location of the vascular anomaly for the purpose of treatment planning. Method In accordance with the International Society for the Study of Vascular Anomalies (ISSVA), vascular anomalies are mainly categorized as slow-flow and fast-flow lesions. Based on the diagnosis and flow dynamics of the vascular anomaly, the recommended periinterventional imaging is described, ranging from ultrasonography and plain radiography to dedicated ultrafast CT and MRI protocols, percutaneous phlebography and transcatheter angiography. Each vascular anomaly requires dedicated imaging. Differentiation between slow-flow and fast-flow vascular anomalies facilitates selection of the appropriate imaging modality or a combination of diagnostic tools. Results Slow-flow congenital vascular anomalies mainly include venous and lymphatic or combined malformations. Ultrasound and MRI and especially MR-venography are essential for periinterventional imaging. Arteriovenous malformations are fast-flow vascular anomalies. They should be imaged with dedicated MR protocols, especially when extensive. CT with 4D perfusion imaging as well as time-resolved 3D MR-A allow multiplanar perfusion-based assessment of the multiple arterial inflow and venous drainage vessels of arterio-venous malformations. These imaging tools should be subject to intervention planning, as they can reduce procedure time significantly. Fast-flow vascular tumors like hemangiomas should be worked up with ultrasound, including color-coded duplex sonography, MRI and transcatheter angiography in case of a therapeutic approach. In combined malformation syndromes, radiological imaging has to be adapted according to the dominant underlying vessels and their flow dynamics. Conclusion Guide to evaluation of flow dynamics in peripheral vascular anomalies, involving vascular malformations and vascular tumors with the intention to facilitate selection of periinterventional imaging modalities and diagnostic and therapeutic approach to vascular anomalies. Key Points:  Citation Format

Black swans and canards in two predator – one prey model

2019 ◽  
Vol 14 (4) ◽  
pp. 408 ◽  
Author(s):  
Elena Shchepakina
Keyword(s):  
Invariant Manifold ◽  
Black Swan ◽  
Black Swans ◽  
Prey Model ◽  
Slow Invariant Manifold

In this paper, we show how canards can be easily caught in a class of 3D systems with an exact black swan (a slow invariant manifold of variable stability). We demonstrate this approach to a canard chase via the two predator – one prey model. It is shown that the technique described allows us to get various 3D oscillations by changing the shape of the trajectories of two 2D-projections of the original 3D system.

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