scholarly journals Singularly perturbed dynamics of the tippedisk

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Simon Sailer ◽  
Remco I. Leine
Keyword(s):  
Global Analysis ◽  
Three Dimensional ◽  
Slow Manifold ◽  
Singularly Perturbed ◽  
Singular Perturbation Theory ◽  
Local Stability Analysis ◽  
Spinning Speed ◽  
Spinning Disc ◽  
Form Condition ◽  
Inversion Phenomenon

The tippedisk is a mathematical-mechanical archetype for a peculiar friction-induced instability phenomenon leading to the inversion of an unbalanced spinning disc, being reminiscent of (but different from) the well-known inversion of the tippetop. A reduced model of the tippedisk, in the form of a three-dimensional ordinary differential equation, has been derived recently, followed by a preliminary local stability analysis of stationary spinning solutions. In the current paper, a global analysis of the reduced system is pursued using the framework of singular perturbation theory. It is shown how the presence of friction leads to slow–fast dynamics and the creation of a two-dimensional slow manifold. Furthermore, it is revealed that a bifurcation scenario involving a homoclinic bifurcation and a Hopf bifurcation leads to an explanation of the inversion phenomenon. In particular, a closed-form condition for the critical spinning speed for the inversion phenomenon is derived. Hence, the tippedisk forms an excellent mathematical-mechanical problem for the analysis of global bifurcations in singularly perturbed dynamics.

scholarly journals Another New Chaotic System: Bifurcation and Chaos Control

2020 ◽  
Vol 30 (11) ◽  
pp. 2050161
Author(s):  
Arnob Ray ◽  
Dibakar Ghosh
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Period Doubling ◽  
Slow Manifold ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Dimensional Parameter ◽  
Lyapunov Coefficient ◽  
The Stability

We propose a new simple three-dimensional continuous autonomous model with two nonlinear terms and observe the dynamical behavior with respect to system parameters. This system changes the stability of fixed point via Hopf bifurcation and then undergoes a cascade of period-doubling route to chaos. We analytically derive the first Lyapunov coefficient to investigate the nature of Hopf bifurcation. We investigate well-separated regions for different kinds of attractors in the two-dimensional parameter space. Next, we introduce a timescale ratio parameter and calculate the slow manifold using geometric singular perturbation theory. Finally, the chaotic state annihilates by decreasing the value of the timescale ratio parameter.

OREGONATOR-BASED SIMULATION OF THE BELOUSOV–ZHABOTINSKII REACTION

2007 ◽  
Vol 17 (12) ◽  
pp. 4337-4353 ◽  
Author(s):  
MO-HONG CHOU ◽  
HSIU-CHUAN WEI ◽  
YU-TUAN LIN
Keyword(s):  
Initial Conditions ◽  
Global Analysis ◽  
Three Dimensional ◽  
Power Spectra ◽  
Stable Limit Cycle ◽  
Fractal Dimensions ◽  
Slow Manifold ◽  
Phase Lag ◽  
Stable Limit ◽  
Belousov Zhabotinskii Reaction

Some observations are made on the Belousov–Zhabotinskii reaction simulated via the Field–Noyes model, also referred to as the Oregonator, and its modification. The simulation is performed with the aid of a cell-to-cell mapping for global analysis. Regarding the standard Oregonator, a two-dimension-like region in the three-dimensional phase space is detected showing the sensitive dependence of short-term ODE integrations on initial conditions. Trajectories with initial conditions closely located in this region may experience a phase lag if they eventually approach the same stable limit cycle connected with a subcritical Hopf bifurcation. When a flow term is added to the Oregonator, chaos can be brought about to mimic the experimental finding by suitably pleating the slow manifold. Coexistent attractors now may have a chaotic member and a fractal separatrix detected by the global analysis. The above mentioned sensitive region is found to play a significant role in shaping the pleating in order for chaos to happen in a manner analogous to the "screw-type" proposed by [Rössler, 1977] as one of the two prototypes for three-variable systems. Some relevant calculations of Lyapunov exponents, fractal dimensions and power spectra are also included.

NUMERICAL COMPUTATION OF CANARDS

2000 ◽  
Vol 10 (12) ◽  
pp. 2669-2687 ◽  
Author(s):  
JOHN GUCKENHEIMER ◽  
KATHLEEN HOFFMAN ◽  
WARREN WECKESSER
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Electrical Potential ◽  
Slow Manifold ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Singularly Perturbed System ◽  
Chemical Systems ◽  
Perturbed System ◽  
Physical And Chemical

Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuation of solutions of boundary value problems. One of these periodic orbits contains canards, trajectory segments that follow unstable portions of a slow manifold. We identify several mechanisms that lead to the formation of these and other canards in this example.

scholarly journals A global analysis of the atmospheric pollutant modeling

2007 ◽  
Vol 22 (1) ◽  
pp. 1-9
Author(s):  
Umberto Rizza ◽  
Jonas C. Carvalho ◽  
Davidson M. Moreira ◽  
Marcelo R. Moraes ◽  
Antônio G. Goulart
Keyword(s):  
Global Analysis ◽  
Turbulent Transport ◽  
Three Dimensional ◽  
Ground Level ◽  
Lagrangian Model ◽  
Linear Superposition ◽  
Atmospheric Pollutant ◽  
Advection Diffusion Equation ◽  
Laplace Transform Technique ◽  
The Laplace Transform

In this article is carried out a comparison between Lagrangian and Eulerian modelling of the turbulent transport of pollutants within the Planetary Boundary Layer (PBL). The Lagrangian model is based on a three-dimensional form of the Langevin equation for the random velocity. The Eulerian analytical model is based on a discretization of the PBL in N sub-layers; in each of the sub-layers the advection-diffusion equation is solved by the Laplace transform technique. In the Eulerian numerical model the advective terms are solved using the cubic spline method while a Crank-Nicholson scheme is used for the diffusive terms. The models use a turbulence parameterization that considers a spectrum model, which is given by a linear superposition of the buoyancy and mechanical effects. Observed ground-level concentrations measured in a dispersion field experiment are used to evaluate the simulations.

scholarly journals Numerical simulation of rainfall with assimilation of conventional and GPS observations over north of Iran

2016 ◽  
Vol 59 (3) ◽  
Author(s):  
Mohammad Ali Sharifi ◽  
Majid Azadi ◽  
Ali Sam Khaniani
Keyword(s):  
Relative Humidity ◽  
Caspian Sea ◽  
Model Simulation ◽  
Global Analysis ◽  
Three Dimensional ◽  
Wrf Model ◽  
Precipitable Water ◽  
Absolute Error ◽  
The Caspian Sea ◽  
North Of Iran

<p>In this work, the effect of assimilation of synoptic, radiosonde and ground-based GPS precipitable water vapor (PWV) data has been investigated on the short-term prediction of precipitation, vertical relative humidity and PWV fields over north of Iran. We selected two rainfall events (i.e. February 1, 2014, and September 17, 2014) caused by synoptic systems affecting the southern coasts of the Caspian Sea. These systems are often associated with a shallow and cold high pressure located over Russia that extends towards the southern Caspian Sea. The three dimensional variational (3DVAR) data assimilation system of the weather research and forecasting (WRF) model is used in two rainfall cases. In each case, three numerical experiments, namely CTRL, CONVDA and GPSCONVDA, are performed. The CTRL experiment uses the global analysis as the initial and boundary conditions of the model. In the second experiment, surface and radiosonde observations are inserted into the model. Finally, the GPSCONVDA experiment uses the GPS PWV data in the assimilation process in addition to the conventional observations. It is found that in CONVDA experiment, the mean absolute error (MAE) of the accumulated precipitation is reduced about 5 and 13 percent in 24h model simulation of February and September cases, respectively, when compared to CTRL. Also, the results in both cases suggest that the assimilation of GPS data has the greatest impact on model PWV simulations, with maximum root mean squares error (RMSE) reduction of 0.7 mm. In the GPSCONVDA experiment, comparison of the vertical profiles of 12h simulated relative humidity with the corresponding radiosonde observations shows a slight improvement in the lower levels.</p>

Dynamic Analysis of Mechanical Systems With Time-Varying Topologies: Part 1 — Dynamic Model and Stability Analysis

1990 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Stability Analysis ◽  
Dynamic Model ◽  
Local Stability ◽  
Global Analysis ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Time Varying ◽  
Discrete Equations ◽  
Local Stability Analysis ◽  
Multiple Collisions

Abstract A model is developed for analyzing mechanical systems with a pair of bodies with topological changes in their kinematic constraints. It is built upon the concept of Poincaré map rather than following the traditional methods of differential equations. The model provides a set of well-defined and naturally-discrete equations of motion and is capable of giving physical insights of dynamic characteristics of deadbeat convergence of multiple collisions and periodic or chaotic responses. The development of dynamic model and a local stability analysis are presented in Part 1, and the global analysis and numerical simulation are discussed in Part 2.

scholarly journals New insight into 3-D mesoscale eddy properties from CMEMS operational models in the western Mediterranean

Ocean Science ◽  
2019 ◽  
Vol 15 (4) ◽  
pp. 1111-1131 ◽  
Author(s):  
Evan Mason ◽  
Simón Ruiz ◽  
Romain Bourdalle-Badie ◽  
Guillaume Reffray ◽  
Marcos García-Sotillo ◽  
...  
Keyword(s):  
Composite Structures ◽  
Global Analysis ◽  
Rapid Evolution ◽  
Mesoscale Eddy ◽  
Three Dimensional ◽  
Western Mediterranean ◽  
Alboran Sea ◽  
Western Mediterranean Sea ◽  
Common Domain ◽  
Alborán Sea

Abstract. Rapid evolution of operational ocean forecasting systems is driven by advances in numerics and data assimilation schemes, and increase of in situ and satellite observations. The Copernicus Marine Service (CMEMS) is a major provider of operational products that are made available through an online catalogue. The service includes global and regional forecasts in near-real-time and reanalysis modes. Here, we apply an eddy tracker to daily sea surface height (SSH) fields from three such reanalysis products from the CMEMS catalogue, with the objective to evaluate their performance in terms of their eddy properties and three-dimensional composite structures over the 2013–2016 period. The products are (i) the Global Analysis Forecast, (ii) the Mediterranean Analysis Forecast and (iii) the Iberia–Biscay–Ireland Analysis Forecast. The common domain between these reanalyses is the western Mediterranean Sea (WMED) between the Strait of Gibraltar and Sardinia. This is a complex region with strong density gradients, especially in the Alboran Sea in the west where Atlantic and Mediterranean waters compete. Surface eddy property maps over the WMED of eddy radii, amplitudes and nonlinearity are consistent between the models, as well as with gridded altimetric data that serve as a reference. Mean 3-D eddy composites are shown only for three subregions in the Alboran Sea. These are mostly consistent between the models, with minor differences being attributed to details of the respective model configurations. This information can be informative for the ongoing development of these CMEMS operational modeling systems. The mesoscale data provided here may be of interest to CMEMS users and in the future could be a useful addition to a more diverse CMEMS catalogue.

scholarly journals Real Geometrical Imperfection of Bow-String Arches—Measurement and Global Analysis

2020 ◽  
Vol 10 (13) ◽  
pp. 4530
Author(s):  
Jaroslav Odrobiňák ◽  
Matúš Farbák ◽  
Jakub Chromčák ◽  
Ján Kortiš ◽  
Jozef Gocál
Keyword(s):  
Numerical Models ◽  
Global Analysis ◽  
Three Dimensional ◽  
3D Scanning ◽  
Research Activities ◽  
Building Information ◽  
Bridge Superstructure ◽  
Geometrical Imperfections ◽  
Theoretical And Numerical Analysis ◽  
Arch Bridges

In order to analyse the buckling behaviour of existing bow-string arch bridges, it is necessary to deal with the imperfections that influence the global stability of their superstructures. Direct quantification of the material imperfections represents an extremely difficult task for this type of structure. On the other hand, the geometrical imperfections can be measured in more detail by using special scanners or high-accuracy surveying instruments. This contribution represents a beginning part of the research activities focusing on the real values of geometric imperfections of existing steel arch bridges using three-dimensional (3D) scanning. The possibility of using these data for further theoretical and numerical analysis based on the finite element method (FEM) and for further creating the building information modelling (BIM) of the bridges is proposed. When verifying the stability of bow-string arch bridges, much higher attention has to be paid to the out-of-plane stability of the arches. The numerical models of an existing bridge superstructure were developed to execute a nonlinear analysis with geometrical imperfections included. Both the theoretical and actual imperfections obtained by 3D scanning were taken into account. The obtained data, their comparison and the applicability of the presented method are finally discussed.

Sign in / Sign up

Export Citation Format

Share Document