scholarly journals Bifurcations with imperfect SO(2) symmetry and pinning of rotating waves

2013 ◽  
Vol 469 (2152) ◽  
pp. 20120348 ◽  
Author(s):  
Francisco Marques ◽  
Alvaro Meseguer ◽  
Juan M. Lopez ◽  
J. Rafael Pacheco ◽  
Jose M. Lopez
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Parameter Space ◽  
Scaling Laws ◽  
Rotating Waves ◽  
Numerical Studies ◽  
Equivariant Dynamical Systems ◽  
Steady Solutions ◽  
Frequency Changes ◽  
The Waves

Rotating waves are periodic solutions in SO(2) equivariant dynamical systems. Their precession frequency changes with parameters and it may change sign, passing through zero. When this happens, the dynamical system is very sensitive to imperfections that break the SO(2) symmetry and the waves may become trapped by the imperfections, resulting in steady solutions that exist in a finite region in parameter space. This is the so-called pinning phenomenon. In this study, we analyse the breaking of the SO(2) symmetry in a dynamical system close to a Hopf bifurcation whose frequency changes sign along a curve in parameter space. The problem is very complex, as it involves the complete unfolding of high codimension. A detailed analysis of different types of imperfections indicates that a pinning region surrounded by infinite-period bifurcation curves appears in all cases. Complex bifurcational processes, strongly dependent on the specifics of the symmetry breaking, appear very close to the intersection of the Hopf bifurcation and the pinning region. Scaling laws of the pinning region width and partial breaking of SO(2) to Z m are also considered. Previous as well as new experimental and numerical studies of pinned rotating waves are reviewed in the light of the new theoretical results.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

scholarly journals Mathematical Analysis of an Epidemic-Species Hybrid Dynamical System

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Jing Hui ◽  
Jian-Hua Pang ◽  
Dong-Rong Lin
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Local Stability ◽  
Orbital Stability ◽  
Species Hybrid ◽  
Hybrid Dynamical System ◽  
Mathematical Analyses ◽  
Existence Of Equilibria ◽  
Impulsive Releasing ◽  
Infected Prey

We consider an epidemic-species hybrid dynamical system. The disease is spread among the prey only and the infected prey can reproduce virus. The predator only eats the infected prey. Mathematical analyses are given for the system with regard to the existence of equilibria, local stability, Hopf bifurcation, and the orbital stability of the Hopf bifurcating limit cycle. We further analyse the system under impulsive releasing of virus and predator.

scholarly journals Two families of elliptical plasma lenses

2019 ◽  
Vol 488 (4) ◽  
pp. 5651-5664 ◽  
Author(s):  
Xinzhong Er ◽  
Adam Rogers
Keyword(s):  
Parameter Space ◽  
Gravitational Lensing ◽  
Low Frequency ◽  
Wavelength Dependence ◽  
Free Electrons ◽  
Density Distributions ◽  
Critical Curves ◽  
Numerical Studies ◽  
Plasma Distribution ◽  
Elliptical Plasma

ABSTRACT Plasma lensing is the refraction of low-frequency electromagnetic rays due to free electrons in the interstellar medium. Although the phenomenon has a distinct similarity to gravitational lensing, particularly in its mathematical description, plasma lensing introduces other additional features, such as wavelength dependence, radial rather than tangential image distortions, and strong demagnification of background sources. Axisymmetrical models of plasma lenses have been well studied in the literature, but density distributions with more complicated shapes can provide new and exotic image configurations and increase the richness of the magnification properties. As a first step towards non-axisymmetrical distributions, we study two families of elliptical plasma lens, softened power law, and exponential plasma distributions. We perform numerical studies on each lens model, and present them over a parameter space. In addition to deriving elliptical plasma lens formulae, we also investigate the number of critical curves that the lens can produce by studying the lens parameter space, in particular the dependence on the lensing ellipticity. We find that the introduction of ellipticity into the plasma distribution can enhance the lensing effects as well as the complexity of the magnification map.

scholarly journals Exploring the phase diagram of fully turbulent Taylor–Couette flow

2014 ◽  
Vol 761 ◽  
pp. 1-26 ◽  
Author(s):  
Rodolfo Ostilla-Mónico ◽  
Erwin P. van der Poel ◽  
Roberto Verzicco ◽  
Siegfried Grossmann ◽  
Detlef Lohse
Keyword(s):  
Phase Diagram ◽  
Reynolds Number ◽  
Couette Flow ◽  
Parameter Space ◽  
Coriolis Force ◽  
Scaling Laws ◽  
Outer Cylinder ◽  
Rotating Cylinders ◽  
Aspect Ratios ◽  
Cylinder Rotation

AbstractDirect numerical simulations of Taylor–Couette flow, i.e. the flow between two coaxial and independently rotating cylinders, were performed. Shear Reynolds numbers of up to $3\times 10^{5}$, corresponding to Taylor numbers of $\mathit{Ta}=4.6\times 10^{10}$, were reached. Effective scaling laws for the torque are presented. The transition to the ultimate regime, in which asymptotic scaling laws (with logarithmic corrections) for the torque are expected to hold up to arbitrarily high driving, is analysed for different radius ratios, different aspect ratios and different rotation ratios. It is shown that the transition is approximately independent of the aspect and rotation ratios, but depends significantly on the radius ratio. We furthermore calculate the local angular velocity profiles and visualize different flow regimes that depend both on the shearing of the flow, and the Coriolis force originating from the outer cylinder rotation. Two main regimes are distinguished, based on the magnitude of the Coriolis force, namely the co-rotating and weakly counter-rotating regime dominated by Rayleigh-unstable regions, and the strongly counter-rotating regime where a mixture of Rayleigh-stable and Rayleigh-unstable regions exist. Furthermore, an analogy between radius ratio and outer-cylinder rotation is revealed, namely that smaller gaps behave like a wider gap with co-rotating cylinders, and that wider gaps behave like smaller gaps with weakly counter-rotating cylinders. Finally, the effect of the aspect ratio on the effective torque versus Taylor number scaling is analysed and it is shown that different branches of the torque-versus-Taylor relationships associated to different aspect ratios are found to cross within 15 % of the Reynolds number associated to the transition to the ultimate regime. The paper culminates in phase diagram in the inner versus outer Reynolds number parameter space and in the Taylor versus inverse Rossby number parameter space, which can be seen as the extension of the Andereck et al. (J. Fluid Mech., vol. 164, 1986, pp. 155–183) phase diagram towards the ultimate regime.

Hopf Bifurcation of a Type of Neuron Model with Multiple Time Delays

2019 ◽  
Vol 29 (12) ◽  
pp. 1950163 ◽  
Author(s):  
Suqi Ma
Keyword(s):  
Hopf Bifurcation ◽  
Parameter Space ◽  
Time Delays ◽  
Neuron Model ◽  
Multiple Time ◽  
Multiple Time Delays ◽  
The Stability ◽  
Geometrical Scheme ◽  
Delay Differential ◽  
Two Parameter

By applying a geometrical scheme developed to tackle the eigenvalue problem of delay differential equations with multiple time delays, Hopf bifurcation of Hopfield neuron model is analyzed in two-parameter space. By the introduction of two new angles, the calculation of imaginary roots is carried out analytically and effectively. By increasing the parameter to cross over the Hopf bifurcation lines, the stability switching direction is confirmed. The method is a useful tool to show the partition of stable and unstable regions in two-parameter space and detect double Hopf bifurcation further. The typified dynamical behaviors based on nearby double Hopf points are analyzed by applying the normal form technique and center manifold method.

Numerical studies into the parameter space conducive to "lock-on" in a GaN photoconductive switch for high power applications

2019 ◽  
Vol 26 (2) ◽  
pp. 469-475
Author(s):  
Animesh R. Chowdhury ◽  
Sergey Nikishin ◽  
James Dickens ◽  
Andreas Neuber ◽  
Ravi P. Joshi ◽  
...  
Keyword(s):  
Parameter Space ◽  
High Power ◽  
Photoconductive Switch ◽  
Numerical Studies

scholarly journals ROBUSTIFICATION OF CHAOS IN 2D MAPS

2011 ◽  
Vol 14 (06) ◽  
pp. 817-827 ◽  
Author(s):  
ELHADJ ZERAOULIA ◽  
J. C. SPROTT
Keyword(s):  
Dynamical System ◽  
Parameter Space ◽  
Piecewise Smooth ◽  
Coexisting Attractors ◽  
Border Collision Bifurcation ◽  
Chaotic Region ◽  
Planar Maps

Robust chaos is defined as the absence of periodic windows and coexisting attractors in some neighborhood of the parameter space since the existence of such windows in the chaotic region implies fragility of the chaos. In this paper, we introduce a new terminology called robustification of chaos, which means creating robust chaos (in the sense of the above definition) in a dynamical system. As a first step, a new chaotification (robustification) method to generate robust chaos in planar maps is presented using simple piecewise smooth feedback to create a border collision bifurcation in the resulting system under some realizable conditions. The results are applied to an elementary example to illustrate the validity of the proposed method.

The Construction for Generalized Mandelbrot Sets of the Frieze Group

2013 ◽  
Vol 756-759 ◽  
pp. 2562-2566
Author(s):  
Feng Ying Wang ◽  
Li Ming Du ◽  
Zi Yang Han
Keyword(s):  
Dynamical System ◽  
Parameter Space ◽  
Mandelbrot Sets ◽  
Group A ◽  
Novel Method ◽  
Definition Of ◽  
Generalized Mandelbrot Sets

By an analysis of symmetric features of equivalent mappings of the frieze group, a definition of their generalized Mandelbrot sets is given and a novel method for constructing generalized Mandelbrot sets of equivalent mappings of frieze group is presented via utilizing the Ljapunov exponent as the judgment standard. Based on generating parameter space of dynamical system, lots of patterns of generalized Mandelbrot sets are produced.

scholarly journals Tracking Nonlinear Oscillations with Time-Delayed Feedback

2006 ◽  
Vol 5-6 ◽  
pp. 417-424
Author(s):  
Jan Sieber ◽  
B. Krauskopf
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Oscillations ◽  
Stability Boundary ◽  
Nonlinear Dynamical ◽  
Friction Oscillator ◽  
Long Time ◽  
Basic Ideas ◽  
The Stability

We demonstrate a method for tracking the onset of oscillations (Hopf bifurcation) in nonlinear dynamical systems. Our method does not require a mathematical model of the dynamical system but instead relies on feedback controllability. This makes the approach potentially applicable in an experiment. The main advantage of our method is that it allows one to vary parameters directly along the stability boundary. In other words, there is no need to observe the transient oscillations of the dynamical system for a long time to determine their decay or growth. Moreover, the procedure automatically tracks the change of the critical frequency along the boundary and is able to continue the Hopf bifurcation curve into parameter regions where other modes are unstable.We illustrate the basic ideas with a numerical realization of the classical autonomous dry friction oscillator.

scholarly journals On the Dynamics of Coupled Parametrically Forced Oscillators

2008 ◽  
Author(s):  
Jeff Moehlis
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Weak Coupling ◽  
Weak Coupling Limit ◽  
Stable Periodic Orbit ◽  
Weakly Coupled ◽  
Forced Oscillators ◽  
Stable Periodic ◽  
Autonomous Dynamical System

It is well known that an autonomous dynamical system can have a stable periodic orbit, arising for example through a Hopf bifurcation. When a collection of such oscillators is coupled together, the system can display a number of phase-locked solutions which can be understood in the weak coupling limit by using a phase model. It is also well known that a stable periodic orbit can be found for a parametrically forced dynamical system, with the phase of the periodic orbit being locked to the forcing. Here we discuss the periodic solutions which occur for a collection of such parametrically forced oscillators that are weakly coupled together.

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