scholarly journals Huygens’ clocks revisited

2017 ◽  
Vol 4 (9) ◽  
pp. 170777 ◽  
Author(s):  
Allan R. Willms ◽  
Petko M. Kitanov ◽  
William F. Langford
Keyword(s):  
Normal Form ◽  
Physical System ◽  
Bifurcation Theory ◽  
Phase Synchronization ◽  
Unified Framework ◽  
System Parameters ◽  
Experimental Systems ◽  
Weakly Coupled ◽  
Equivariant Bifurcation Theory ◽  
Phase Motion

In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. By contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here, it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, but also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather.

Application of Bifurcation Theory to Axial Flow Compressor Instability

1989 ◽  
Vol 111 (4) ◽  
pp. 426-433 ◽  
Author(s):  
F. E. McCaughan
Keyword(s):  
Parameter Space ◽  
Bifurcation Theory ◽  
Qualitative Difference ◽  
Axial Flow ◽  
Complete Information ◽  
Rotating Stall ◽  
Numerical Work ◽  
System Parameters ◽  
Parametric Effects ◽  
Flow Compressor

When a compression system becomes unstable, the mode of response depends on the operating and system parameters, such as throttle setting and B parameter. Previous numerical work on the model developed by Moore and Greitzer has provided a limited picture of the parametric effects. Applying bifurcation theory to a single-harmonic version of the model has supplied much more complete information, defining the boundaries of each mode of response in the parameter space. Specifically this is shown in a plot of B versus throttle setting, which compares well with the corresponding map produced experimentally. We stress the importance of the shape of the rotating stall characteristic. The analysis shows the qualitative difference between classic surge and deep surge.

Global Analysis of the Double Impact Oscillator Dynamics

1999 ◽  
Author(s):  
František Peterka
Keyword(s):  
Global Analysis ◽  
Impact Oscillator ◽  
System Parameters ◽  
Impact Oscillators ◽  
Real Value ◽  
Simulation Results ◽  
The Impact ◽  
Impact Motion ◽  
Double Impact ◽  
Phase Motion

Abstract The double impact oscillator represents two symmetrically arranged single impact oscillators. It is the model of a forming machine, which does not spread the impact impulses into its neighbourhood. The anti-phase impact motion of this system has the identical dynamics as the single system. The in-phase motion and the influence of asymmetries of the system parameters are studied using numerical simulations. Theoretical and simulation results are verified experimentally and the real value of the restitution coefficient is determined by this method.

WIGNER MEASURES AND MOLECULAR PROPAGATION THROUGH GENERIC ENERGY LEVEL CROSSINGS

2003 ◽  
Vol 15 (10) ◽  
pp. 1285-1317 ◽  
Author(s):  
CLOTILDE FERMANIAN KAMMERER
Keyword(s):  
Energy Transfer ◽  
Normal Form ◽  
Energy Level ◽  
Point Of View ◽  
Time Dependent ◽  
Type B ◽  
Unified Framework ◽  
Level Crossings ◽  
Normal Form Theorem ◽  
Uniformly Bounded

We study the time-dependent Schrödinger equation with matrix-valued potential presenting a generic crossing of type B, I, J or K in Hagedorn's classification. We use two-scale Wigner measures for describing the Landau–Zener energy transfer which occurs at the crossing. In particular, in the case of multiplicity 2 eigenvalues, we calculate precisely the change of polarization at the crossing. Our method provides a unified framework in which codimension 2, 3 or 5 crossings can be discussed. We recover Hagedorn's result for wave packets, from Wigner measure point of view, and extend them to any data uniformly bounded in L2. The proof is based on a normal form theorem which reduces the problem to an operator-valued Landau–Zener formula.

scholarly journals Equivariant bifurcation theory

Scholarpedia ◽  
2007 ◽  
Vol 2 (9) ◽  
pp. 2511 ◽  
Author(s):  
Jeff Moehlis ◽  
Edgar Knobloch
Keyword(s):  
Bifurcation Theory ◽  
Equivariant Bifurcation ◽  
Equivariant Bifurcation Theory

Application of Double Impact Oscillator in Forming

2001 ◽  
Author(s):  
František Peterka
Keyword(s):  
Numerical Simulations ◽  
Single System ◽  
Impact Oscillator ◽  
System Parameters ◽  
Impact Oscillators ◽  
Simulation Results ◽  
The Impact ◽  
Impact Motion ◽  
Double Impact ◽  
Phase Motion

Abstract The double impact oscillator represents two symmetrically arranged single impact oscillators. It is the model of a forming machine, which does not spread the impact impulses into its neighborhood. The anti-phase impact motion of this system has the identical dynamics as the single system. The in-phase motion and the influence of asymmetries of the system parameters are studied using numerical simulations. Theoretical and simulation results are verified experimentally.

Stability and Hopf bifurcation analysis of a new four-dimensional hyper-chaotic system

2020 ◽  
Vol 34 (29) ◽  
pp. 2050327
Author(s):  
Liangqiang Zhou ◽  
Ziman Zhao ◽  
Fangqi Chen
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Equilibrium Points ◽  
Chaotic Attractors ◽  
Local Dynamic ◽  
Chaotic Motions ◽  
System Parameters ◽  
Lyapunov Coefficient

With both analytical and numerical methods, local dynamic behaviors including stability and Hopf bifurcation of a new four-dimensional hyper-chaotic system are studied in this paper. All the equilibrium points and their stability conditions are obtained with the Routh–Hurwitz criterion. It is shown that there may exist one, two, or three equilibrium points for different system parameters. Via Hopf bifurcation theory, parameter conditions leading to Hopf bifurcation is presented. With the aid of center manifold and the first Lyapunov coefficient, it is also presented that the Hopf bifurcation is supercritical for some certain parameters. Finally, numerical simulations are given to confirm the analytical results and demonstrate the chaotic attractors of this system. It is also shown that the system may evolve chaotic motions through periodic bifurcations or intermittence chaos while the system parameters vary.

scholarly journals Computation and consequences of high order amplitude- and parameter-dependent tune shifts in storage rings for high precision measurements

2019 ◽  
Vol 34 (36) ◽  
pp. 1942011 ◽  
Author(s):  
Adrian Weisskopf ◽  
David Tarazona ◽  
Martin Berz
Keyword(s):  
Phase Space ◽  
Normal Form ◽  
High Precision ◽  
Storage Ring ◽  
Differential Algebra ◽  
Storage Rings ◽  
High Order ◽  
Precision Measurements ◽  
System Parameters ◽  
Parameter Dependent

Nonlinear effects of the various electric field and magnetic field components of storage rings to confine the particles and bend their trajectory can cause substantial amplitude-dependent tune shifts within the beam. Furthermore, tune shifts are often sensitive to variations of system parameters, e.g. total particle momentum offsets [Formula: see text]. Such amplitude- and parameter-dependent tune shifts influence the dynamics and stability of a beam in particle storage rings. Thus, it is critical for high precision measurements to analyze and understand these influences. On this basis, we present normal form methods for the calculation of high order amplitude and system parameter dependencies of the horizontal and vertical tunes in storage rings using the differential algebra (DA) framework within COSY INFINITY. A storage ring is simulated using COSY INFINITY to generate a DA Poincaré return map describing the transverse phase space behavior after each revolution in the storage ring. The map is expanded around the parameter-dependent closed orbit of the system before transforming the resulting map into normal form coordinates to extract the high order tune dependencies on the phase space amplitude and variation in the system parameters. As a specific example, a storage ring similar to the Storage Ring of the Muon [Formula: see text]-2 Experiment at Fermilab (E989) is investigated.

Sign in / Sign up

Export Citation Format

Share Document