Multistability, quasiperiodicity and chaos in a self-oscillating ring dynamical system with three degrees of freedom based on the van der Pol generator

2021 ◽  
Vol 148 ◽  
pp. 110978
Author(s):  
Sergey Astakhov ◽  
Oleg Astakhov ◽  
Natalia Fadeeva ◽  
Vladimir Astakhov
2018 ◽  
Vol 27 (07) ◽  
pp. 1850066
Author(s):  
Payel Mukhopadhyay ◽  
K. Rajesh Nayak

Carter's constant is a nontrivial conserved quantity of motion of a particle moving in stationary axisymmetric spacetime. In the version of the theorem originally given by Carter, due to the presence of two Killing vectors, the system effectively has two degrees of freedom. We propose an extension to the first version of Carter's theorem to a system having three degrees of freedom to find two functionally independent Carter-like integrals of motion. We further generalize the theorem to a dynamical system with [Formula: see text] degrees of freedom. We further study the implications of Carter's constant to superintegrability and present a different approach to probe a superintegrable system. Our formalism gives another viewpoint to a superintegrable system using the simple observation of separable Hamiltonian according to Carter's criteria. We then give some examples by constructing some two-dimensional superintegrable systems based on this idea and also show that all three-dimensional simple classical superintegrable potentials are also Carter separable.


1971 ◽  
Vol 10 ◽  
pp. 110-117
Author(s):  
Claude Froeschle

AbstractDynamical systems with three degrees of freedom can be reduced to the study of a four-dimensional mapping. We consider here, as a model problem, the mapping given by the following equations: We have found that as soon as b ≠ 0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).


2021 ◽  
Vol 11 (24) ◽  
pp. 11943
Author(s):  
Wael S. Amer ◽  
Tarek S. Amer ◽  
Seham S. Hassan

The focus of this article is on the investigation of a dynamical system consisting of a linear damped transverse tuned-absorber connected with a non-linear damped-spring-pendulum, in which its hanged point moves in an elliptic path. The regulating system of motion is derived using Lagrange’s equations, which is then solved analytically up to the third approximation employing the approach of multiple scales (AMS). The emerging cases of resonance are categorized according to the solvability requirements wherein the modulation equations (ME) have been found. The stability areas and the instability ones are examined utilizing the Routh–Hurwitz criteria (RHC) and analyzed in line with the solutions at the steady state. The obtained results, resonance responses, and stability regions are addressed and graphically depicted to explore the positive influence of the various inputs of the physical parameters on the rheological behavior of the inspected system. The significance of the present work stems from its numerous applications in theoretical physics and engineering.


2018 ◽  
Vol 51 (13) ◽  
pp. 372-377 ◽  
Author(s):  
Juan E. Andrade García ◽  
Alejandra Ferreira de Loza ◽  
Luis T. Aguilar ◽  
Ramón I. Verdés

Author(s):  
A. H. S. Iyer ◽  
M. H. Colliander

Abstract Background The trend in miniaturisation of structural components and continuous development of more advanced crystal plasticity models point towards the need for understanding cyclic properties of engineering materials at the microscale. Though the technology of focused ion beam milling enables the preparation of micron-sized samples for mechanical testing using nanoindenters, much of the focus has been on monotonic testing since the limited 1D motion of nanoindenters imposes restrictions on both sample preparation and cyclic testing. Objective/Methods In this work, we present an approach for cyclic microcantilever bending using a micromanipulator setup having three degrees of freedom, thereby offering more flexibility. Results The method has been demonstrated and validated by cyclic bending of Alloy 718plus microcantilevers prepared on a bulk specimen. The experiments reveal that this method is reliable and produces results that are comparable to a nanoindenter setup. Conclusions Due to the flexibility of the method, it offers straightforward testing of cantilevers manufactured at arbitrary position on bulk samples with fully reversed plastic deformation. Specific microstructural features, e.g., selected orientations, grain boundaries, phase boundaries etc., can therefore be easily targeted.


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