Conditions for the absence of cycles of the second kind in continuous and discrete systems with cylindrical phase space

2014 ◽  
Vol 47 (3) ◽  
pp. 105-114
Author(s):  
V. B. Smirnova ◽  
N. V. Utina ◽  
A. I. Shepeljavyi ◽  
A. A. Perkin
Keyword(s):  
Phase Space ◽  
Discrete Systems ◽  
Cylindrical Phase ◽  
Cylindrical Phase Space

FRACTALS AND CLOSURES OF LINEAR DYNAMICAL SYSTEMS EXCITED STOCHASTICALLY BY TEMPORAL INPUTS

2001 ◽  
Vol 11 (03) ◽  
pp. 755-779 ◽  
Author(s):  
RYOICHI WADA ◽  
KAZUTOSHI GOHARA
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Vector Fields ◽  
Invariant Set ◽  
Poincaré Section ◽  
Poincare Section ◽  
Linear Dynamical Systems ◽  
Cylindrical Phase ◽  
Iterated Functions ◽  
Cylindrical Phase Space

Fractals and closures of two-dimensional linear dynamical systems excited by temporal inputs are investigated. The continuous dynamics defined by the set of vector fields in the cylindrical phase space is reduced to the discrete dynamics defined by the set of iterated functions on the Poincaré section. When all iterated functions are contractions, it has already been shown theoretically that a trajectory in the cylindrical phase space converges into an attractive invariant set with a fractal-like structure. Calculating analytically the Lipschitz constants of iterated functions, we show that, under some conditions, noncontractions often appear. However, we numerically show that, even for noncontractions, an attractive invariant set with a fractal-like structure exists. By introducing the interpolating system, we can also show that the set of trajectories in the cylindrical phase space is enclosed by the tube structure whose initial set is the closure of the fractal set on the Poincaré section.

First-Kind Cycles of Systems with Cylindrical Phase Space

2020 ◽  
Vol 248 (4) ◽  
pp. 457-466
Author(s):  
S. S. Mamonov ◽  
A. O. Kharlamova
Keyword(s):  
Phase Space ◽  
Cylindrical Phase ◽  
Cylindrical Phase Space

scholarly journals On the Gardner Problem for the Phase-Locked Loops

2019 ◽  
Vol 489 (6) ◽  
pp. 541-544
Author(s):  
N. V. Kuznetsov ◽  
M. Y. Lobachev ◽  
M. V. Yuldashev ◽  
R. V. Yuldashev
Keyword(s):  
Phase Space ◽  
Order System ◽  
Phase Locked Loops ◽  
Stability Criteria ◽  
Third Order ◽  
Classical Stability ◽  
Cylindrical Phase ◽  
Lock In ◽  
Cylindrical Phase Space

This report shows the possibilities of solving the Gardner problem of determining the lock-in range for multidimensional phase-locked loops systems. The development of analogs of classical stability criteria for the cylindrical phase space made it possible to obtain analytical estimates of the lock-in range for third-order system.

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