Method for Constructing Periodic Solutions of a Controlled Dynamic System with a Cylindrical Phase Space

2020 ◽  
Vol 59 (2) ◽  
pp. 139-150
Author(s):  
L. A. Klimina
Keyword(s):  
Phase Space ◽  
Dynamic System ◽  
Periodic Solutions ◽  
Cylindrical Phase ◽  
Cylindrical Phase Space

NONEXISTENCE OF PERIODIC SOLUTIONS IN A CLASS OF DYNAMICAL SYSTEMS WITH CYLINDRICAL PHASE SPACE

2005 ◽  
Vol 15 (04) ◽  
pp. 1423-1431 ◽  
Author(s):  
YING YANG ◽  
ZHISHENG DUAN ◽  
LIN HUANG
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Linear Matrix ◽  
Specific Kind ◽  
Nonlinear Dynamical ◽  
Cylindrical Phase ◽  
Cylindrical Phase Space

This paper investigates the nonexistence of a specific kind of periodic solutions in a class of nonlinear dynamical systems with cylindrical phase space. Those types of systems can be viewed as an interconnection of several simpler subsystems with the interconnecting structure specified by a permutation matrix. Frequency-domain conditions as well as linear matrix inequalities conditions for nonexistence of limit cycles of the second kind are established. The main results also define the frequency range on which cycles of the second kind of the system cannot exist. Based on this LMI approach, an estimate of the frequency of cycles of the second kind can be explicitly computed by solving a generalized eigenvalue minimization problem. Numerical results demonstrate the applicability and validity of the proposed method and show the effect of nonlinear interconnections on dynamical behavior of entire interconnected systems.

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
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