Time-Scale Model Reduction of Microgrid Based on Singular Perturbation Theory

Author(s):  
Zhikang Shuai
2005 ◽  
Vol 15 (11) ◽  
pp. 3411-3421 ◽  
Author(s):  
JOHN GUCKENHEIMER ◽  
KATHLEEN HOFFMAN ◽  
WARREN WECKESSER

Relaxation oscillations are periodic orbits of multiple time scale dynamical systems that contain both slow and fast segments. The slow–fast decomposition of these orbits is defined in the singular limit. Geometric methods in singular perturbation theory classify degeneracies of these decompositions that occur in generic one-parameter families of relaxation oscillations. This paper investigates the bifurcations that are associated with one type of degeneracy that occurs in systems with two slow variables, in which relaxation oscillations become homoclinic to a folded saddle.


1977 ◽  
Vol 44 (1) ◽  
pp. 25-30 ◽  
Author(s):  
S. Weinbaum ◽  
L. M. Jiji

This paper treats the problem of the inward solidification at large Stefan number 1/ε, ε = CP(Ti − Tf)/L, of a finite slab which is initially at an arbitrary temperature Ti above the melting point. The face at which the heat is removed is maintained at a constant temperature below fusion while the opposite face is either (a) insulated or (b) kept at the initial temperature. Perturbation series solutions in ε are obtained for both the short-time scale characterizing the transient diffusion in the liquid phase and the long-time scale characterizing the interface motion. The asymptotic matching of the two series solutions shows that to O(ε1/2) the short-time series solution for interface motion for the insulated Case (a) is uniformly valid for all time. A singular perturbation theory is, however, required for the isothermal Case (b) since the interface motion is affected to this order by the inhomogeneous temperature distribution in the liquid phase.


2021 ◽  
Vol 24 (1) ◽  
pp. 5-53
Author(s):  
Lihong Guo ◽  
YangQuan Chen ◽  
Shaoyun Shi ◽  
Bruce J. West

Abstract The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.


2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.


1994 ◽  
Vol 16 (6) ◽  
pp. 409-417 ◽  
Author(s):  
N. Yorino ◽  
H. Sasaki ◽  
Y. Masuda ◽  
Y. Tamura ◽  
M. Kitagawa ◽  
...  

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