Exact solutions for the quadratic mixed-parity Helmholtz–Duffing oscillator by bifurcation theory of dynamical systems

2015 ◽  
Vol 81 ◽  
pp. 68-77 ◽  
Author(s):  
Yixiang Geng
2017 ◽  
Vol 28 (08) ◽  
pp. 1750104 ◽  
Author(s):  
Youssef Khmou

This short paper is focused on the bifurcation theory found in map functions called evolution functions that are used in dynamical systems. The most well-known example of discrete iterative function is the logistic map that puts into evidence bifurcation and chaotic behavior of the topology of the logistic function. We propose a new iterative function based on Lorentizan function and its generalized versions, based on numerical study, it is found that the bifurcation of the Lorentzian function is of second-order where it is characterized by the absence of chaotic region.


2020 ◽  
Author(s):  
Belle Liu ◽  
Alexander James White ◽  
Chung-Chuan Lo

AbstractOne of the most intriguing observations of recurrent neural circuits is their flexibility. Seemingly, this flexibility extends far beyond the ability to learn, but includes the ability to use learned procedures to respond to novel situations. Here, we report that this flexibility arises from the synergistic interplay between recurrent mutual excitation and recurrent mutual inhibition. Specifically, we show that mutual inhibition is critical in expanding the functionality of the circuit, far beyond what feedback inhibition alone can accomplish. By taking advantage of dynamical systems theory and bifurcation analysis, we show mutual inhibition doubles the number of cusp bifurcations in the system in small neural circuits. As a concrete example, we build a simulation model of a class of functional motifs we call Coupled Recurrent inhibitory and Recurrent excitatory Loops (CRIRELs). These CRIRELs have the advantage of being multi-functional, performing a plethora of functions, including decisions, switches, toggles, central pattern generators, depending solely on the input type. We then use bifurcation theory to show how mutual inhibition gives rise to this broad repertoire of possible functions. Finally, we demonstrate how this trend also holds for larger networks, and how mutual inhibition greatly expands the amount of information a recurrent network can hold.


Author(s):  
Albert C. J. Luo ◽  
Fuhong Min

In this paper, the analytical conditions for the controlled pendulum synchronizing with periodic motions in Duffing oscillator is developed through the theory of discontinuous dynamical systems. The conditions for the synchronization invariant domain are obtained, and the partial and full synchronizations are illustrated for the analytical conditions.


2006 ◽  
Vol 16 (04) ◽  
pp. 925-943 ◽  
Author(s):  
JIBIN LI ◽  
MINGJI ZHANG ◽  
SHUMIN LI

By using the bifurcation theory of planar dynamical systems and the method of detection functions, the bifurcations of limit cycles in a Z2-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 7 are studied. An example of a special Z2-equivariant vector field having 50 limit cycles with a configuration of compound eyes are given.


2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yongan Xie ◽  
Shengqiang Tang

We study a class of high dispersive cubic-quintic nonlinear Schrödinger equations, which describes the propagation of femtosecond light pulses in a medium that exhibits a parabolic nonlinearity law. Applying bifurcation theory of dynamical systems and the Fan sub-equations method, more types of exact solutions, particularly solitary wave solutions, are obtained for the first time.


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