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Nonlinear Systems
Published in Jitendra R. Raol, Ramakalyan Ayyagari, Control Systems, 2020
Jitendra R. Raol, Ramakalyan Ayyagari
As the parameters of nonlinear dynamic systems are changed, i.e. as the coefficients of the governing differential equation are changed, the solutions of f(x,t) obviously change, and hence, the stability of the equilibrium point; even the number of equilibrium points can change. This phenomenon of bifurcation, i.e. quantitative change of parameters leading to qualitative change of system properties, is the topic of bifurcation theory. Values of these parameters at which the qualitative nature of the system’s motion changes are known as critical or bifurcation values.
Bifurcations
Published in Arturo Buscarino, Luigi Fortuna, Mattia Frasca, ®and Laboratory Experiments, 2017
Arturo Buscarino, Luigi Fortuna, Mattia Frasca
As has been previously discussed, the bifurcation theory studies systems characterized by equations that qualitatively change their behavior depending on one or more parameters. In particular, it has been shown that, when in the space of parameters some singularity like a cusp occurs, even a small change in the parameters may lead to the appearance or disappearance of equilibrium points or to a change of the nature of the equilibrium point, from attractor to repellor, for instance. These scenarios, generally characterized by well-defined geometrical structures, lead to the catastrophes. The theory was introduced by René Thom in the 1960s and great contributions to the topic were given by Christopher Zeemen in the 1970s [95]. We will discuss it with an outstanding example reported by Steven Strogatz in his book [84], that is well conceived to introduce the reader to the problem.
Bifurcation, chaotic and multistability analysis of the $(2+1)$-dimensional elliptic nonlinear Schrödinger equation with external perturbation
Published in Waves in Random and Complex Media, 2022
Samina Samina, Adil Jhangeer, Zili Chen
In this section, the preceding model is transformed into a dynamical structure and all possible phase graphs of the system are presented. By using the Galilean transformation on Equation (32), we obtain the following dynamical structure: where and . It is demonstrated that the three equilibrium points can be seen using the bifurcation theory [30] of phase portraits of the dynamical system (35). According to the idea of planar dynamical systems, we divide equilibrium points in our systems into different types: Centre points ( if and ),Saddle points ( if J<0),Node ( if J>0 and ),Zero point ( if J = 0)Poincar index ( if P = zero).
Dynamics of ion-acoustic waves in nonrelativistic magnetized multi-ion quantum plasma: the role of trapped electrons
Published in Waves in Random and Complex Media, 2022
The bifurcation theory of nonlinear waves in plasmas has been used to describe the systems in which the total entropy is different from the sum of the entropies of its parts [22]. many authors work on bifurcations [22], [31–35]. They reported through perturbative and non-perturbative approaches. EL-Monier and Atteya added the bifurcation theory to the planar dynamical system of the generalized KdV-Burgers equation to establish the existence of solitary and periodic traveling wave solutions [22]. The derived mKdV- Burgers equation for a system composed of warm ions and trapped electrons was manipulated mathematically by the bifurcation theory to the dynamical system [36]. The phase portraits of the traveling wave solutions of both dissipative and non-dissipative systems are analyzed. Recently, the effects of ion loss rate and ion-neutral collision frequency on the solitary waves were investigated [33].
Stability and Bifurcation Characteristics of a Forced Circulation BWR Using a Nuclear-Coupled Homogeneous Thermal-Hydraulic Model
Published in Nuclear Science and Engineering, 2018
Dinkar Verma, Subhanker Paul, Pankaj Wahi
Researchers have used many methods and codes to carry out bifurcation analysis of the models. The center manifold theory is applied to the March-Leuba et al. model3 and captures the limit cycles using the Hopf bifurcation theory in conjunction with variational methods.17 Tsuji et al.6 exploited the computer codes DERPAR (Ref. 18) and BIFOR2 (Ref. 19) for the stability analysis using bifurcation theory. A detailed bifurcation analysis (with and without nuclear feedback) is performed with the help of the BIFDD code19 (successor of BIFOR2), and a parametric study investigates the effect of coupling between neutronics and thermal hydraulics on the nonlinear characteristics of a BWR model.20 Utilizing BIFDD (Ref. 19) along with the MATCONT continuation code (Ref. 21), Refs. 10 and 11 study BWR stability. Reference 12 applies the method of multiple time scales (MMTS) to a simplified lumped-parameter model to carry out the bifurcation analysis.