Search results for "bifurcation"
showing 10 items of 204 documents
Desingularization Theory and Bifurcation of Non-elementary Limit Periodic Sets
1998
In the study of the Bogdanov-Takens unfolding, we introduced in 4.3.5.2 the following formulas of rescaling in the phase-space and in the parameter space: $$ x = {r^2}\bar x,y = {r^3}\bar y,\mu = - {r^4},\nu = {r^2}\bar \nu . $$
Transition to turbulence in toroidal pipes
2011
AbstractIncompressible flow in toroidal pipes of circular cross-section was investigated by three-dimensional, time-dependent numerical simulations using a finite volume method. The computational domain included a whole torus and was discretized by up to ${\ensuremath{\sim} }11. 4\ensuremath{\times} 1{0}^{6} $ nodes. Two curvatures $\delta $ (radius of the cross-section/radius of the torus), namely 0.3 and 0.1, were examined; a streamwise forcing term was imposed, and its magnitude was made to vary so that the bulk Reynolds number ranged between ${\ensuremath{\sim} }3500$ and ${\ensuremath{\sim} }14\hspace{0.167em} 700$. The results were processed by different techniques in order to confirm…
Socioemotional Wealth and Networking in the Internationalisation of Family SMEs
2021
In internationalisation processes, international expansion exposes family SMEs to external networking and the risks such expansion entails, and perceived threats to their socioemotional wealth (SEW) might restrain their willingness to take these actions. However, very few studies measure SEW and associate it with internationalisation. Considering SEW preservation’s prominence in family SMEs and SMEs’ heavy dependence on networking during internationalisation, we hypothesise that SEW preservation has a negative association and networking has a positive association with the family SMEs’ degree of internationalisation (DOI). We reconstruct four SEW constructs that carry significance for family…
Dynamic instability in absence of dominated splittings.
2006
We want to understand the dynamics in absence of dominated splittings. A dominated splitting is a weak form of hyperbolicity where the tangent bundle splits into invariant subbundles, each of them is more contracted or less expanded by the dynamics than the next one. We first answer an old question from Hirsch, Pugh and Shub, and show the existence of adapted metrics for dominated splittings.Mañé found on surfaces a $C^1$-generic dichotomy between hyperbolicity and Newhouse phenomenons (infinitely many sinks/sources). For that purpose, he showed that without a strong enough dominated splitting along one periodic orbit, a $C^1$-perturbation creates a sink or a source. We generalise that last…
From deterministic cellular automata to coupled map lattices
2016
A general mathematical method is presented for the systematic construction of coupled map lattices (CMLs) out of deterministic cellular automata (CAs). The entire CA rule space is addressed by means of a universal map for CAs that we have recently derived and that is not dependent on any freely adjustable parameters. The CMLs thus constructed are termed real-valued deterministic cellular automata (RDCA) and encompass all deterministic CAs in rule space in the asymptotic limit $\kappa \to 0$ of a continuous parameter $\kappa$. Thus, RDCAs generalize CAs in such a way that they constitute CMLs when $\kappa$ is finite and nonvanishing. In the limit $\kappa \to \infty$ all RDCAs are shown to ex…
On the construction of lusternik-schnirelmann critical values with application to bifurcation problems
1987
An iterative method to construct Lusternik-Schnirelmann critical values is presented. Examples of its use to obtain numerical solutions to nonlinear eigenvalue problems and their bifurcation branches are given
Oscillation spectra of semilinear photorefractive coherent oscillator with two pump waves
2002
The transition of the single-frequency oscillation of a semilinear photorefractive coherent oscillator for sufficiently large coupling strengths into two-frequency oscillation is predicted and is observed experimentally. The critical value of coupling strength at which the bifurcation occurs is a function of pump-intensity ratio and cavity losses. For certain combinations of these parameters, the critical coupling strength for spectrum bifurcation becomes smaller than the threshold coupling strength: in these cases double-frequency oscillation appears at the threshold. The supercritical bifurcation in the oscillation spectrum is analogous to the second-order phase transition.
Experimental approach to transverse wave-number selection in cavity nonlinear optics
2004
Spontaneous transverse pattern formation is experimentally studied in a ${\text{BaTiO}}_{3}$ photorefractive oscillator under degenerate four-wave mixing conditions. A near self-imaging resonator of high Fresnel number and quasi-one-dimensional in the transverse plane is used. A fine control technique of the cavity detuning, $\ensuremath{\Omega}$, is described. It allows a precise study of the relation of $\ensuremath{\Omega}$ with the transverse wave number ${k}_{\ensuremath{\perp}}$ of the roll patterns selected by the system. The law ${k}_{\ensuremath{\perp}}^{2}=\ensuremath{-}\ensuremath{\Omega}∕a$ is verified, which evidences that wave-number selection is mainly dictated by the cavity …
Control of the Biodegradation of Mixed Wastes in a Continuous Bioreactor by a Type-2 Fuzzy Logic Controller
2009
Abstract Type-2 fuzzy logic control is proposed for nonlinear processes characterized by bifurcations. A control simulation study was conducted for a bioreactor with cell recycle containing phenol and glucose as carbon and energy sources in which a pure culture of Pseudomonas putida is carried out. The model developed by Ajbar [Ajbar, A. (2001). Stability analysis of the biodegradation of mixed wastes in a continuous bioreactor with cell recycle. Water Research, 35 (5), 1201–1208] was used for the simulations. The particular dynamics of the bioreactor, characterized by two saddle-node bifurcations, makes its control difficult, since it may become unstable also for small variations of some p…
Hidden Strange Nonchaotic Attractors
2021
In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic at…