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# Topic: Bifurcation diagram

###### In the News (Fri 19 Jul 19)

 From simple rules to order, fluctuations, chaos The parameter k is plotted along the x-axis of the bifurcation diagram (from 1 on the right to 4 on the left). For small and medium k values (to the left) what the bifurcation diagram shows is that (as we found earlier), the population reaches a single, stable population size (which increases with increasing values of k). The bifurcation diagram is constructed by taking a particular value for the starting population size (x=0.125 is the default value, as shown below the staircase diagram) and then, for each value of k, iterating it a number of times (70 is the default value, as shown in the left slider above the diagram). serendip.brynmawr.edu /complexity/newton/rules5.html   (1005 words)

 Bifurcation Diagram For all these reasons, a bifurcation diagram  is a particularly powerful method for studying the attractors in the quadratic map. Recall that a bifurcation diagram is a plot of an asymptotic solution on the vertical axis and a control parameter on the horizontal axis. Much of the structure in the bifurcation diagram can only be understood by keeping track of both the stable attracting solutions and the unstable repelling solutions, as we did in constructing the orbit stability diagram (Figure 2.13). www.drchaos.net /drchaos/Book/node39.html   (666 words)

 Bifurcation Diagram for the cubic map and a realistic model for population dynamics -- from Wolfram Library Archive Bifurcation Diagram for the cubic map and a realistic model for population dynamics -- from Wolfram Library Archive Bifurcation Diagram for the cubic map and a realistic model for population dynamics The notebooks compute the bifurcation diagrams for the cubic map and a realistic population dynamics model. library.wolfram.com /infocenter/MathSource/6614   (145 words)

 Bifurcation diagram for quadratic maps There is a good way to trace bifurcations of period of attracting orbit on the (x, c) plane by the bifurcation diagram of f (it is very similar to the "logistic bifurcation map"). At the lower part of the bifurcation diagram you see chaotic bands and white narrow holes of windows of periodic dynamics. Bifurcation map (the right picture) compliments the "spider" and shows its interior. www.ibiblio.org /e-notes/MSet/Logis.htm   (377 words)

 Bifurcation Diagram For all these reasons, a bifurcation diagram  is a particularly powerful method for studying the attractors in the quadratic map. Recall that a bifurcation diagram is a plot of an asymptotic solution on the vertical axis and a control parameter on the horizontal axis. Much of the structure in the bifurcation diagram can only be understood by keeping track of both the stable attracting solutions and the unstable repelling solutions, as we did in constructing the orbit stability diagram (Figure 2.13). cnls.lanl.gov /People/nbt/Book/node39.html   (666 words)

 Bifurcation diagram - Definition, explanation An example is the bifurcation diagram of the logistic map. The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the Feigenbaum constant. Hence this bifurcation diagram demonstrates a nice example of the importance of chaos theory in even very simple non-linear systems. www.calsky.com /lexikon/en/txt/b/bi/bifurcation_diagram.php   (187 words)

 1.3 Universality As was shown in the diagrams, subregions within the bifurcation diagram look remarkably similar to each other and to the diagram as a whole. In the quadratic case, the bifurcation diagram was finite and ended at a parameter value beyond which all orbits escaped to infinity. As expected, the orbits gave a bifurcation diagram nearly identical to that for the quadratic map, but with a bit of a twist. hypertextbook.com /chaos/13.shtml   (1000 words)

 Definition of Bifurcation diagram The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. The ratio of the values of r where bifurcation occurs is called the Feigenbaum constant. Hence the bifurcation diagram is a nice example of the importance of chaos theory in even very simple non-linear systems. Indeed, the bifurcation diagram is fractal, since any of the 'arms' in the periodic regions, when magnified, appear similar to the whole diagram. www.wordiq.com /definition/Bifurcation_diagram   (349 words)

 Brazilian Journal of Physics - Low frequency oscillations and bifurcation diagram in semi-insulating GaAs samples   (Site not responding. Last check: ) The bifurcation diagrams were obtained from the sequence of minima as a function of the applied bias. A global view of the underlying dynamics is presented through bifurcation diagrams which are convenient to investigate the periodicity and to classify the bifurcations of the current oscillations versus a control parameter. Bifurcation diagrams were reconstructions from the minima sequence as a function of the applied bias. www.scielo.br /scielo.php?script=sci_arttext&pid=S0103-97332006000300006&lng=es&nrm=iso   (1487 words)

 Bifurcation - Scholarpedia One of the principal objectives of bifurcation theory is to prove the structural stability of normal forms. One of the principal uses of bifurcation theory is to analyze the bifurcations that occur in specific families of dynamical systems. Switching to the continuation of a periodic orbit at an Andronov-Hopf bifurcation or to the continuation of a saddle homoclinic bifurcation curve from the Bogdanov-Takens bifurcation are examples of such starting techniques based on normal form computations. www.scholarpedia.org /article/Bifurcation   (1222 words)

 Generic Cell Cycle Model Bifurcations can help us to understand how the system reacts for parameter changes: A set of differential equations with a parameter set determine how the system will move in time from an initial condition, how the concentrations of the different regulators change. Bifurcation theory can tell how the attractors of a system change for parameter modifications, so it can be used to understand how the transitions (bifurcations) between cell cycle phases (attractors) depend on certain parameters. A one dimensional cut of this diagram is the one dimensional wild-type bifurcation diagram, and another cut is the one dimensional bifurcation diagram of the corresponding mutant. mpf.biol.vt.edu /research/generic_model/main/pp/intro.php   (1423 words)

 WOB Examples A main part of the bifurcation diagram is shown in Figures 1, 2. The solid curve in this branching diagram represents oscillations, with phase diagrams being symmetric with respect to the origin as in Figure 4. Accordingly, the phase diagrams of these ``secondary" branches are asymmetric with respect to the origin; Figure 6 shows one such phase plot. www.bifurcation.de /exb3/exb3comx/exb3.com.html   (462 words)

 Bifurcation Diagram A Bifurcation Diagram is a visual summary of the succession of period-doubling produced as r increases. The next figure shows the bifurcation diagram of the logistic map, r along the x-axis. Bifurcations occur at r=3, r=3.45, 3.54, 3.564, 3.569 (approximately), etc., until just beyond 3.57, where the system is chaotic. www.vanderbilt.edu /AnS/psychology/cogsci/chaos/workshop/BD.html   (231 words)

 Lab 6.5 - To the Hopf Bifurcation The bifurcations discussed in Lab 6 (pitchfork (super and subcritical), transcritical, fold/saddle node) come about when a single eigenvalue passes through zero as a result of a single parameter change. This is a supercritical Hopf bifurcation with an exchange of stability from the critical point to the periodic orbit. The physical relevance of the subcritical Hopf bifurcation is seen when the bifurcation parameter is moved from where there exists the stable steady state to where there exists only a large amplitude periodic solution. www.caam.rice.edu /~bpeercy/math468/lab6_5.html   (533 words)

 wind Bifurcation diagram for the reduced gravity shallow water model on a beta-plane, with continental geometry and idealized wind forcing, using the Ekman number E as control parameter. Bifurcation diagram for both the quasi-geostrophic model (drawn) and the reduced gravity shallow water model (dashed) in a rectangular basin, using the Ekman number E as control parameter Bifurcation diagram for the 1.5 layer quasi-geostrophic model in a 1000 x 1000 km basin, using the Reynolds number Re as control parameter. www.phys.uu.nl /~hakan/pionier/wind.html   (807 words)

 Chip Ross: "Complete" Bifurcation Diagram   (Site not responding. Last check: ) The orbit diagram shows for which values of c the function f(x)=x^2+c has an attracting cycle, and for which values of c the function exhibits chaotic behavior. The two diagrams coincide whenever the orbit diagram shows an attracting cycle; where they don't coincide the bifurcation diagram shows repelling cycles. Point D represents a period doubling bifurcation also, as points on an attracting 3-cycle (in yellow) become repelling as an attracting 6-cycle is born. abacus.bates.edu /~sross/colorbif.html   (380 words)

 The Dispatch - Serving the Lexington, NC - News   (Site not responding. Last check: ) In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values (equilibria/fixed points or periodic orbits) of a system as a function of a bifurcation parameter in the system. The bifurcation parameter r is shown on the horizontal axis of the plot and the vertical axis shows the possible long-term population values of the logistic function. The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a period-doubling bifurcation. www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Bifurcation_diagram   (225 words)

 Visualising bifurcations in high dimensional systems: The spectral bifurcation diagram Bifurcation diagrams which allow one to visualise changes in the behaviour of low dimensional nonlinear maps as a parameter is altered are common. Three techniques are considered; the first two, density and max/min diagrams, are analogous to the traditional bifurcation diagrams used for maps. These diagrams are generally more difficult to interpret than the corresponding diagrams of maps, however, due to projection effects and the continuous nature of the flow. www.lse.ac.uk /collections/cats/abstracts/VisualisingBifurcations.htm   (217 words)

 Algorithms for Molecular Biology | Full text | Inverse bifurcation analysis: application to simple gene systems For instance, mathematical and symbolic bifurcation analysis has led to an understanding of the possible dynamical behaviors that may arise out of simple gene systems (for a monograph, see [2], examples of more recent papers dealing with natural, designed, and model systems are [3-7]). For cell cycle models, bifurcation diagrams have given biologists a systems-level perspective of the roles played by the various constituent modules, as well as providing the ability to predict the behavior of mutant cells [8,9]. For this particular model, the inverse bifurcation problem is to map geometric as well as topological relationships in the bifurcation diagrams to conditions on the parameters. www.almob.org /content/1/1/11   (4468 words)

 Bifurcation of equilibrium point The symbols ,  and   represent codimension two, three and four bifurcations which are the points of intersection of double Hopf bifurcations, D-type of branching and Hopf bifurcation, and double D-type of branchings, respectively. A schematic bifurcation diagram is shown in Fig. Figure6: Bifurcation diagram corresponding to the curve l in Fig. www.eng.kagawa-u.ac.jp /~kitaji/revised/node8.html   (338 words)

 Egwald Mathematics — Geometry, Linear Algebra, Optimal Control, Statistics and Econometrics, Nonlinear Dynamics, ... The phase diagrams show the solution trajectories for various values of the parameter α of the model's difference equation. The bifurcation diagram shows the period doubling cascade path to chaos. In Hopf bifurcations, a stable limit cycle emerges — growing from a fixed point (supercritical Hopf bifurcation), or an unstable limit cycle disappears — diminishing towards a fixed point. www.egwald.ca /mathematics   (1464 words)

 DynCyc97 - 4. Farmers-Bandits interaction under fixed authority The analysis of the above given bifurcation diagram in the (z,E) space, allows us to derive the bifurcation diagram of the subsystem (2a-b) in the space (x,y,z) (where we must not forget, that z is a parameter) for fixed values of E. All these diagrams are qualitative diagrams, both because the curves have been deformed for the sake of illustration, and because the dimension of the limit cycle cannot obviously be ascertained from the above bifurcation diagram. This bifurcation diagram given for an value of E equal to E2, is very similar to the above one, the first in this series of bifurcation diagrams in the space (x,y,z). www.cg.tuwien.ac.at /studentwork/VisSem97/DynCyc97/chapter4.html   (838 words)

 Bifurcation diagram - meaning of word The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs convergent series to the Feigenbaum constant. It is an orbit diagram (which is unfortunately sometimes incorrectly referred to as a bifurcation diagram). A bifurcation diagram shows all cycles, attracting or otherwise, and does not include points that have not yet converged to a cycle, since it is not generated by iteration. www.wordsonline.org /Bifurcation_diagram   (461 words)

 Bifurcation Diagram for the cubic map and a realistic model for population dynamics -- from Mathematica Information ... Bifurcation Diagram for the cubic map and a realistic model for population dynamics The notebooks compute the bifurcation diagrams for the cubic map and a realistic population dynamics model. Bifurcation diagram, cubic map, realistic population dynamics model, cycle of period three, chaotic behaviour library.wolfram.co.jp /infocenter/MathSource/6614   (133 words)

 Falling all the way: Cascade diagrams There is clearly a sizeable degree of self-similarity in the waterfall of bifurcations. In all cases, the limiting ratio of the gaps between bifurcations turned out to be 4.6692.... Amazingly, the bifurcation diagram shows that there are moments of tranquility beyond the threshold of chaos. www.math.okstate.edu /mathdept/dynamics/lecnotes/node54.html   (757 words)

 Classifying Spiral Plants   (Site not responding. Last check: ) Above is the fixed-point bifurcation diagram that represents a dynamical system for plant pattern formation. A change in the horizontal axis on this bifurcation diagram corresponds to a different divergence angle in the lattice representation. An interesting fact about the bifurcation diagram is that every point on the highlighted part of the diagram represents a different type of Fibonacci phyllotaxis formation. sophia.smith.edu /~jaconnol/Math/page04.htm   (127 words)

 Cornell REU 2002: Dynamical Systems -- Judith Hubbard   (Site not responding. Last check: ) This paper is simply a continuation of the previous one; it examines certain portions of the period 3 bifurcation diagram more closely to clear up questions which are listed at the beginning of the paper. This paper examines a symmetry-breaking bifurcation on the bifurcation diagram of period 3 orbits and follows that asymmetric curve. Bifurcation diagram of period 3 symmetric solutions for various epsilon, in fl and white www.mathlab.cornell.edu /~judithh   (403 words)

 Directory of open access journals Our goal is to obtain a bifurcation diagram; i.e., to divide the parameter space into regions within which the system has topologically equivalent phase portraits and to describe how these portraits are transformed at the bifurcation boundaries. We present the whole bifurcation diagram and among others, we prove existence of more difficult bifurcations and existence of unstable cycles. Phase portrait ; bifurcation ; central manifold ; topological equivalence ; structural stability ; bifurcation diagram ; limit cycle. www.doaj.org /doaj?func=abstract&id=118343&recNo=1&toc=1   (162 words)

 WOB Examples The data of the bifurcation point and four turning points (TP) are listed in the Table. The solution profiles of the bifurcation point are shown in Figures 4 and 5; notice the symmetries. This bifurcation point is straddled between the two turning points with asymmetric solutions, which are depicted in Figures 2 and 3. www.bifurcation.de /exb2/exb2comx/exb2.com.html   (172 words)

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