Bifurcation Diagram


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Bifurcation Diagram

A graph that shows the critical points where bifurcation occurs, and the possible solutions that exist at that point.
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Assuming the fact that [L.sub.n] = L, [[omega].sub.n] = [omega] and [[phi].sub.n] = 0, a bifurcation diagram was built, see Figure 1, this was developed varying the normalized traffic light cycle.
They cover general functional analytic setup and an example that forces chaos, validated numerics for equilibria of analytic vector fields: invariant manifolds and connecting orbits, continuation of solutions and studying delay differential equations with rigorous numerics, computer-assisted bifurcation diagram validation and applications in materials science, dynamics and chaos for maps and the Conley index, and rigorous computational dynamics in the context of unknown nonlinearities.
To better understand how the transition to chaos happens, one may consider a so-called bifurcation diagram. Figure 6 shows, for every value of r, the corresponding set of points that are reached after 1,000 iterations.
The bifurcation diagram and global phase portraits of system (4) for parameters a, b, c, m in all cases are shown in Figure 1.
Figure 11 gives the bifurcation diagram of system (2) with the parameter a = 0.0000001.
From the bifurcation diagram, it can be seen that the honeycomb sandwich plate can have periodic and multiperiodic motions when parametric amplitude [P.sub.1] changes.
Figure 3 describes the bifurcation diagram on the plane ([x.sub.1], [[OMEGA].sub.1] when [[OMEGA].sub.1] varies in the interval (0,1.5).
We plot in Figure 2 a two-parameter (a, b) bifurcation diagram depicting the dynamical behaviors of system (1a), (1b), and (1c).
The bifurcation diagram for the state variable y(t) and the Lyapunov exponents are presented in Figure 2 with the change of the system parameter [xi].
With reference to the bifurcation diagram of Figure 5 the classical forward or backward continuation of parameter b is obtained with the following initial conditions [x.sub.1](0) = 3; [x.sub.2](0) = [x.sub.3](0) = [x.sub.4](0) = [x.sub.5](0) = 0 and [x.sub.1](0) = 1; [x.sub.2](0) = [x.sub.3](0) = [x.sub.4](0) = [x.sub.5](0) = 0.
Caption: FIGURE 1: Bifurcation diagram of forced van der Pol oscillator for w = 0.45 and [eta] = 1 using (a) Poincare map and (b) largest nonzero Lyapunov exponent.
Caption: Figure 2: (a) The bifurcation diagram for forward direction ([[beta].sub.1] = 0.1961, [[beta].sub.2] = 0.0101); (b) the bifurcation diagram for backward direction ([[beta].sub.1] = 0.1961, [[beta].sub.2] = 0.4); cross mark denotes stable equilibrium and dot mark denotes unstable equilibrium.