Merge pull request #2445 from rveltz/patch-2

Update bifurcation_diagram_computation.md, add periodic orbits comput…

Author: ChrisRackauckas

docs/src/tutorials/bifurcation_diagram_computation.md

@@ -110,12 +110,9 @@ bprob = BifurcationProblem(osys,
     jac = false)
 
 p_span = (-3.0, 3.0)
-opt_newton = NewtonPar(tol = 1e-9, max_iterations = 20)
-opts_br = ContinuationPar(dsmin = 0.001, dsmax = 0.05, ds = 0.01,
-    max_steps = 100, nev = 2, newton_options = opt_newton,
+opts_br = ContinuationPar(nev = 2,
     p_max = p_span[2], p_min = p_span[1],
-    detect_bifurcation = 3, n_inversion = 4, tol_bisection_eigenvalue = 1e-8,
-    dsmin_bisection = 1e-9)
+    )
 
 bf = bifurcationdiagram(bprob, PALC(), 2, (args...) -> opts_br; bothside = true)
 using Plots
@@ -128,3 +125,26 @@ plot(bf;
 ```
 
 Here, the value of `x` in the steady state does not change, however, at `μ=0` a Hopf bifurcation occur and the steady state turn unstable.
+
+We compute the branch of periodic orbits which is nearby the Hopf Bifurcation. We thus provide the branch `bf.γ`, the index of the Hopf point we want to branch from: 2 in this case and a method `PeriodicOrbitOCollProblem(20, 5)` to compute periodic orbits.
+
+```@example Bif2
+br_po = continuation(bf.γ, 2, opts_br,
+    PeriodicOrbitOCollProblem(20, 5);
+    )
+
+plot(bf; putspecialptlegend = false,
+    markersize = 2,
+    plotfold = false,
+    xguide = "μ",
+    yguide = "x")
+plot!(br_po,  xguide = "μ", yguide = "x", label = "Maximum of periodic orbit")
+```
+
+Let's see how to plot the periodic solution we just computed:
+
+```@example Bif2
+sol = get_periodic_orbit(br_po, 10)
+plot(sol.t, sol[1,:], yguide = "x", xguide = "time", label = "")
+```
+