Several answers address the bifurcation problem dx/dt=f(x,a) and suggest integrating in time to steady state for each a, then plotting x as a function of a.  Do NOT do this.  There are two reasons: first, it is very computationally expensive; secondly, you can only ever find a small part, the stable states, of the bifurcation diagram.

Instead,  solve the algebraic equations f(x,a)=0 via some good solver (in large problems the good solver should invoke some Krylov subspace method for efficiency), then you are empowered to plot both stable and unstable states as a function of a, their connections and Hopf bifurcations.  Easiest is to use the package AUTO, or it your problem fits XPP+AUTO.

A bifurcation does not necessarily lead to "chaos".  The saddle node, or Hopf bifurcations of vector fields on the plane are, in principle, fairly straight forward to study, and you can draw the  corresponding bifurcation diagramms by hand. If you are reffering speciffically to the period doubling bifurcation of mappings (of the line? of the plane? something else?) then you should be in position to solve equations like f^n(x)=x. In any case, I find the book of Hirsch-Smale-Devaney invaluable for someone who wants to get acquainted with these concepts.    

By running the dynamical system (differential equations or iterations) for different initial conditions, at each value of the control parameter,  discarding the transient part-to be determined-and plotting the rest, as function of the control parameter.

Indeed, it's not correct that a period doubling bifurcation accompanies the onset of chaos-in the phase where there is periodic behavior, there isn't any chaos. 

Dear Stavros Anastassiou, What is the  Hopf bifurcations?

Indeed, period doubling bifurcations are not the only route to chaos, but a period doubling cascade is one such route.

If your question is about real maps of the form

xn+1=f(xn; b)

where b is the parameter, then the bifurcation diagram shows you the asymptotic value(s) of the map as a function of this parameter, b.

If your question is how to build one I can give you two methods: for a simple method there is a funcion called orbits

Function: orbits (F, y0, n1, n2, [x, x0, xf, xstep], ...options...);

which you can use in wxMaxima which is Free Open Source software native to GNU/Linux but you may install it in Windows and Mac as well. Also, for better control, I built a Python program myself to plot these diagrams and you can obtain it from my page here:

https://www.researchgate.net/publication/267703912_Canonical_Cubic_Map_Discrete_Dynamical_System_Iterator_%28Python_program%29

The map used in this program is cubic, and actually what is termed the "Canonical Cubic Map". You can find the definitions and all the references to the basic theory of popular books (like Devaney's) on discrete dynamical systems pretty much self contained from my Master's thesis work:

https://www.researchgate.net/publication/266026010_Stability_and_chaos_in_real_polynomial_maps

or look at

https://www.researchgate.net/publication/271771292_Stability_of_Real_Parametric_Polynomial_Discrete_Dynamical_Systems

for the corresponding condensed paper. I hope it helps.

Data Canonical Cubic Map Discrete Dynamical System Iterator (Pyth...

Thesis Stability and chaos in real polynomial maps

Article Stability of Real Parametric Polynomial Discrete Dynamical Systems

Dear Adnene Arbi,

Hopf bifurcation is one of the simplest bifurcations of vector fields.  It is described, along with the other basic bifurcations, in any good textbook on dynamical systems (although some textbooks pay more attention to mappings, or more attention to vector fields and you should choose one matching your needs). In any case, understanding the basic definitions and properties of an object (see also the answer above) is probably the best way to start before trying to compute it. Hope it helps.

Thank you sirs

I suggest that you read the book:

@book{Kuzn2004,

author = {Yu. A. Kuznetsov},

edition = {3},

pages = {631},

address = {New York},

publisher = {Springer-Verlag},

series = {Applied Mathematical Sciences},

title = {Elements of Applied Bifurcation Theory},

volume = {112},

year = {2004}

}

Study the following books can be useful:

Hopf bifurcations:

Applied Nonlinear Dynamics Analytical Computational and Experimental Methods

bifurcation diagram:

Nonlinear ODE, Jordan, Smith

Dear Prof Barreira, 

regarding your comment on the period of a periodic orbit and equivalence. Isn't it true that under the relation of topological conjugacy, change of the period implies bifurcation, but under the relation of topological equivalence (which allows reparametrization of time) change of the period only does not imply bifurcation? I had that impression... 

Stavros. 

Dear Adnène Arbi,

During the numerical evaluation of the system dynamics, just identify the local maxima with varying control parameter and plot them. Thats it.

Thank you.

Debabrata Biswas 

Dear Debabrata Biswas 

I just identify the local maxima of  which function?!

Thank you.

Adnène Arbi

Dear Adnène Arbi,

You have to evaluate the functions that modelizes your situation. As for example in this paper: 

Ángel Alberto Magreñán, José Manuel Gutiérrez, Real dynamics for damped Newton’s method applied to cubic polynomials, Journal of Computational and Applied Mathematics, Volume 275, February 2015, Pages 527-538, ISSN 0377-0427, http://dx.doi.org/10.1016/j.cam.2013.11.019.

You can see that we study the bifurcation diagram of the damped Newton's method applied to different polynomials.

If you need anything else just contact me and I will help you.

Thanks you.

Alberto.

Bifurcation diagram is a graphical representation of response (Amplitude vs control parameter) . Answer is hidden in why do you do bifurcation diagrams? Operating zones of the system may be identified. 

Bifurcation is steady the state response. Transients could be avoided by ignoring the initial response data , say 10000 points, depending on the system. why do you need bifurcation? that is the answer.

What the set of independent system parameters is? For example, an ordinary missel is of 17 such parameters, each governs the bifurcation.

Dear Adnène Arbi,

The evaluation of a "bifurcation diagram" (i.e. its reliability) assumes to know some fundamental points by considering the problem in a more general aspect than that of the Dim1 parameter space. Note that the study of a model in the form of an ODE, after the classical methods of discretization, and a Poincaré section, leads to a discrete form (recurrence relationship model, or map), generally numerically formulated.

- For example in a Dim2 parameter plane, the Dim1 "bifurcation diagram" is the section of a cascade of bifurcation curves. For the discrete form, the first one is a fold curve, associated with a multiplier (or eigen value) S=+1 for a fixed point, or a cycle (periodic solution of period k), followed by a sequence of flip curves (S=-1), obtained by period doubling, with period k2n, n=1,2, …..

- The bifurcation structure is generally fractal of “box-within-a-box” (or “embedded boxes") type, cf. :

http://www.researchgate.net/publication/267747567_WHAT%27S_THE_FRACTAL_BOX-WITHIN-A-BOX_BIFURCATIONS_STRUCTURE

and sec. 5 of

http://www.researchgate.net/publication/263889910_Nonlinear_maps__From_the_Toulouse_colloquium_%281973%29_to_NOMA%2713

- The set of bifurcation curves in a parameter plane must be considered as giving only a first coarse information about the discrete equation dynamics. Indeed this set only represents the "skeleton" of a larger information permitting to follow the dynamical evolution in presence of parameter variations. This evolution can be identified by considering the parameter plane as made up of sheets, each sheet being associated with a well defined stable, or unstable, cycle (with its period k and an index j characterizing the permutation of its k points after k iterations). Then the parameter plane is said foliated, and can be associated with a three-dimensional representation of the sheets. Covering areas of several sheets, associated with stable cycles (or more complex attractors), gives rise the multistability phenomenon, cf. sec. 6 of

http://www.researchgate.net/publication/263889910_Nonlinear_maps__From_the_Toulouse_colloquium_%281973%29_to_NOMA%2713

- From the variation of a third parameter, the qualitative bifurcation structure of the Dim2 parameter plane may change (cf. references of the above quotation).

- Ignoring these points, a numerical study of the Dim1 bifurcation diagram may lead to jumps of a sheet of the foliated bifurcation structure to another sheet (for example the multistability phenomenon), and to a wrong interpretation of the numerical study.

Some examples in my Research Gate page:

Behavior of Harmonics Generated by a Duffing Type equation with a Nonlinear Damping: Part I.

Behavior of Harmonics Generated by a Duffing Type equation with a Nonlinear Damping: Part II.

Bifurcation Structures Generated by the Nonautonomous Duffing Equation

Reducible fractional harmonics generated by the nonautonomous Duffing-Rayleigh equation. Pockets of reducible harmonics and Arnold’s tongues

Fractional Harmonic Synchronization in the Duffing-Rayleigh Differential Equation

Technical Report WHAT'S THE FRACTAL BOX-WITHIN-A-BOX BIFURCATIONS STRUCTURE

Conference Paper Nonlinear maps: From the Toulouse Colloqium (1973) to NOMA’13

About the examples I gave in my last answer:

Reducible fractional harmonics generated by the nonautonomous Duffing-Rayleigh equation. Pockets of reducible harmonics and Arnold’s tongues

http://www.researchgate.net/publication/265832477_Reducible_fractional_harmonics_generated_by_the_nonautonomous_Duffing-Rayleigh_equation._Pockets_of_reducible_harmonics_and_Arnolds_tongues

Behavior of Harmonics Generated by a Duffing Type equation with a Nonlinear Damping: Part I.

http://www.researchgate.net/publication/220264701_Behavior_of_Harmonics_Generated_by_a_Duffing_Type_equation_with_a_Nonlinear_Damping_Part_I

Behavior of Harmonics Generated by a Duffing Type equation with a Nonlinear Damping: Part II.

http://www.researchgate.net/publication/220264742_Behavior_of_Harmonics_Generated_by_a_Duffing_Type_equation_with_a_Nonlinear_Damping_Part_II

Bifurcation Structures Generated by the Nonautonomous Duffing Equation

http://www.researchgate.net/publication/242919477_Bifurcation_Structures_Generated_by_the_Nonautonomous_Duffing_Equation

Bifurcation Structure in a Model of a Frequency Modulated CO2LASER

http://www.researchgate.net/publication/260544009_Bifurcation_Structure_in_a_Model_of_a_Frequency_Modulated_CO2LASER

Article Reducible fractional harmonics generated by the nonautonomou...

Article Behavior of Harmonics Generated by a Duffing Type equation w...

Article Behavior of Harmonics Generated by a Duffing Type equation w...

Article Bifurcation Structures Generated by the Nonautonomous Duffing Equation

Article Bifurcation Structure in a Model of a Frequency Modulated CO2LASER

The simplest answer is to use the package AUTO embedded in the software XPP (just download, install and run the free XPP software).  XPP (with its inbuilt AUTO) can generate a complete bifurcation diagram numerically.

To generate algebraic understanding of one or more bifurcations, study my recent book A. J. Roberts. Model emergent dynamics in complex systems. SIAM, Philadelphia, Dec 2014.

Dear Adnène Arbi,

Suppose you have the system equation $\dot{x}=af(x)$. then integrate it with suitable algorithm. During the time evaluation of the system, leave transients and identify the local maxima with varying control parameter (say $b$ here). Plot these in whole range of $b$. Thats all.

Hope this will help you.

Thank You.

Hi Adnène,

I have done so much work on panel flutter, and in my work, bifurcation diagram is a main tool for observation and evolution of complex dynamics for the fluttering panel.

In my experience, the time response used to pot the bifurcation diagram has to be get its steady value in a long enough time. So here in the time intergration, you can use the solution of the last time step as the initial condition for the next time step. In that case, the time response will converge much faster. And then, you just need to choose a certain range of the final steady response to plot the bifurcation diagram.

If you need more information about this technique, maybe you can check my paper:

Dan Xie, Min Xu, Honghua Dai, Earl H. Dowell. Observation and evolution of chasos for a cantilever plate in supersonic flow, Journal of Fluids and Structures, 2014, 337, 

which you can find in my ResearchGate page.

Hope this helps.

Dan

Several answers address the bifurcation problem dx/dt=f(x,a) and suggest integrating in time to steady state for each a, then plotting x as a function of a.  Do NOT do this.  There are two reasons: first, it is very computationally expensive; secondly, you can only ever find a small part, the stable states, of the bifurcation diagram.

Instead,  solve the algebraic equations f(x,a)=0 via some good solver (in large problems the good solver should invoke some Krylov subspace method for efficiency), then you are empowered to plot both stable and unstable states as a function of a, their connections and Hopf bifurcations.  Easiest is to use the package AUTO, or it your problem fits XPP+AUTO.

Yes I understand. Thank you  Dr.  Dan Xie and Dr. A. J. Roberts for your answers and suggestions.      

One of the main interesting bifurcation diagram which can be easily understanding is the one of the discrete logistic equation depending on the parameter. For some values of the parameter against the solution (graph of r and a solution) you can see how the solution behaves, global attractor, period two solution, period 4 solution and according to Sharkovsky theorem this will  leads to chaos.  

Remember that the method that you use for integration directly influences the bifurcation pattern and create false steady state values.

Thanks for all giving valuable answers and suggestions 

See

https://www.youtube.com/watch?v=k8jb0jHDilI

https://www.youtube.com/watch?v=-KaJpeIvTFA

https://www.youtube.com/watch?v=B555tBiLogg

https://www.youtube.com/watch?v=b9ISbtVKgnQ

https://www.youtube.com/watch?v=0br85iAH8d8

http://www.ucl.ac.uk/~ucesgvd/hopf.pdf

http://www.math.colostate.edu/~shipman/47/volume3b2011/M640_MunozAlicea.pdf

https://www.ndsu.edu/pubweb/~novozhil/Teaching/484%20Data/17.pdf

http://www.math.harvard.edu/archive/118r_spring_05/handouts/text.pdf

www.scilab.in/textbook_companion/generate_book/27

http://authors.library.caltech.edu/25053/1/Hopf.pdf

http://homepages.ulb.ac.be/~dgonze/TEACHING/nonlinear.pdf

http://www.thphys.uni-heidelberg.de/~biophys/PDF/Skripte/NonlinearDynamics.pdf

SOFTWARE

for software (pplane) to visualise dynamical systems see

http://math.rice.edu/~dfield/dfpp.html

http://math.rice.edu/~dfield/index.html

for scilab

http://www.scilab.org/download/latest

for XPPAUT

http://www.math.pitt.edu/~bard/xpp/xpp.html

This should get you started.

General info on dynamical systems and critical points

http://www.math.utah.edu/~gustafso/f2010/dynamicalSystems.pdf

https://people.ok.ubc.ca/rtyson/Teaching/Math225/LectureNotes/Nullclines.pdf

http://faculty.jsd.claremont.edu/jmilton/Math_Lab_tool/Labs/Lab5.pdf

see https://www.youtube.com/watch?v=tv7N9F6Ar_Y&list=PLqOZ6FD_RQ7lnGZ7fkn503y_7U4rrJ-Se&index=4

2) Then you need to find the steady state and linearise by finding the Jacobian at the steady state

see https://www.youtube.com/watch?v=cHnToGonVyY&list=PLqOZ6FD_RQ7lnGZ7fkn503y_7U4rrJ-Se&index=5

and

https://www.youtube.com/watch?v=z1ghu3lbT3A&list=PLqOZ6FD_RQ7lnGZ7fkn503y_7U4rrJ-Se&index=8

https://www.youtube.com/watch?v=C3qjJ8OQAWM&index=22&list=PLqOZ6FD_RQ7lnGZ7fkn503y_7U4rrJ-Se

and see

https://www.youtube.com/watch?v=sSPDYLhHdco&index=23&list=PLqOZ6FD_RQ7lnGZ7fkn503y_7U4rrJ-Se

https://www.youtube.com/watch?v=0OlapzDrJ9w&index=24&list=PLqOZ6FD_RQ7lnGZ7fkn503y_7U4rrJ-Se

https://www.youtube.com/watch?v=hGCDbMc0vmA&list=PLqOZ6FD_RQ7lnGZ7fkn503y_7U4rrJ-Se&index=25

https://www.youtube.com/watch?v=kwRrL5pIorI&list=PLqOZ6FD_RQ7lnGZ7fkn503y_7U4rrJ-Se&index=26

Then you should be able to understand criteria for a limit cycle which depends on your parameters. If you find a Hopf bifurcation it is crucial to know whether you have a subcritical Hopf system or a supracritical one. References on the top will help.

Poincaree Bendixon can only (for practicle purposes) be used in a 2D dynamical system.