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When dealing with the complex nonlinear aeroelastic systems, sometimes we encounter with the supercritical or subcritical Hopf bifurcation. Usually the subcritical bifurcation is more dangerous. Is there an efficient way to predict the bifurcation type of a very complex (can not be studied analytically) nonlinear aeroelastic system?
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Numerical experiments are mostly the common way to detect or qualitatively comment on the type of bifurcation, particularly if a pitchfork type bifurcation is super or sub-critical. In higher dimensions this becomes a periodic orbit or a limit cycle in the absence of any external forces (limit cycles are self sustained).
One method could be to observe any form of hysteresis in the observed large amplitude oscillations. If while changing the parameter (eg. α) in one direction; say increasing, there persists a stable fixed point which becomes unstable at αc and a larger limit cycle appears. Now while decreasing the parameter from α > αc (ensure the initial condition is near the larger limit cycle), the larger limit cycle disappears at α=αc and the initial fixed point reappears; then this is supercritical hopf. However if the larger limit cycle is still stable below αc and looses stability at a lower value of αnc jumping to the initial fixed point; this is a signature of sub-critical bifurcation. The larger limit cycle looses stability by means of saddle node type bifurcation for cycles. This also means that a local unstable limit cycle exists between αn and αc located between the fixed point and the larger limit cycle which engulfs the larger limit cycle at the saddle node bifurcation point αn
One method to detect this unstable limit cycle is to use the method of pseudo arc-length continuation or path following algorithm. Packages like COCO exist which use the method of continuation to detect these unstable limit cycles for dynamical systems.
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Using MATLAB, how to draw the bifurcation diagram for a chaotic system?
Can you kindly share any *.m file (MATLAB code) for this? How to do this?
Can you illustrate the bifurcation analysis with any classical system like for example, the Lorenz system?
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Parameter continuation is a powerful tool to do that. You can use it to study the bifurcation of fixed points and periodic orbits. Further, one can use parameter continuation to obtain stable/unstable manifolds of the dynamical system. There are a few continuation packages, including AUTO, MATCONT, and COCO.
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Whether the existence of periodic window affects the encryption efficiency or not?
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You would want to avoid these periodic windows when you are designing encryption algorithms because as the term "periodic" implies, your chaotic map will behave in a predictable, periodic manner. This means it will alternate between several values rather than visiting the entire phase space. Always pick the control parameter values that maximize the phase space and is not within the periodic window.
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for a non-autonomous system, for example
x''=-ax'-bx-cx^3+f*sin(wt)
In the bifurcation software AUTO07, we can model the excitation force sin(wt) by add another oscillator
Y'=Y+w*Z-Y*(Y^2+Z^2)
Z'=-w*Y+Z-Z*(Y^2+Z^2)
then I can analyze the bifurcations of the system. now, i find a branch, and there is a period-doubling bifurcation point (PD) on this branch. but how can i solve another branch with setting this PD as a start point? 
I try to use the branch switch, but i cannot obtain another branch, the branch solved is identical with the formal one. 
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You can see this preprint.
I solved a non-autonomous equation (Pendulum equation) using Piecewise Analytic Method (PAM)
You can use PAM with systems of equations also.
.........................................
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What is the effect of non-associated flow rule on bearing capacity of a strip footing using finite element method?
In classical soil mechanics, one of the simplest problems is the bearing capacity of a strip footing on a frictional soil obeying Mohr-Coulomb failure criterion. The analytical solutions to determine the ultimate bearing capacity make use of upper bound/lower bound theorems of limit analysis where the fundamental assumption is the associated flow rule i.e. the dilation angle is equal to the friction angle.
However, it is known that soils usually have a dilatancy angle less than the friction angle. This leads to a non-associated flow rule and the analytical solutions are no longer valid.
I want to use finite element method to determine the ultimate bearing capacity of a footing on a frictional soil obeying Mohr-Coulomb failure criterion with a non-associated flow rule.
BUT I find in the literature that there is not a unique ultimate bearing capacity when a non-associated flow rule is used. As you can see in the attached pages from Varn Baars (2018) and Loukidis et al. (2008), the load-displacement curve starts to oscillate and it does not converge to a unique value. The oscillations become more significant as the difference between friction angle and dilatancy angle increases.
I came across theories of bifurcation /strain localization, ….regularization… but none have helped me to understand how I can calculate the bearing capacity of a strip footing using a non-associated flow rule. Does it physically exist? What is the effect of the level of non-associativity on the ultimate bearing capacity?
References:
Loukidis, D., Chakraborty, T., and Salgado, R. (2008) Bearing capacity of strip footings on purely frictional soil under eccentric and inclined loads, Canadian Geotechnical Journal 45, pp 768-787
S Van Baars, 2018 - 100 YEARS OF PRANDTL'S WEDGE.
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Dear Mustafa
there is an article about "Bearing capacity of strip footings by incorporating a non-associated flow rule in lower bound limit analysis" that found out clearly the increase in the magnitude of bearing capacity factors with an increase in the magnitudes of dilative coefficient (η).You might want to check it out, it may be useful. kind regards
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Bifurcation diagrams are very useful to evaluate the dynamical behavior of nonlinear dynamical systems. In chaos literature, I notice that some authors draw bifurcation diagrams by removing the first 1000 seconds of data. I like to understand the reason behind this. Any help on this is highly appreciated. Thank you!
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As you mentioned, the bifurcation diagram allows us to analyze the dynamics of the systems, in this case, chaotic systems. The idea in eliminating a certain amount of data in the time series (no matter if are 1000 secs, 1500 points in the time series, or whatever) before constructing the bifurcation diagrams, lies in eliminating the transitory state of the system and focus on what it is known as steady-state dynamics. I think that is what is referred to in the articles you describe, where the amount of time referred to is not so relevant, and it would suffice to say that the dynamics of the steady-state is analyzed, or that they have not been considered the transitory states for the construction of the bifurcation diagrams.
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I'm trying to draw a bifurcation plot, Poincare Map and Lyapunov exponent for a ODE problem. I should know how to do it in MATLAB!
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Hi,
I've a dynamical systsem (Aircraft) with 8 state variables (U1,U2,U3..U8) (and eight first order ODEs ...F(1),F(2),..F(8)) and three parameters (PAR1, PAR2,PAR3). I used one of those as continuation parameter (PAR1) and the other two (PAR2 and PAR3) as free parameters, while i performed continuation and bifurcation analysis, i could only see a partial set of states (6)
Hope the figures below explain my question better. In the PyPlaut and bifurcation diagram, i could only see six states(U(1)..U(6)) displayed. But when i enter 'print start(1)' as shown in the attached picture, i see U(1)...U(7), U(8). I have 8 states in my model. I'm not able to understand where is it storing the data about how to access the same. That data is not there in fort.7 and fort.8.
Thanks,
Kumara
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Once you extract the AUTO files and before building using ./configure go to the path ..../auto/07p/src and open the file io.f90, in the file in the subroutine WRLINE you can find the lines nearly at the beginning of subroutine (line 1052 in auto07p-0.8 & around line 1095 in auto07p-0.9)
IF(N1.GT.7)THEN
N1=7
N2=0
ELSEIF(N1+N2.GT.7)THEN
N2=7-N1
ENDIF
Here the no.7 limits the size of your output printing (L2-NORM + 6 States (i guess)) increase to 10 i guess you can have 9th order system. in general increase 7 to n+1, n being the dimension of your state vector and then install using ./configure and make.
The attached file is an execution of auto in Windows using MinGw of an 8th order system with matrix being [BR, PT, TY, LAB, P(1), L2-NORM, U(1), U(2), U(3), U(4), U(5), U(6), U(7), U(8)]
Hope this is helpful.......
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When plotting a bifurcation diagram in nonlinear dynamics, the axis x displays a given phase parameter. Are there examples in which the phase parameter stands for time passing (for example, from the value T0 to the value T200 seconds, or months, or years)?
Thanks!
To make an example, I was thinking to something like the one in the Figure below, concerning the phase transitions among liquids, solids and gases: if you leave, e.g,., that the temperature raises of one degree every second, can we say that the axis x displays time (apart temperature values)?
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see my theory
In which I discuss the possibility of rapid movement given specific situations arise in my theory... You speak of some in your question...
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The bifurcation curve generated in MatCont is displayed in the same color(blue). The stability of the equilibrium or limit cycle can only be tracked in the layout of "numeric" in "Window" by Eigenvalues or Multipliers. Does anyone know how to display the stability of the curve graphically by color or any other ways(e.g. line style)?
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We are going to release a new version that supports this functionality, expect something like March 2019.
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How these two are related? What kind of bifurcation leads to chaos?
Let me quote Wikipedia here:
Chaos :
Robert L. Devaney says that, to classify a dynamical system as chaotic, it must have these properties:
  1. it must be sensitive to initial conditions
  2. it must be topologically mixing
  3. it must have dense periodic orbits
Bifurcation:
A bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior.
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The dynamic system and chaos can be related as follows: when a dynamical system, described by a set of parameterized differential equations, changes qualitatively, as a function of an external parameter, the nature of its long-time limiting behavior in terms of fixpoints or limit cycles, one speaks of a bifurcation.
On the types of bifurcations and chaos can be found in more detail in the book: Gros, Claudius. Complex and adaptive dynamical systems. Chapter 2
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I have studied literature on bifurcation of various systems. I observe two types of bifurcation diagrams, one was a curve(which is easy to interpret) but the second type is like a shaded region, which I am unable to interpret. For example look at the attached diagram. Interestingly, while drawing bifurcation for my model (Predator-prey model) I get something different.
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Naked-eye visual study of bifurcation diagrams can be tricky. One should also consider the impact of numerical simulation tools (see https://www.researchgate.net/publication/305682196_Bifurcation_Diagram_Behavior_for_Discrete_models_of_Rossler_system_depending_on_timestep) and possible initial conditions effect. It is always helpful to apply additional methods of analysis. E.g. plot the lyapunov spectrum beneath the bifurcation diagram to see where the oscillation mode really changed. Another interesting tool is phase shift diagrams, because amplitudal bifurcation diagrams are not that representative for some specific cases of chaos, e.g. Gavrilov-Shilnikov case and spiking systems. A further step to a deep analysis is the dynamical map.
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In bifurcation diagrams, generally, fixed points are plotted against a particular parameter. However, people also plot such diagrams using experimentally obtained time responses. I am not able to understand exactly what is plotted against that parameter using the time response. In some of the papers, the y-axis is labeled as 'Amplitude'. Can anyone elaborate on this? What is the standard or right way to plot a bifurcation diagram using time series for different values of parameter?
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Hi Pranav,
One way is to plot the Poincaré section of the envelope of the experimental time response as a function of the bifurcation parameter.
I did something similar in this paper if you want to take a look:
Best,
G.
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Hello,
I want to find tangent modulus or acoustic tensor during a uniaxial tensile test. The experiment is on sheetmetal specimen. Abaqus 3D model is dynamic/explicit.
For formulation, you can refer to Haddag et al. (2009). I also attach the online pdf link. I want to find element of Equation 73.
I believe those parameters are not default in Abaqus. But they must be found on the background. Should I use VUMAT to get them? Are the parameters readily available or should I define them? Can you suggest an documents, guidelines?
  1. Haddag, B., Abed-Meraim, F., & Balan, T. (2009). Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage. International journal of Plasticity, 25(10), 1970-1996.
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Hi Volkan Ili,
I am afraid that the answer to your first question is yes, the linear perturbation is purely static, so any dynamics/temporal effects are lost, you will see only the linear stiffness effects = the tangent modulus.
On the subject of comparing bifurcation and perturbation, I can't really suggest studies I'm afraid, and I'm not that familiar with your field to give you good advice without reading up on it. But I wish you good luck with your work!
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In nonlinear systems, we know several bifurcations (i.e. Saddle-node, Pitchfork, Transcritical, and Hopf). The question is: does there exist a specific bifurcation that merges two "stable" limit cycles to one "stable" (and probably with larger amplitude) limit cycle?
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Try x'' = ax - x^3 + (1 - x'^2)x' in the vicinity of a = 1.87525 (the Rayleigh-Duffing two-well oscillator).
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On what parameters the selection of continuation method depends ?
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We have proposed a numerical method (denoted as MEM) for solving the
Nonlinear Dynamic problems. If you would like, you can view our published articles in this field entitled "dynamic analysis of SDOF systems using modified energy method". By the way, the presented idea is also generalized to MDOF systems in another study, which is available on my research-gate account.
if you need more information, please let me know.
Regards,
Jalili
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Hi,
I have a 3D figure of around 1000 points and the shape of the figure is very similar to a bifurcation. The question is how can I generate the related dynamic equation of these points?
Please let me know if you have any idea about using MATLAB codes as well.
Thanks
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I suggest to fit a multiple (3-pl) regression equation to the 1000 points in 3D by least squares methods or by orthogonal polynomials.
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In different literature, I have seen that they have construct the border lines of Arnold tongue using xppaut auto. Can anyone provide the step for calculating Arnold tongue using xppaut for an oscillatory system?
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Dear Andreas Otto, Thank you for your reply
My system is a circadian oscillator described by ODE
x'(t)=f(x(t))+forcing signal
I need to construct 1:1 entrainment region (forcing period vs forcing amplitude) using xppaut. I tried with matlab by comparing the period of forced oscillator and forcing signal, but it is not smooth.
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In the paper attached, titled "Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment" . I ask about the way to do the figures related with bifurcations and vectors field.
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The MATLAB function "quiver" can be used to plot vector fields. MATLAB program to plot the bifurcation diagram of a logistic map are available in the MATHWORKS website. You can modify them for the map which you are dealing with.
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Hello All,
As we plot a frequency response curve for a forced vibrating system after applying the perturbation method (MMS). I would like to ask whether the ampliutude at any frequency (as represented in frequency response curve) is exaclty the same as that which will be obtained by integrating the equation of motion? Or is it just a representation of how the amplitude of the motion will vary as the forcing frequency will cross/move aorund the natural frequancy?
I'am attaching a represntative frequancy response curve for your ready reference.
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Thank all,
Yes the amplitude represented by frequency response curve is same as that shall be obtained from numerical integration of equation of motion.
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I've been started to study about CHAOS, and my main major is EEG signal processing.
I'm working on a bigger project that is about detecting and classifying emotion, attention or memorization in EEG signals.
My main question is that, if you friends can help me find an articles in this concept; which relates "Chaos" with "Emotion or attention or memorization"?
Best recommendations would be articles with rich and available data-sets on internet.Preferably for last three years.
Even a single "key word" or "article name" will help me, although much better to have article it self. 
Don't spare your thought from me. even single words could be help full.
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With my pleasure. I am sure that you will find answers to your questions in the manuscripts (below).
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Is there any program capable of simulating partial differential equations formulated in curvilinear coordinates, which uses these coordinates internally for the computations (i.e., no transform to cartesian coordinates)? And is there any program doing bifurcation analyses for such equations? And, finally, is there any program which can do both?
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Dear Mario, I use Mathematica and for solving pde in curvilinear coordinates (e.g. spherical or cylindrical) first I transform equation from cartesian to curvilinear coordinates, then I (hopefully) solve it using Mathematica internal built-in functions like NDSolve; I think that these internal functions apply numerical methods directly to curvilinear coordinates. See link below for an example from Quantum Mechanics. Gianluca
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In non linear system problem
I got origin is an asymptotically stable equalibrium point using liapunov function whereas when i solved respective  linearized system then i got same point is stable but not asymptotically stable equli. point.
Is there any problem like this?
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The nonlinear system can be asymptotically stable at 0 even though the linearised system is not, for instance:
x' = -x^3 
is asymptotically stable at 0, the Lyapunov function V(x) = x^2 works, but the linearised system x' = 0 is only stable, not asymptotically stable.
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The following two equations are Richard's equation, but with different signs in the K (hydraulic conductivity) term. What makes the difference? Why it is so ? Most of the literature used Richard's equation with negative sign in K term.
Reference for the equations:
1. The equation with positive sign is from Wikipedia
2. Another equation with negative sign is from
Numerical solution for one-dimensional Richards’ equation using differential
quadrature method
by Jamshid Nikzad, Seyed Saeid Eslamian, Mostafa Soleymannejad, and Amir Karimpour
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Suppose you introduce a new independent variable u=-z.   Then your negative sign system becomes the positive sign system.  The difference is just a matter of choosing the spatial coordinate direction; there is no discrepancy.
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Dear researchers, I need some matlab code to draw bifurcation diagram for fractional-order system.
I want to draw a bifurcation diagram for a fractional-order chaotic system. Unfortunately, I could not find any appropriate code to do this. Is there anyone who could help me to afford of this problem,
Thanks,
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Dear Dr. Markus
Thanks a lot.
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HAUTUS, M. L. J., Controllability and observability conditions of linear autonomous systems. Nederl. Akad. Wetensch., Proc., Ser. A 72, 443-448 (1969).
Thanks in advance.
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Thank you very much. However, I am already having the attached one. I need the original one.
--Vikas
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When we calculate the equilibrium in one or two dimensional systems our purpose is to find the steady state of the system, but in a chaotic system, steady state is independent of equilibrium points. Then what is the purpose of calculating the equilibrium?
Thanks in advance
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Could be that the  non transversal intersections of the invariant manifolds of the equilibrium point generate the chaotic behavior.
Example Shilnikov singularities
BW
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Dear All,
I am working on understanding the transition from laminar to turbulent behaviour in Navier-Stokes equations by the approach of dynamical systems theory.
I have derived a three-dimensional dynamical system model on the level surface of some energy function from the NS equations (with minimal dissipation hypothesis). Based on this model, I obtained a devil's staircases behaviour similar to those from the circle map. However, quite noticeable, which is in contrast to the conventional devil's staircase observed from the sine circle map, is the overlaps between the quasiperiodic curve and the phase locking plateaus, see attached figures. However, such an overlap is only apparent, indicating a successive hop between periodic and quasiperiodic behaviours. Moreover, the decrease in the values of the phase locking plateaus does not follow the Farey sequence, instead of following (...4/5,4/6,4/7,4/8.4/9.4/10...). While increasing the nonlinear coupling of the system, the sequence becomes (...8/10,8/11,8/12,8/13,8/14,8/15,8/16...). I would like to know if similar behaviours have been observed in experiments, or it is merely mathematical creatures? Even in the sense of mathematics, these seem against the dynamical system theory, yet following some certain rules. Could anyone point me out, please?
In the plot, Re stands for Reynolds number, and Ro stands for the rotation number, which is defined in a limit sense for an angle variable phi, where phi is the angle between velocity and vorticity vectors (subjected to a coordinate shift).
Thank you.
Very kind wishes,
Wang Zhe
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Maybe you are observing twice as many phase locking intervals because of your degree of accuracy.
There is a geometrical picture that naturally produces all of the rational numbers in REDUCED FORM which you may find useful.
step one: put unit  diameter  tangent circles resting on the horizontal axis touching at integers values.
step two: put  big as possible smaller tangent  to previous circles inside the curved triangular regions described by these circles and the axis.By symmetry these touch at rational numbers: odd integers divided by two.their diameters are one quarter or the square of the denominator of the fraction at the resting point namely one fourth.
step three : repeat step two putting in largest circles in each of the created curved triangular regions. now the resting points are obtain by "farey addition" of  adjacent fractions, namely add numerators to get the numerator, add denominators to get the denominator. this process always gives reduced fractions. now the denominators equal three and the diameters are one ninth.
 continue recursively with step four , step five ,......  etc
by examining any fixed circle one sees a 1/x sequence of circles going down towards the tangent resting point.
this fits with your graphs.
this beautiful figure is part of the  ancient apolllonian packing whose hausdorff dimension 1.3.... is mentioned numerically in mandelbrot,s book on fractals and which is discussed rigorously in the sense of proof, in my paper
" Hausdorff measures old and new and limit sets of geometrically finite kleinian groups"
which also has nice pictures.
i would need more information about the dynamics  you use to venture a dynamical systems comment.
dennis sullivan
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could anyone provide the source code to plot basin of attraction of a given nonlinear ODE system?
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If your are looking for source code
  1. you must indicate at least the language
  2. probably this is not the proper forum
  3. http://lmgtfy.com/?q=plot+basin+attraction+code
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Any suggestion lectures and references on Nonlinear weakly stability analysis and type of bifurcations.
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Dear Shaker,
A nice review of bifurcation theory for physicist is: Crawford, J. D. (1991). Introduction to bifurcation theory. Reviews of Modern Physics, 63(4), 991.
and if you wish to learn finer details, I would suggest this book:  Manneville, P. (1995). Dissipative structures and weak turbulence. In Chaos—The Interplay Between Stochastic and Deterministic Behaviour (pp. 257-272). Springer Berlin Heidelberg.
Cheers,
Carles
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I mean is there any code that help me to draw it by using Mathematica or Matlap.
Regards,
Zenab
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Well you first need to compute the bifurcations before you can plot them, DDE-BIFTOOL is a Matlab package that supports that numerical analysis. Then plotting is the next, but much easier step. You can find it (with manual and tutorials) at: sourcefore:ddebiftool.sourceforge.net
It is mainteained by Jan Sieber (Exeter, UK).
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Hi 
I'm working on chaotic systems. I write MATLAB code with RK4 and plot the lyapunov  exponent and phase space and poincare section, but I have a problem in drawing bifurcation diagram.
Could you please help me.
Bests
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You should specify more details about your problem.
The general principle of preparing bifurcation diagrams (bifurcation parameter vs. state variable) is as follows:
On x-axis put a bifurcation parameter and on y-axis the values of the state variable of interest, you have obtain while doing N Poincaré sections (e.g. N=100) .
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I want to know if we can talk about a strange attractor and a chaotic behavior in the three dimensional phase portrait when we obtain a strange attractor  in the corresponding two dimensional phase portrait ?
Thank you
Best regards,
Afef 
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You cannot have a strange attractor in the 2-D system if time is continuous, but you can in 3-D. If the predator and prey are x and y, add a third variable z = wt that is the phase of the periodic forcing function. Then there is a third equation dz/dt = w that is periodic in z with period 2*pi/w, and the strange attractor is something like a fuzzy torus.
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Hello,
I was investigating the incompressible Navier-Stokes equations analytically and hopefully being able to extract its bifurcation characteristics. During which I obtained a set of periodic solutions on the level surfaces of the helicity density: h=u*w. I realized that these solutions may be interpreted as the periodic exchange between the kinetic and vortical energy while preserving the local helicity density.
1.) I was wondered is this solution sort of related to the direct and reverse energy cascade?
2.) I am particularly interested in the physical interpretation of limit cycles on the surface of h=0. Will the limit cycles shrink down to a point, the origin?
3.) I am equivalently intrigued by the possibility of solution trajectories flip across that particular surface h=0.
Thank you.
Wang Zhe
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Use this forum it can help you. Good luck.
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Dear all,
I am planning to investigate laminar-turbulent phase transition in fluid dynamics via studying the bifurcation characteristics of the Navier-Stokes equation. I would like to know if there is any theory which details the variation of an index at a bifurcations points during the phase transition. 
Thank you.
Wang Zhe
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You have two different equations: an algebraic one for computing fixpoints (in the framework what you call "static bifurcation") and a boundary value problem for computing periodic solutions (your "dynamic bifurcation"). Considering Poincare maps, the latter may be turned into an algebraic equation, too. Both algebraic equations are quite different and each yields its own index value as the sign of its own Jacobian determinant. Putting the index values of both equations together is like comparing apples and oranges.
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It's possible to calculate the bifurcation angle.
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many thanks
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I would like to solve Delay PDE and Stochastic Delay PDE. Moreover, I would like to draw the bifurcation diagram depending on the associated parameters with the delay (e.g. \tau say!). 
It would be really appreciable if anyone would help me with this.
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I suggest mathematician Peter Kloeden from Frankfurt University.
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Dear colleagues.I have one  nonlinear suspension system .In this system i checked two different condition.
1-I eliminated the weight with static spring forces of system.
2-I didn't eliminat the weight with static spring force of system so in the equation of spring, for displacement i had to add static displacement of spring in this condition.
then i solved the questions of system then drew bifurcation diagram of system and saw many different for state 1 and 2.I attached this bifurcation diagram for you..... and my question is
***Why did i see this difference in bifurcation diagram?**
Thanks
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These two cases should result in the same solution if I understand your question correctly. If weight loaded the system to a different position on force-deflection curve then you would expect different solutions. What method did you use to solve the equations, it may be an artefact of accuracy of your numerical solver. May I suggest to check your equation to see if you correctly manipulated them. Check to see if you accounted for the correct deflection accounting the nonlinearity.
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At first assumed that we have a suspension system with nonlinear characters, then with special initial condition chaotic vibration appeared in system in this time what is the response of chaotic system.......for example can we see separation of tire from the road or other phenomenon in this system?
(please explain about other phenomena that appear in this conditions)
thank you
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Dr. Atarodi
We investigated nonlinear dynamics of farm tractor. Generally, farm tractors don’t have any suspension system, only k and c of tires play a role of suspension.
At that case, very clear deterministic chaos was observed in field experiments and numerical experiments such as subharmonic resonance and bifurcation diagram etc.
In the field experiments, we measured almost 10G (max-min) of vertical acceleration under seat position. At that time, all fours wheels were separated from the ground; just as bouncing ball model. You can down load from RG. Hope it would be useful for your research.
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I want to calculate the Liapunov exponent for my system (please see the attached file).
also, how can I draw the chaos diagram for the system?
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I think you are asking the wrong question.  A Lyapunov function is used to determine if an equilibrium point (the origin,, without loss of generality), is (asymptotically) stable.  The Lyapunov exponent is a different thing. It measures the rate of variation of the distance between two trajectories.  It cannot be obtained analytically, but numerically.  See https://en.wikipedia.org/wiki/Lyapunov_exponent  for a definition.  Check http://sprott.physics.wisc.edu/chaos/lyapexp.htm to get an idea how to compute it.   Here is some code to compute it using Matlab. http://www.mathworks.com/matlabcentral/fileexchange/4628-calculation-lyapunov-exponents-for-ode
Have a look on XPP/AUTH, has nice features to study chaos.  http://www.math.pitt.edu/~bard/bardware/tut/xppchaos.html
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Dear colleagues at first assumed that we have one chaotic system and in this system strange attractor so How can i find basin of attraction of it.
can you introduce some sources about it?
thanks
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1. calculate the trace of Jacobian of your system.
if negative run your system to infinity!
if positive you most inverse the time of your system and run to infinity!
the trace of Jacobian of your system specify the power of  volume in state space!
2. attractor and repulsive attractor it's very important. if you have repulsive attractor and  negative jacobi you must solve the big problem!
you can run your program with enough random initial condition!
3. in the end, before run program everyone should analysis the system with applications : 1 Jacobian,2 Lyaponuv Ex ,3 poincare section and 4return map and extra .
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k.hemachandra reddy,researcher in jntu
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Dear Sanjoy Roy.at the first you must know about bifurcation definition,that means that the division of something into two branches or parts then you have to know that when you want apply bifurcation in one system you have to have one variable parameter in your equations,then you investigate affect of this variation to your answer of equation.
in power system if you want investigate about bifurcation and chaos you can work on many field ....i suppose this Link help you
----Bifurcation and chaos in power systems----
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i wanna calculate  Lyapunov exponent and i dont know any thing about it can you introduce for me some good source about it??
thanks 
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In graduate courses on analysis, you are taught that, if you integrate a differential equation of a mechanical system with two initial conditions that are close enough, the solution to the differential equation remains the same: This is the Cauchy criterion.
In chaotic systems, instead, it happens the exact opposite: if you integrate the system with two arbitrarily close initial conditions, the solutions diverge with time. If you observe this divergence in a very short time window, the divergence is exponential. The Lyapunov exponent is a measure of this divergence, in the sense that the two solutions diverge like exp(lambda*t), where lambda is the Lyapunov exponent, and t the time.
I don't know what system you are dealing with, but the basic idea behind is this.
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At the first i investigated vibration of vehicle suspension system then i choose parameter of system till chaos appeared in response then when i drew diagram for acceleration of sprung mass i saw very high acceleration so i changed my system and repeated this work  again see high acceleration and now its for me a question that,high acceleration is one consequence of chaos in this system.
**other question ***
when i use zero initial conditions and i dont have any excitation but suspension system show nonzero  response......i dont know WHY??
(i attached acceleration diagram too)(units----m/s^2,s)  
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The high acceleration is normally seen when natural frequency of the sprung mass matches to that of excitation frequency. ( Resonance frequency.) Check by allowing more time.
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Dear colleagues .we have one mechanical system and proved that vibration of this system was chaotic then i gonna use nonlinear  control method to control this system. is it possible or no??
Can you introduce some sources for help me.. .......
thanks
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First of all, what is the aim of your controller? I guess, the compound system of your mechanical system and the controller should not be chaotic,  maybe no movement at all, or maybe a oscillation with a fixed base frequency. Chaotic behaviour is characterized by the occurrence of frequencies in a wide range, including very large and very small ones, the frequencies on both ends cannot be neglected. The controller must suppress them all (except the targeted frequency) if I guessed correctly.  In terms of PID controllers, the D part may be present but should be small because of the high frequencies in the given system, similarly for the I part. Adaptive control techniques mentioned in a previous answer are a way to treat the very low frequencies.
Anyway, controlling a nonlinear chaotic system does not necessarily require a nonlinear controller: the compound system remains nonlinear even in case of a linear controller. And investigating the behaviour of the compound system still needs techniques from nonlinear system theory (e.g. the guidelines to design a PID controller are rules of thumb at best).
Sorry, I am not able, to give a more specific answer.
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 I drew bifurcation diagram of a mechanical system and saw jumping in this diagram,i want know that what is the meaning of this and when this phenomenon appear in bifurcation diagram?
(for example in 2.6 Hz we have jump) 
if it was possible introduce some source for helping . 
thank you
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By definition a bifurcation is a point where a system undergoes a qualitative change in its structure, and a bifurcation diagram is intended to help identify such points. Bifurcations can be classified into two types: continuous (also called subtle or supercritical) and discontinuous (also called catastrophic or subcritical). Discontinuous bifurcations usually exhibit hysteresis. Those that don't are called "explosive" (Smale 1967). It is also possible that your system has multiple attractors, and that a fixed initial condition will cross a basin boundary as the boundary moves in response to the change in parameters. In that case, the location of the jump will depend on your chosen initial conditions. This problem is usually avoided by changing the parameter slowly without returning to the original initial conditions, in which case hysteresis is observed by comparing the bifurcation diagram for increasing values of the parameter with the one for decreasing values of the parameter. In the region where hysteresis occurs, you should find (at least) two coexisting attractors.
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How do I Plot Bifurcation diagram by using origin if I have data and control parameter ?
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You obtain the quasi-continuous plot for many
values of control parameter. If the values of
the parameter are thin then you have to expand
your data.
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Consider a 2D (or a higher dimension) chaotic map like Tinkerbell map:
x(n+1) = x(n)2 -y(n)2 +a x(n) +b y(n)
y(n+1) = 2x(n)y(n) +c x(n) +d y(n)
Suppose that we don’t know the above equations and we only have access to the time series of “x” (long enough, clean). Can we calculate the Largest Lyapunov Exponent?
In addition:
Can we calculate the all of the Lyapunov Exponents?
What is the effect of noise?
What is the effect of short time series?
I appreciate any references, codes, and comments.
Thank you in advance
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Dear Sajad,
Some comments and caveats on Wolf and Benettin algorithms:
1)Augustova, P., Beran, Z., and Celikovsky, S. (2015). ISCS 2014: Interdisciplinary Symposium on Complex Systems, Emergence, Complexity and Computation (Eds.: A. Sanayei et al.), chapter On Some False Chaos Indicators When Analyzing Sampled Data, pages 249{258. Springer.
2) Tempkin, J. and Yorke, J. (2007). Spurious Lyapunov exponents computed from data. SIAM Journal on Applied Dynamical Systems, 6(2):457{474.
WBR
  NK
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For example, Bifurcation diagram for Delayed Lorenz system by using DDE-BIFTOOL?. Bifurcation parameter is taking as delay.
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You can also use XPPAUT to draw bifurcation diagram.
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Hi
I am trying to look at material instabilities in Lee Fenves non-associated plasticity model (aka Concrete damaged plasticity in abaqus) in form of $\det(E_{t})=0$
looking at element stiffness matrix, I am suspicious that the values provided by abaqus is really  a tangential stiffness matrix because:
  1. I expect a lower rank stiffness matrix in peak regime
  2. In softening, the stiffness matrix shouldn't be positive-definite
  3. the stiffness matrix shouldn't be symmetric
what I get from abaqus is in contradiction with my expectations. I get a symmetric positive definite matrix with the highest rank possible (18 for case of C3D8 element) which is not the case that i am looking for. That is why i am trying to get DDSDDE matrix from abaqus, which is in constitutive level, but I don't have idea about extracting it as an output. I would appreciate it if anybody can help.
PS: I attached my abaqus input file
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Shahriyar
abaqus mtxasm job=matrix_outX3.sim
[oldjob=matrix_outX3.sim]
[text]
is working for versions 6.13 and later but for earlier versions, I found the way. I've attached the input file.
I changed it this way:
*MATRIX GENERATE, STIFFNESS,ELSET=element_middle
*ELEMENT MATRIX OUTPUT,
ELSET=element_middle, STIFFNESS=YES,
OUTPUT FILE=USER DEFINED,FILE NAME=reza12
the option "file name" helps us format the output in .mtx instead of .SIM 
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In a closed nonlinear system, after a Hopf bifurcation, is there a variation in entropy?
Thanks!
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The entropy of a system undergoing a phase transition increases if the phase transition is towards higher internal energy (e.g., melting) and decreases if the phase transition is towards lower internal energy (e.g., freezing). The change in the entropy of the surroundings is of opposite sign. If the phase transition occurs quasistatically at thermodynamic equilibrium, then the total entropy change of the system plus surroundings combined is zero, otherwise the total entropy change is positive.
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I can think of many situations in which the assumptions of the Implicit Function Theorem do not hold but most of the results of the theorem can be recovered. The example of f(x,a) = a + x^3 comes to mind. At (x,a) = (0,0) the derivative is zero and therefore the implicit function theorem cannot be used. On the other hand, there is a unique continuous solution given by x = (-a)^1/3 for all real a.
I am looking for a review of some literature in which the authors have either developed a theory or have worked with a specific cases in which they side-step the Implicit Function Theorem to recover similar results.
My particular situation is as follows:
I have a nonlinear operator between a Banach spaces whose Frechet derivative is injective but is not bounded below and therefore it cannot have a bounded inverse. I am looking to show that a zero of the function can be continued for a small parameter value but because the derivative is not invertible at this specific point I cannot use the Implicit Function Theorem.
I am familiar with Nash-Moser Theory although through quite a bit of work I am convinced it cannot be applied in this situation.
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Dear Jason,
The following remarks do not solve your problem, but suggest constructing a convex or concave continuous solution y=f, being given the equation F(x,y)=0, where F is convex and continuous. Under certain additional assumptions, one defines f(x)=sup{t; F(x,t)<=0}, g(x)=inf{t; F(x,t)<=0}. Then f is concave, while g is convex, both of them being continuous under my assumptions. The graph of f is the upper part of the surface (or curve) {F=0}, while the graph of g is the lower part. Some details could be found in my paper  "An implicit function theorem for convex not necessary differentiable functions", Scientific Bulletin of the Polytechnic Institute of Bucharest, Electrical Engineering, 53, 1-2(1991), pages 19-23. No electronic copy of this paper is available. The method works in a quite general setting.
Best regards, Octav
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I know that critical transitions occur at catastrophic bifurcations, namely abrupt changes in the qualitative behavior of a system that occur at specific thresholds in external conditions. Catastrophic bifurcations arise in systems with alternative stable states (or, in general, alternative attractors).
So for example, a saddle-node bifurcation is catastrophic. Also subcritical Hopf bifurcations are catastrophic, therefore I think they should give rise to critical transitions.
Supercritical pitchfork and Hopf bifurcations are noncatastrophic, therefore I think they should be related to non-critical transitions. Nevetheless, in some articles the authors say that critical transitions occur also at supercritical Hopf bifurcations. Why? Are they wrong? Is there something I do not understand?
Many thanks in advance for your help!
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Dear Dr Fasoli,
According to page 257 of the J.M.T. Thomson book "Nonlinear Dynamics and Chaos" (Wiley, 2002), dangerous/safe stability boundaries are synonyms of catastrophic/non-catastrophic bifurcations.
The concept of "catastrophe" is due to René Thom his book "Stabilité structurelle et morphogénèse" (W.A. Benjamin, Inc., 1973), who was unaware the Andronov' School contribution to the qualitative theory of nonlinear dynamics, developed in the former Soviet Union (more particularly in Gorki, now Nizhniy Novgorod). It is also the case of the notion of structural stability, introduced by Aleksandr Andronov and Lev Pontryagin in 1937 under the name "systèmes grossiers", or rough (or inert) systems (cf. https://en.wikipedia.org/wiki/Structural_stability).
Sincerely
C. Mira
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Hi I have a system which seems to present transient chaos and I am beginner in this domain. Please what are the elements used to characterize transient chaos in a system?? About the finte-time Lyapunov exponent can you please help me with its definition??
The time evolution I have is shown by the figure attached.
Thank you for your answers
Patrick
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Transient chaos is very common, especially when initial conditions are chosen near the boundary of the basin of attraction or when the system is very close to a bifurcation point. Your plot shows clearly such an example, although you don't say what is being plotted. A good thing to plot is the largest Lyapunov exponent, which is a running average of the rate of exponential separation of two nearby orbits. For transient chaos, it will appear to converge to a positive value, but then after some time it will rather suddenly start heading toward zero. The finite-time LE is just what the name suggests -- it's the LE calculated over a moving window of finite time, long enough to average out the usually large fluctuations but short enough to see clearly the transition from chaos to periodicity as it goes from positive to zero. Chaotic transients can be very long, and in fact one can never be 100% confident that any observed chaotic system is not a transient.
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How to estimate parameters for a set of delay ordinary equations and make bifurcation analysis for certain parameters?
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You can also use dde23 function in MATLAB to obtain numerical solution of your delay differential equations and after this calculate bifurcation diagram at varying of equation parameters. Solution of your equations at a different values of parameters can be executed in parallel by using 'parfor' fuction:
If numerical solution of your differential equation takes a lot of time or you need to find the solution for many values of the parameters it is a very convenient to save obtained numerical solution into .txt or .csv file. And after this read data from file and calculate bifurcation diagrams, Lyapunov exponents etc.
For calculating bifurcation diagrams you need to determine the values of a local maximums in your obtained numerical solution as is done in the work, reference to which is is given below:
and after this plot the values of these maximus on the values of parameters for which they are defined.
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I am aware of some examples of such cases but they are very rare. For example the attached figure shows period-doubling route to chaos based on the real data (for more details about this figure see [1, 2]).
Can you introduce me some references demonstrating “period-doubling route to chaos” in a REAL biological system? I especially appreciate if someone can help in obtaining proper dataset.
Thank you,
Sajad
[1] D.W. Crevier, M. Meister, Synchronous period-doubling in flicker vision of salamander and man, Journal of Neurophysiology, 79 (1998) 1869-1878.
[2] R. Falahian, M.M. Dastjerdi, M. Molaie, S. Jafari, S. Gharibzadeh, Artificial neural network-based modeling of brain response to flicker light, Nonlinear Dynamics, 81 (2015) 1951-1967.
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A good example with some clear graphics is found here:  Controlling Cardiac Chaos (Science 257, 1230 (1992))
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In Mathematics the Lyapunov exponent of a dynamical systems is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Is there some new algorithms  for calculate the Lyapunov exponent?
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Lots of people just plot a projection of the trajectory in some 2-D space, and if it looks scrambled, they declare that the system is chaotic, and they are usually correct. However, if you do this, iterate the equations for quite a while before you begin plotting the results since chaotic transients are quite common for systems that eventually attract to a limit cycle or stable equilibrium point. If you plan to publish your claim of chaos in a reputable journal, you really need to show that the largest Lyapunov exponent is positive and statistically significant.
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Can anyone suggest some examples for bifurcations and chaos without delay? I want to know whether we frame an equation or system of equations with chaos without delay. Can anybody help with some examples which exhibit chaos or bifurcation? Can you also give any particular examples of which influences cause the system to become chaotic?
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There are a lot of dynamical systems without delay which exhibit chaos and bifurcations. See for example the attached file about an interesting 2D discrete system: the Tinkerbell map.
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kindly attach some relevant journals for further understanding
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Thanks Elman.
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I have a system of autonomous differential equations whose dynamical behavior I wish to analyze. The matrix is {{0,-omega,0,0},{omega,0,0,0},{0,0,0,0},{0,0,0,0}}. A reference would be very helpful, or even a name for the associated bifurcations. Another case of interest would be {{0,-omega,0,0},{omega,0,0,0},{0,0,-1,0},{0,0,0,0}}
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Hello Greg,
It seems that in the first case you have the Jacobian of a 'Zero-Zero-Hopf' bifurcation, which is codimension 3 - that is, an interaction of two steady-state bifurcations with a Hopf bifurcation. To the best of my knowledge this bifurcation is unstudied because of how complicated it is and that the study of it hasn't been properly motivated by a physical/mathematical problem yet.
The second case leads to a Zero-Hopf bifurcation which is quite well studied. Bifurcation textbooks such as Kuznetsov's and Guckenheimer and Holme's give detailed overviews of the unfolding of this bifurcation. This bifurcation is codimension 2 and the vector field can be appropriately reduced to get rid of the strongly contracting coordinate via a Centre Manifold Reduction.
Hope this helps!
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There are some lines in bifurcation diagram that the density of points there, is more than other spaces. Do these lines have Sinusoidal manner?
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Your observation is correct, though what you might expect are polynomial curves in bifurcation diagrams. Search in google for "polynomial curves in bifurcation diagrams" and you will find plenty of material on this topic.
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Consider a 3-dimensional dynamical system exhibiting a saddle-node bifurcation for a known value of control parameter. Is it expected that all three variables of the system undergo a jump when the bifurcation happens? 
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To give a more complete answer to your case: in a 3D system, the bifurcation point is
Po=(Xo,Yo,Zo,Ko)
where two fixed points are merged/split (depending on the direction of perturbation for the control parameter). The bifurcation occur at a point where the the implicit function theorem condition is violated (non invertible jacobian).
The jump occur because two solutions are merging (taking one direction of perturbation of the control parameter, K). So where you jump to - is a global property depending on the other fixed points of the  system while the analysis of the saddle-node is local analyzing the curve (X(K),Y(K),Z(K)), i.e., a single fixed point.     
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Hello all,
I have developed a model for a 9 dimensional dynamical system. I intend to perform bifurcation analysis on the aforementioned system in MatCont. I should add that, the system is highly nonlinear with very large equations in its Jacobian and Hessian matrices. The continuation on the stationary branch of solutions is easily performed and occurrence of several bifurcations can be observed. However, when I try to start the continuation study in the periodic orbits emerged from a Hopf bifurcation, MatCont returns the error of "Current step size is too small (point 1)". It means that the continuation algorithm does not reach convergence in calculating the second point of continuation on the periodic branch of solutions. Could anyone suggest me how to deal with this issue?
P.S.
I have tried to manipulate the amplitude, initial step size, Min step size, Max step size and tolerance values of the numerical algorithm. Can it be the case that I have not tried enough?
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Dear Matt,
 I want to thank you for your attention and response. The problem is that the integration window you mentioned is open to manipulation for time simulations, i.e. finding the equilibrium of the system. For bifurcation analysis, which is a continuation process, there are only two windows, Starter window (for initial conditions) and Continuer window (for options regarding the continuation), in none of which there is an option for choosing the solver. I somehow need to change the solver that MatCont uses by default.
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I have a function in terms of a vector x, and a parameter a, say F(x,a). I also know that there exists a smooth unique parametrized curve x(a) such that F(x(a),a) = 0 for all a. I want to know if the Jacobian matrix of F differentiated with respect to x is nonsingular when evaluated on the curve x(a).
In other words, if we have a dynamical system can we say that local solvability of an equilibrium guarantees that stability cannot change assuming a Hopf bifurcation does not take place?
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The Jacobian matrix of F with respect to x may be singular.
Let F(x,a) = (x-a)^2.
The only solution of equation F(x(a),a)=0 is the curve x(a)=a, and the derivative of F with respect to x on the curve vanishes.
For an example in several variables, take F(x_1,...x_n,a) = ((x1-a)^2,...(x_n-a)^2), then x(a)=(a,...a) and the Jacobian matrix of F with respect to x vanishes on the curve.
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AUTO can be used to study the continuation of periodic solutions of the system of ODEs.
When I study the frequency response of the following Duffing equation by using AUTO.
 
d2u/dt2+2*xi*omega0*du/dt+omega02*u+epsilon*omega02*u3=F*cos(omega*t),
epsilon=-0.1, xi=0.01,F=0.1,omega0=1
I found that if epsilon>0 or epsilon<0 but close to zero, say epsilon=-0.01 , it is easy to get the frequency response. However, if epsilon is very small, say epsilon=-0.1 , there will be abnormal termination. I have adjusted DS, DMAX, NTST, NCOL, NMX in the constant file as well as the computation direction, yet still no reasonable results obtained. I wonder whether AUTO-07p can be used to study the frequency response of the strong softening nonlinearity. How can I derive the frequency response using AUTO-07p if epsilon=-0.1 .
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Hi Amaechi
Thank you for your recommendation. Using Runge-Kutta method is a quite good approach to obtain the stable branches of the frequency response cureve. However, how to get the unstable part by this method?
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In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied.  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.   How to evaluate a bifurcation diagram ?
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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.
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For glow dischage plasma system period doubling bifurcation are demonstated as a route to low dimensional chaos. Why it is low dimensional ? can anyone plz explain the physical difference between low dimensional and high dimensional chaos. From the two chaotic time series how can i distinguish them? Will i consider lyapunov exponent value?
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Dear Vramori Mitra:
High-dimensional chaos is the term used to describe systems which possess chaos with more than one positive Lyapunov exponent, while on the contrary Low-dimensional chaos correspond to only one positive Lyapunov exponent. This might be the simple explanation for the case you mention about the glow discharge plasma. I do not think there is a simple physical interpretation on this. Perhaps you can gain some insights in: http://chaos1.la.asu.edu/~yclai/papers/PRE_99_HL.pdf
Best,
Miguel
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Hello all,
I am using Bifurcation and Continuation Software AUTO-07p, for my thesis. I need to plot some of the variables of my system, against time. However, solutions are only plotted for the time interval of [0,1]. I was wondering if there is a way to extend the time axis and obtain the time solution of my variables in a larger interval.
Thanks in advance for your kind attention.
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If you wish to solve your equations for an arbitrary time duration T0, a neat way would be re-scale your time variable from "t" to, say, "tr" such that tr=t/T0. If you then re-cast your equations in terms of "tr" rather than "t" then all you need to do is to change T0 before you begin your Auto run to change the duration since Auto will solve for tr=[0,1], hence t=[0,T0]. With this technique, you could also conceivably introduce T0 as a continuation parameter.
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In other words, do nonautonomous dynamical systems have a steady-state response?
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Hi Nasser,
what you need is Poincaré sampling. In general, define a Poincaré section and then, as you know, a stable cycle will become a fixed-point on the Poincaré section. Then you can proceed as for autonomous systems.
In your case in which excitation is periodic, you can perform stroboscopic sampling, which is a particular case of Poincaré sampling. To do this all you need is to define a fixed phase for the forcing function and take a sample of the output every time the input has that phase. Different phases will provide different-looking diagrams but they will be 100% equivalent. WARNING, in order to avoid numerical problem, make sure your sampling time (and integration interval) is such that you will have an INTEGER number of samples within a period of your input.
This works very well once you take notice of such details.
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In Forced system problem some problem to define time.
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Keep in mind that many nonautonomous dynamical systems can be easily transformed into autonomous dynamical systems. The way to do that is really simple: in Duffing you may have a sin(w*t). Let's consider that it is the solution of an harmonic oscillator:
dz/dt = - w*u
du/dt = w*z
Then, you Duffing non-autonomous oscillator becomes an 4D-autonomous dynamical system. You can use for bifurcation analysis any software in this case. I have built one for Mathematica in 2007 for 3D autonomous dynamical systems. You can adapt it for 4D 5D and so on...
Best Regards.
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Recently we completed a book chapter on non-equilibrium self-assembly (preprint available upon request). and we stumbled across many claims of bi-stability but after analysis no serious one remained. We would appreciate your suggestions for a working example, preferably a simple one.
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That is indeed catalytic action. I am looking for an experimental system that does self-assembly in that manner: catalytic and not using an interface.
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As for an  amplitude equation of Hopf bifurcation
$$dA/dt=(a+i) A+b|A|^2A$$, 
If the real part $Re(b)<0$, the Hopf bifurcation is supercritical for $Re(a)>0$, there exists a stable limit cycle.
If the real part $Re(b)>0$, whether Hopf bifurcation is subcritical and there exists a unstable limit cycle?
Whether the subcritical  bifurcation need the condition both $Re(b)>0$ and $Re(a)<0$ hold?
As for $Re(b)>0$,  whether we need furthermore expand the amplitude equation to five order to see the stability of the limit cycle? As far as I'm concerned, If $Re(b)>0$, the limit cycle is not stable, thus the expansion to five order can not change the stability of the limit cycle. Only when $Re(b)=0$, the expansion of  five order can help to judge the stability.
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it will be good for you to get the answers by yourself. To do so just set A=Rexp(i\phi). 
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I want to do Bifurcation analysis for reactor dynamics, If any one using MATLAB for this  analysis then please help or suggestion some paper or text. Thanks in advance
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