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# Bifurcation Analysis - Science topic

Explore the latest questions and answers in Bifurcation Analysis, and find Bifurcation Analysis experts.

Questions related to Bifurcation Analysis

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?

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?

Whether the existence of periodic window affects the encryption efficiency or not?

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.

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 This leads to a non-associated flow rule and the analytical solutions are no longer valid.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 768-787S Van Baars, 2018 - 100 YEARS OF PRANDTL'S WEDGE.

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!

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!

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

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)?

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)?

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:

- it must be sensitive to initial conditions
- it must be topologically mixing
- 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. 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.

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?

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?

- 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.

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?

On what parameters the selection of continuation method depends ?

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

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?

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.

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.

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.

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?

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?

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

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,

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.

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

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

could anyone provide the source code to plot basin of attraction of a given nonlinear ODE system?

Any suggestion lectures and references on Nonlinear weakly stability analysis and type of bifurcations.

I mean is there any code that help me to draw it by using Mathematica or Matlap.

Regards,

Zenab

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

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

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

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

It's possible to calculate the bifurcation angle.

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.

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

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

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?

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

i wanna calculate Lyapunov exponent and i dont know any thing about it can you introduce for me some good source about it??

thanks

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)

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

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

How do I Plot Bifurcation diagram by using origin if I have data and control parameter ?

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

For example, Bifurcation diagram for Delayed Lorenz system by using DDE-BIFTOOL?. Bifurcation parameter is taking as delay.

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:

- I expect a lower rank stiffness matrix in peak regime
- In softening, the stiffness matrix shouldn't be positive-definite
- 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

In a closed nonlinear system, after a Hopf bifurcation, is there a variation in entropy?

Thanks!

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.

I know that

**occur at***critical transitions***, 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).***catastrophic bifurcations*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!

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

How to estimate parameters for a set of delay ordinary equations and make bifurcation analysis for certain parameters?

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.

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?

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?

kindly attach some relevant journals for further understanding

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}}

There are some lines in bifurcation diagram that the density of points there, is more than other spaces. Do these lines have Sinusoidal manner?

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?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?

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?

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.

*d*^{2}u/dt^{2}+2*xi*omega0*du/dt+omega0^{2}*u+epsilon*omega0^{2}*u^{3}=F*cos(omega*t),

*epsilon=-0.1, xi=0.01,F=0.1,omega0=1*I found that if

**or***epsilon>0***but close to zero, say***epsilon<0***, it is easy to get the frequency response. However, if***epsilon=-0.01***is very small, say***epsilon***, 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***.***epsilon=-0.1*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 ?

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?

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.

In other words, do nonautonomous dynamical systems have a steady-state response?

In Forced system problem some problem to define time.

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.

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.

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