Science topic

# Nonlinear Dynamics - Science topic

The study of systems which respond disproportionately (nonlinearly) to initial conditions or perturbing stimuli. Nonlinear systems may exhibit "chaos" which is classically characterized as sensitive dependence on initial conditions. Chaotic systems, while distinguished from more ordered periodic systems, are not random. When their behavior over time is appropriately displayed (in "phase space"), constraints are evident which are described by "strange attractors". Phase space representations of chaotic systems, or strange attractors, usually reveal fractal (FRACTALS) self-similarity across time scales. Natural, including biological, systems often display nonlinear dynamics and chaos.

Questions related to Nonlinear Dynamics

Greetings,

I would appreciate any guidance on obtaining the natural frequency of a cantilever plate modeled using the third-order shear deformation theory.

I applied the Galerkin method to calculate the natural frequency and used the modes referenced in the attached article. I have included both the extraction file and the numerical code used for this purpose.

Thank you in advance for your assistance.

For modeling non-linear data, splines are quite a handy tool. However, it might get confusing to decide the most suitable spline type to use. Based on your data, what is the rule of thumb to select between natural cubic spline, b-spline or penalized splines? What things you take into account while selecting a suitable spline for modeling your data?

Imagine training a neural network on data like weather patterns, notoriously chaotic and unpredictable. Can the network, without any hints or constraints, learn to identify and repeat hidden periodicities within this randomness? This question explores the possibility of neural networks spontaneously discovering order in chaos, potentially revealing new insights into complex systems and their modeling through AI.

in designing and simulation of a nonlinear dynamics of 2 dof robotic manipulator using model reference adaptive control system, how can we select the reference model to the robot? or what can we use in the reference model for such nonlinear dynamic systems.

I am trying to use engineering stress-strain data for nonlinear modelling of the isotropic material in APDL. There is of course no problem in elastic region. In plastic region, we have strain hardening till necking followed by strain softening zone till rupture. But how do I exaclty know when softening starts?

**Does softening always starts after necking?**a) If No. How can I confirm the softening zone from engineering stress-strain curve?

b) If Yes, then I can simply change the engineering curve into true curve and use available hardening model till necking. Then what about softening? Well, I tried to use damage critetia to model the softening zone. As we know, it needs true stress-strain data as an input to find diffetent damage parameters.

**So, can we use the conversion relation beyond necking and find damage parameters?**Any suggestions are welcomed and Thank You in advance.

Dear Colleagues ; I am interested in studying advanced techniques for AC machines and i need to answers some questions they are as follows:

1/ Which basic distinctions set observation techniques apart from control approaches?

2/ Does the application of the type of observation impact the performance of the AC machines when hybridization between advanced techniques approaches occurs?

3/ Is it possible to apply both strategy to the machine by applying different techniques for each strategy?

3/ If we improve the rates of each strategy using advanced artificial intelligence techniques, will this increase the data of the controlled system?

4/ What's the difference between the MRAS, LGI (Kalman filter) and SMO. Are they limited to a specific time for machine systems?.

5/ Is it logical to apply a technique subject to a linear strategy to a nonlinear model of a machine like the Kalman filter or the extended-based adaptive observation?

Is there any solver available in MATLAB, MATHEMATICA or PYTHON PACKAGE?

Greeting Researchers

It is well known that within the linear range, the in-plane and transverse motion of a plate are independent of each other i.e. the equations governing the in-plane and transverse motions are uncoupled. The stiffening/softening effect of the in-plane loads on the transverse vibrations of the plates is then accounted for by considering the work done by the in-plane loads during the transverse motion. In FEM, the work done by the in-plane loads is used to obtain the geometric stiffness matrix. The FEM equations can be given as

Ma + (K + Kg)u = f ... (1)

where u is the vector of nodal displacements, a is the vector of nodal accelerations, K is the stiffness matrix and Kg is the geometric stiffness matrix.

When the range of motion is no longer linear, the equations of motion for the in-plane and transverse motion are inherently coupled. In this case, incorporating geometric nonlinearity in the Von Karman sense, the FEM equations may be written as

Ma + (K + Knl)u = f ... (2)

where Knl is the nonlinear stiffness matrix.

Now my question is

**whether it is necessary to include Kg in eq. (2) i.e. whether the correct dynamic equation of motion with the incorporation of nonlinearity is as shown below in eq. (3)**Ma + (K + Kg + Knl)u = f ... (3)

My opinion is that since in the nonlinear case, the equations of motion are inherently coupled and thus, there is no need for the inclusion of matrix Kg as done in eq. (3). The coupling is incorporated through the matrix Knl and eq. (2) is the correct dynamic equation of motion. In the linear case, since the equations of motion are uncoupled, it is necessary to add the matrix Kg to incorporate the effect of the in-plane loads on the dynamics of the transverse motion.

I would like to have your valuable opinions on the same.

Thank you for your time.

Best Regards,

Jatin

I'm currently doing a nonlinear model in Perform3D and I need to calculate the mode shape. I create the structure first in ETABS and import it on Perform3D but I noticed that there's a difference in the modal mass participation ratio. Therefore, I know that the story response plot that I extracted in ETABS is different from the Perform3D.

Now I don't know where to extract that data. Can anyone help me with this? For research purposes only.

Thank you.

Software like SAP 2000 shows the same modal periods for linear and non-linear models.

**Analyze the isothermal and kinetic results using nonlinear models and calculate the chi square for each model.**

I want to analyze a tall building for performance based wind design, How and where can I get the time-history wind load function ?.

Dear all,

I am currently dealing with a problem of spurious oscillations in wave propagation in elastoplastic bodies.

The problem in question is a 1D column subjected to a 10Hz sine wave acceleration input motion at the bottom of the mesh. Abaqus/Standard is used for the dynamic analysis.

Generally, the stress strain behavior looks fine, the same with displacements and velocities. The oscillations are most pronouncedly experienced in the calculated accelerations. They appear in time when the element/Gauss point is subjected to a sharp change in stiffness, e.g. from elastic branch into plastic one when no smooth hyperbolic relationship is used. The oscillations are of much higher frequency than the input motion [say around 100Hz].

I am wondering where these oscillations come from. I have tried to include numerical damping in the HHT direct time integration scheme however this did not influence the observed oscillations. I am wondering if further “play” with alpha, beta, gamma parameters in the HHT method could result in damping out the oscillation I experience. So far I have only used set of parameters as suggested in Abaqus Manual for “application=moderate dissipation” option.

I have also tried the effects of time and space discretization, non of the two was effective [for space discretization going for much finer solution than the minimum of 10 nodes per wave length as typically advised]. The problem remains and is insensitive to mesh or time step refinement. The only way of removing the oscillations is applying the Rayleigh damping, however this seems to be artificial way of removing the problem since the constitutive model is elastoplastic and deemed to be capable of accounting for material damping.

Generally, Abaqus manual says

*“The principal advantage of these operators is that they are unconditionally stable for linear systems; there is no mathematical limit on the size of the time increment that can be used to integrate a linear system”*so I understand that maybe this scheme is not stable or inaccurate for the nonlinear dynamic problems or a family of nonlinear dynamic problems. Would some another commonly used direct time integration scheme, such as lets say Bathe be more accurate here?Anyone has experienced maybe a similar problem? Where is it I can look for the reason of the oscillations?

Thanks in advance for any help and advice on that.

In his name is the judge

Hi

I think the best way to assign the nonlinear effect of steel components like beams that to model it with a plastic hinge.

So To build a nonlinear model of a one-story steel structure I use the Modified Ibarra – Krawinkler model to model my beam. In fact, I model a beam with two hing (zero length element) at the start and end and one elastic beam in the middle.

I used material developed by Lignos and Krawinkler named uniaxial material Bilin and here is more information about this material

After all, when I extract the output belonging to the stress-strain of this material under seismic loading like Kobe I get something like this diagram. (kobe0.5g) based on

*"Ibarra L.F., and Krawinkler, H. (2005). “Global collapse of frame structures under seismic excitations”, Rep. No. TB 152, The John A. Blume Earthquake Engineering Center, Stanford University, Stanford, CA."*

named article and this diagram (ref1) the stress-strain shows the right modeling and performance. consider that my record is near fault and the diagram of reference also belongs to near-fault loading.

If I use the 1g scale for the Kobe record I get this diagram (kobe1g) and here is the diagram of the 2g scale for the Kobe record (kobe2g) So my best shot is this when the record multiply and the ground accel become higher the material reaches the capacity of its deterioration very more quickly(In fact after a short time the strain of material switch between 1e10 an-1e10 and strain is between about -2 and 2) but it's a little non-sens for me in the other hand there is one benchmark( in the reference is named it before this benchmark is for standard loading) diagram for this material stress-strain (logos) and my diagram is different with it.

In the end here are the 1M$ questions

1- why when I give more scale to record the diagram become like that?

2- and why my diagram is different from the benchmark stress-stress of material?

Any help is greatly appreciated.

Take refuge in the right.

I have modelled Railway track bridge System, but while solving it shows Solver was unable to converge the non linear model

Please Help solving this error.

In his book "The Book of Why", Pearl says “Keep in mind also that the regression-based adjustment works only for linear models, which involve a major modeling assumption. With linear models, we lose the ability to model nonlinear interactions, such as when the effect of X on Y depends on the level of M. ” Does this mean that the regression model Y = a + b*X + c*M + d*X*M + e*Z is wrong (where X is the independent variable, M is the moderating variable and Z is the control variable)? I mean, the inclusion of the control variable Z in the regression model does not serve the purpose of blocking the backdoor path (if Z is the only confounder)?

Hi,

I want to ask you:

**'Which programming languages do you prefer to use for implementing software in Structural Engineering? (PYTHON, C++, or anything else)'**Recently I have heard about PYTHON and its application in my major, but still, I have doubts about choosing it or C++.

If you think there is a better programming language

**for implementing software in Structural Engineering**, please share your opinion with me regarding the mentioned question. I will be more than happy to receive your suggestions.Thanks

My nonlinear model in state space is

xdot= Ax + Bu+ Phi(x), where phi(x) is nonlinearity in the system

I am looking for papers/articles concentrated on indirect MRAC design. Any directions are appreciated.

Hello Researchers,

I am solving a nonlinear eigenvalue problem for a plate. The total stiffness matrix is the sum of the linear stiffness matrix and the nonlinear stiffness matrix. The linear stiffness matrix is symmetric but the nonlinear stiffness matrix is asymmetric due to which the total stiffness matrix becomes asymmetric. The mass matrix as usual is symmetric. The

**'eig'**command in MATLAB is used only for symmetric matrices. Can anyone suggest an eigenvalue solver for asymmetric matrices?Hi

I'm working on a research for developing a nonlinear model (e.g. exponential, polynomial and...) between a dependent variable (Y) and 30 independent variables ( X1, X2, ... , X30).

As you know I need to choose the best variables that have most impacts on estimating (Y).

But the question is that can I use Pearson Correlation coefficient matrix to choose the best variables?

I know that Pearson Correlation coefficient calculates the linear correlation between two variables but I want to use the variables for a nonlinear modeling ,and I don't know the other way to choose my best variables.

I used PCA (Principle Component Analysis) for reduce my variables but acceptable results were not obtained.

I used HeuristicLab software to develop Genetic Programming - based regression model and R to develop Support Vector Regression model as well.

Thanks

Two dimensional discrete-time nonlinear dynamical Henon chaotic map generates pseudo-random binary sequence which has been described as below Xn+1 = 1+Yn – aXn^2 and Yn+1 = bXn,where n=0, 1, 2… ..Here, the parameters, a and b are prime importance as the dynamic behaviour of system depends on these values. The system cannot be chaotic unless the value of a and b are 1.4 and 0.3 respectively.

Linear stability analysis fails to determine the local stability property of a non-hyperbolic equilibrium point as there is a emergence of a centre subspace (other than stable and unstable subspace) of the linearized system corresponding to the eigenvalue whose real part is zero. Centre manifold of the corresponding nonlinear system may not be unique. So, what is the exact procedure to analyze such kind of situation?

How can I numerically solve coupled polynomial or transcendental equations using MATHEMATICA?

Hello everyone,

I've implemented a FEM code in

*matlab*, and now I'd like to make it faster. Since I have to perform nonlinear dynamic analysis, I have a lot of iterations, each one requiring to asseble the full stiffness matrix K.What I would like to do thus is to parallelize the assembling procedure of K. Till now I tried

**parpool**and**spmd**, the latter with poor results, the first one performing nicely (speedup factor x2... despite using 10 cores...) but only under a certain number of elements. Overcome a certain treshold, parallel computation (14 cores) would take as much as 10 times the single core version.I understand this may be related to overheads in the comunication between "master" and workers and/or slicing procedures, but it seems I cannot get the hang of it...

Does anyone have suggestions and/or can point me to some useful material on this specific matter?

Thank you all in advance,

Jacopo

I have been reading papers on asymmetric effects using panel nonlinear ARDL approach but in most cases, I do not see control variables in the model

Hello

Recently I have heard that just SAP should be used for dynamic analysis of building and the results of ETABS are not valid for nonlinear dynamic analysis. I have done nonlinear dynamic analysis with Etabs software. Are the results acceptable?

Is there any evidence for this?

Thanks in advanced

Specifically, I am using generalized additive models with penalized splines in R - gam() and gamm4(). The smooth term tells me the F-statistic and p-value, but I would like to get the standardized betas so the results can be more comparable to the linear models within my study.

Can someone point me in the right direction? I am coming from a limited mathematical background so resources that go beyond a pure theoretical explanation would be most beneficial for me.

In Uncoupled modal response history analysis developed by Prof. Anil K. Chopra, we get the Nonlinear governing equations of inelastic SDOF systems.

After that, how to solve these Nonlinear governing equations when we know the relationship "force–deformation" (F

_{sn}/L_{n}− D_{n}) to get the response-time history of D_{n}(t) and F_{sn}(t) ?And what software can be used?

I am doing 3pt bending test in LS-Dyna with a glassfibre-epoxy sample. So it is a non-linear dynamic (quasi static) problem. I assume it is a implicit problem since the event is slower and the effects of strain rates are minimal. But my professor used explicit control card to solve this. what difference can it give?

Given that the 4th-order non-strict feedback nonlinear dynamics (also, including the transcendental terms)

x_dot=f(x)+g(x).

**u(x)**, ..........................................(1)where x=[x1,x2,x3,x4],

f(x)=[ f11(x2,x4) ; f22(x2,x3,x4); f33(x4); f44(x2,x3,x4) ],

where f22(x2,x3,x4)=atan(x2,x3,x4), and f44(x2,x3,x4)=atan(x2,x3,x4),

g(x)=[0;b1;0;b2].

I am trying to find the solution for above differential equation using the following steps

**step 1:**convert the dynamics (1) into linear strict feedback using feedback linearization.

z_dot=p(z)+m(z).

**v(z)**..........................................(2)where z=[z1;z2;z3;z4], p(z)=[p11(z1,z2); p22(z1,z2,z3);p33(z1,z2,z3,z4); p44(z1,z2,z3,z4)],

m(z)=[0;0;0;v].

**step 2:**apply the backstepping,

**v(z)**, where

**v**is the function of

**u(x)**.

Note: elements (in order) in closed-loop system are consist of (a) input; then (b) controller,

**v(z);**(c) original dynamics (1); (d) converted dynamics.**step 3:**But the result is showing steady-state error in converted dynamics i.e., especially for z2, z4 (also, very sensitive to initial conditions). but all states of original dynamics are converging very well on zero equilibrium for any initial conditions.

Please help me about some questions as given below

(1) So what should I do for removing this error?

(However I am thinking about applying the Integral backstepping, adaptive backstepping, Dynamic Surface Control (DSC) ).

(2) Should I only concentrate on original dynamics (because converging the original dynamics with the help of converted dynamics)? But my aim is that I want to converge both of the dynamics.

(3) Will it be useful if apply backstepping for converted dynamics (2) or it is better to apply sliding mode directly on original dynamics (1)?

Hello everyone!

I am using dlnm package in R (by A. Gasparrini) and I would like to perform qAIC in order to choose the model with the best fit.

Since I am not experienced in R, is there an efficient way to do so?

Thank you in advance!

Can any body suggest me a straightforward way with low computational cost, for obtaining LCO amplitude and frequency for multi-dimensional phase space?

I tried to solve an equation of Van-der-Pole using linearizations in the current point of the phase space (state-space). This method produced damped oscillations whereas the nonlinear model produced stable oscillations. Is there a method to obtain a solution closed to exact using linear models with some kind of linearizations of something like that? I would like to have a deal with eigenvalues and aigenvectors of the linear equations produced by the former nonlinear equations.

I am trying to model the shear link in sap2000/ etabs by I do not know how to model the web stiffened beam in this software because for the shear link we have to use web stiffness.

Please suggest how to do it.

I want to perform a nonlinear dynamic analysis of this model.

I'm working on the FE analysis of nonlinear vibration of circular plate in ansys APDL. I considered the Mooney-Rivlin model of hyperelastic materials. So i want to know that how to find natural frequency of this problem in ansys APDL?

I have a dataset of 5 variables of quantitative continuous type: 4 independent and 1 dependent (see attached). I tried using linear multiple regression for this (using the standard

*function in R), but no statistical significance was obtained. Then I decided to try to build a nonlinear model using the***lm***function, but I have relatively little experience in this. Could you help me, please: how to choose the right "equation" for a nonlinear model? Or maybe I'm doing everything wrong at all? So far I have used the standard linear model in the "non-linear" model.***nls**I would be very grateful for your help.

If you do not have the opportunity to open the code and see the result, I copy it here:

------

*library(XLConnect)*

*wk <- loadWorkbook("base.xlsx")*

*db <- readWorksheet(wk, sheet=1)*

*INDEP <- NULL*

*DEP <- NULL*

*DEP <- as.numeric(db[,1])*

*for(i in 1:4){*

*INDEP[[i]] <- as.numeric(db[,i+1])*

*}*

*MODEL <- NULL*

*SUM <- NULL*

**MODEL<-nls(DEP ~ k0 + INDEP[[1]]*k1 + INDEP[[2]]*k2 + INDEP[[3]]*k3 + INDEP[[4]]*k4, start=list(k0=0,k1=0,k2=0,k3=0,k4=0))***SUM <- summary(MODEL)*

-----

The result is:

-----

*Formula: DEP ~ k0 + INDEP[[1]] * k1 + INDEP[[2]] * k2 + INDEP[[3]] * k3 +*

*INDEP[[4]] * k4*

*Parameters:*

*Estimate Std. Error t value Pr(>|t|)*

*k0 6.04275 1.30085 4.645 6.41e-06 ****

*k1 0.03117 0.01922 1.622 0.107*

*k2 -0.02274 0.01663 -1.367 0.173*

*k3 -0.01224 0.01717 -0.713 0.477*

*k4 -0.01435 0.01541 -0.931 0.353*

*---*

*Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1*

*Residual standard error: 1.418 on 186 degrees of freedom*

*Number of iterations to convergence: 1*

*Achieved convergence tolerance: 2.898e-08*

-----

I'm trying to linearize a non-linear model and write it in state-space form (derive state-space matrices) for control and optimization purposes. The nonlinear model is: x(k+1)=f(x(k),u(k),d(k)), and it contains the minimum of three terms:

f(x(k),u(k),d(k))=x(k)+min(a.u(k),b.x(k),c.d(k))+...

a,b and c are constants and the three terms inside the min() function are all linear. How do I go about linearizing this model? Note that I want to use the linearized model as a prediction model for MPC.

Thank you in advance.

I have a model that consist of two ODEs and one PDE which are all coupled and non-linear. When I linearize the system around the steady state and simulate the system with initial values that are different from steady state I get a stationary deviation in one of the solutions. Due to non-linearities I expect a quite different behavior for the linear system. However, if I multiply some the elements of the system matrix with a constant (relaxation or dampning coefficient?), the linear system now resembles the non-linear system much better in terms of transient behavior and that I no longer have a stationary deviation which I had before. Is this method for fitting the linear model allowed and have some theory that can justify the use of such relaxation constants? Or is it just a 'cheat' which is not valid as the system does change due to the use of such constant?

What is the stability of a fixed point (non-isolated), which has zero eigenvalues? Is there any analytical procedure for finding stability of these type of fixed points. If numerical analysis can do the job, then how to do that?

In the case of fitting nonlinear models, is the COD (R

^{2}) sufficient^{for the selection of the best fitting model or should other criteria be used such as Akaike's information criteria?}Dear researchers

As you know, one of the challenges of using nonlinear procedures is to determine the behavior of plastic hinges of members with deformation controlled action that this behavior is assigned to the plastic hinge by a force-deformation curve and its relations using parameters modeling. various researches has shown that the uncertainties in these modeling parameters significantly affect the structural responses.

Also, the acceptance criteria of different performance levels relating to the mentioned force-deformation curve are needed for performance-based design of structures.

There are two questions now:

1- Are force-deformation curves presented in ASCE 41-13 suitable only for nonlinear static analysis (push over)? or also is applicable for nonlinear dynamic analysis?

2- Given that the acceptance criteria presented in ASCE 41-13 are derived based on the mentioned force-deformation relations in this code (a, b and c modeling parameters), what acceptance criteria can be used to evaluate the structure at the IO, LS and CP performance levels if the other force-deformation relations presented in the technical literature (such as Lignos and Hartloper relations for beams and columns of moment frames, respectively) are utilized for concentrated plasticity modeling?

The mentioned curves (Lignos and Hartloper relations) are mostly used in structural modeling to study the structural collapse, in which the collapse is determined by the criteria mentioned in FEMA p-695 and as a result, acceptance criteria in accordance with these behavior curves have not been researched.

Given a system of First Order ODE`s of type y'=Ay+B(y) where B(y) has each of its elements composed by biquadratic polinomials of y.

Do alternatives exist (to the classic Runge-Kutta or Adams-Bashforth / Adams-Moulton methods) to resolve this type of systems improving their computational time and/or their accuracy?

Hello!

I'm currently studying the

**influence of adding a mass in a cantilever tip, near to a bifurcation.**In order to do it, I simplified a model of the cantilever to a 1DOF mass(m)-spring(k)-damper(b) system, where there is a stiffness linear and nonlinear (koX + k1X^3) so the system can be descrived as a Duffing like behavior ( for large amplitudes).

I wrote the equation of motion of the system as in the Image I annexed and then I made it nondimensional.

Now I wrote the differential equations (velocity, position) just to look at the

**phase portrait and the poincaré mapping**. (For analyse the reponse frequency I have to apply Method of multiple scales and obtain amplitude and phase but for now just want to analyse the system before it).In order to analyse the phase portrait, etc. I used a

**matlab program**made by Housam Binous, for forced Duffing systems and I tried to adapt to my case ( Basically instead of Force*cos(Wt) and used the electrostatic force as described in the image) but I am not getting the results I expected in the plots, they are very strange).Could you help me solving this challenge? I'm struggling with it, but It should be simple .

I annexed the matlab files and the image.

Thank you very much !

Kind Regards,

Bruno Silva

Hi all,

Can you please explain the "Dynamic Pull-in" concept in electrostatic parallel plate MEMS actuators? I would like to know "in detail" about its causes and effects. How is it related to the "Static pull-in"?

Regards,

Raghu

- Theory of chaos describes behaviour of dinamic systems evolution which are sensitive on starting conditions. Chaos implies nonlinearity. Nonlinear relationship are a necessary condition for chaotic systems. Existance of nonlinearity alone does not make a chaotic system. All natural processes are nonlinear. Human brain is the most chaotic system on world. Is chaos organisational form of nature?

When I am trying code:

AIC(model) an error occurs:

Error in UseMethod("logLik") :

no applicable method for 'logLik' applied to an object of class "c('cesEst', 'list')"

The model is a fitted CES function. So basically I am also unsuccessful in obtaining log likelihood of the model.

Would highly appreciate your help!

Hello,

my question might be more theoretical. The datasets I currently analyse are not normally distributed (according to shapiro-wilk and kolgomorov-smirnov @ p = 0.05). Therefore, I prefer plotting them as boxplots. However, I would actually like to fit a nonlinear model (in that case a hill equation) to the (non-existing) mean in order to extract some parameters.
So, is there any way and is it allowed to fit my function to the median instead of the mean values? I haven't found publications where this is done and I assume that this is for a good reason. I just don't know why and what else to do with my data.
Thanks for your help!
Philipp

Eccentrically Braced Frame building of 6 story modeled in SAP 2000. Conducting a non-linear dynamic response to earthquakes.

I would be grateful for suggestions to solve the following problem.

The task is to fit a mechanistically-motivated nonlinear mathematical model (4-6 parameters, depending on version of assumptions used in the model) to a relatively small and noisy data set (35 observations, some likely to be outliers) with a continuous numerical response variable. The model formula contains integrals that cannot be solved analytically, only numerically. My questions are:

1. What optimization algorithms (probably with stochasticity) would be useful in this case to estimate the parameters?

2. Are there reasonable options for the function to be optimized except sum of squared errors?

Dear Colleagues,

I have recently graduated with a BSc in Mechanical Engineering. During my BSc, I assisted research and projects on a variety of fields ranging from nanomechanics of advanced materials (experimental), predictive analysis of stochastics data input for control (MATLAB), human balance control (theoretical), dynamical modeling of fluid/solid coupling problems, and corresponding CFD in OpenFOAM, computational aerodynamics with HPC. Upon my graduation, I joined a research team at ETH Zurich as a scientific assistant to work on vortex kinematics (theoretical and computational).

**My main interest areas are:**- Nonlinear Dynamics and Chaos, Stochastic Systems, Machine Learning of Dynamical Systems and Fluid Dynamics, Prediction, Nonlinear Control
- Computational Finance, Financial Analytics
- Numerical Methods, Computing and Algorithm Development

Clearly, all of the fields mentioned above require a decent knowledge of mathematical modeling, analysis, and computation (mostly by parallel computing over HPCs). One can also argue that these areas are not really far from each other as they can be all classified into an umbrella field of Dynamical Systems Theory.

I will soon start my MSc in Computational Science and Engineering at ETH Zurich. However, I am struggling to decide which specialization area I should choose.

**As a part of the program I have to enroll at least in two of the following CORE SUBJECTS:**- Advanced Numerical Methods for CSE
- Optimization for Data Science
- Computational Statistics
- Advanced Systems Lab (Fast Numerical Codes)

Of this, I am planning to take all as they are rich in content, relevant to my multidisciplinary taste, and beneficial for my future plans. They are also fairly complementary to one another.

**I will also have to take two mandatory subjects as a part of the admission requirement:**- Numerical Methods for CSE
- High-Performance Computing Lab for CSE

*The program requires me to take 5 courses in my selected specialization area. The rest of the credits necessary to graduate can be chosen freely from any department.

ETH is a top-notch institute for education and research in all three of Control & Robotics, Fluid Dynamics, and Applied/Computational Mathematics. This at least ensures that whatever I choose I will still get a quality education and have a chance to do quality research.

As we all know, modern areas such as robotics, data science, software engineering, neuroscience, computational biology and etc. have rather well-defined career paths. These people would not have as many troubles as a multidisciplinary guy (e.g. my MSc program) to decide what subjects to take and what to focus on.

Now, I lost 2 lost years between the high school and university and I believe this has eliminated some of my flexibility in this kind of decision, especially given that I am in a distance relationship of which I have to also take care of. It is likely that I will prefer to stay at ETH for my Ph.D. or work some time here before my Ph.D. I may also choose to do my Ph.D. in one of the other top schools.

I really appreciate your opinions and advice!

Thank you for your time and patience!

Kind Regards

By dynamical systems, I mean systems that can be modeled by ODEs.

For linear ODEs, we can investigate the stability by eigenvalues, and for nonlinear systems as well as linear systems we can use the Lyapunov stability theory.

I want to know is there any other method to investigate the stability of dynamical systems?

Hello to all friends and researchers.

In the next few weeks I will have a webinar on "Adaptive Fuzzy Sliding mode Control for Nonlinear Dynamic Chemical Processes". But I have trouble presenting content.

If any of you know the information or article or any reference in this field, I will be happy to share it with me and use your comments. Thanks

Can we model nonlinear behavior of area elements for in-plane stresses?

Wolf's and Rosenstein's algorithms does not seem to include the multidimensional scenario (if I understand them correctely). I want to measure the nonlinear dynamics of the liquid state machine's dynamical system.

Hi All

in phase space reconstruction and with a aim of unfolding a system dynamic (non linear) first we must determine 2 parameter , time lag and number of dimension. my main question is if our dimension more than 3 how can these dimensions be drawn in MATLAB?

Can anybody tell me the steps to be fallowed in sap2000 to define and assign to obtain the hysteresis curves for a nonlinear dynamic analysis model.

Hi > my name is reda kriker .. studying at master course on civil engineering .... doing an etabs model for a G+4 building ... considering progressive design and modification based on GSA ...

I need to know what type of analysis did you conduct and if nonlinear dynamic how did you model the plastic hinges ?

I am trying to model nonlinear, cyclic behaviour of a cantilever beam, I wish to apply a cyclic load at the free end. And I want the midpoint of the beam to be displacement controlled (ie. reverse loading direction once the displacement at midspan reaches predetermined values).

How can I do this?

If we have a nonlinear dynamical system how are we supposed to perform parameter estimation, for different parameters and with respect to which output.

How are we going to come up with an optimization scheme, and choose which set of parameter to optimize the with respect to which output/s.

I have a stochastic mixed-integer non-linear model and it is difficult to be solved. So I want to write the constraints with uniform distribution in the sub-model that is found in the whole model. what are the steps for this? Thanks

Consider the famous fractal sets, Mandelbrot and Julia sets. They are based on the idea of choosing two complex numbers Z(0) and C with proper run time and escape-region. They are achieved by repeatedly evaluating the following equation:

Z(n+1) = Z(n)^2+C

For example, in Mandelbrot set, consider a 400×400 mesh when x is in [-2.5 1.5], y is in [-1.5 1.5], run-time is 32 and the escape region is 2.

The final plot is as follows

The yellow part in that figure corresponds to the points in which the value of the function never reaches the escape region. However, the different spectrum of the blue points corresponds to the iterations in which the function crosses the escape region

I have two questions:

a) Is there any study on the transient part of such process (and not steady state)

b) What happens when we don’t consider escape-region?

ANN, which has the ability to model nonlinear systems, could be used to predict BW from age.

One of the salient phenomena of nonlinear dynamics called antimonotonicity (creation followed by destruction of bifurcation bubbles) is widely observed in the literature. However, a crucial question remains unanswered: that of knowing its real practical utility. Answers and suggestions will be welcome.

I am studying the rheological behaviour of some pastes at different temperatures. A common approach to modelling the influence of temperature is to consider its effect on the apparent viscosity of the material. Other studies have separately modelled the parameters of the Hershel-Bulkley model as a function of temperature. This was achieved by fitting the data obtained for each parameter (σ0, k, and n) of the Hershel-Bulkley model to the Arrhenius equation separately. From this, estimates for the pre-exponential factor and activation energy, Ea, were obtained. Since the Arrhenius equation is a non-linear model, I assume these estimates were obtained through non-linear regression.

Questions:

1. Is my assumption correct?

2. If yes, should I retain values of 8.314 J/mol.K for R and 273.15 K for absolute temperature?

Suggestions would be greatly appreciated.

Thank you!

is VIF also suitable to check multicollinearity for non-linear models? is there other methods to measure the nonlinear dependency between the predictors ?

In the case of storage at uncontrolled temperatures, in order to establish shelf life.

I am working with the Weibullian model, any other that I can use?

**How is linear algebra used to represents nonlinear models in 3d game programming or in real coordinate space applications?**

I ran an abaqus model of nonlinear static analysis and there is no error message. The message file actually said the analysis completed in 54 increments. However when I loaded odb to abaqus for visualization, there are no increments to show for the step. Any reason why?