Nonlinear Systems

Nonlinear Systems

  • Zhengdong Du asked a question:
    Is it possible to detect the hysteresis effect using the Monte Carlo simulation method?
    A common numerical method for computing bifurcation diagrams is the Monte Carlo method. Using this method, for a fixed parameter value, a set of initial points is chosen by random number generator, then the flow is computed and the data is recorded and plotted after a long time of transients. Then the parameter is increased slightly and this process is repeated until the parameter reaches the maximum value. My question is: is it possible to detect hysteresis effect using this? I guess we cannot because the initial points corresponding to each parameter value is independent to the result corresponding to the previous parameters. Thus when you decrease the parameter and compute the diagram, you would get the same result. But I am not sure if this is correct. I would be very grateful if someone can explain to me on this.
  • Zhen Wang added an answer:
    Can anyone answer my questions on the Lyapunov number and Lyapunov exponent?
    In the book "Chaos: An introduction to dynamical systems" (J.A. Yorke) on page 107, where the Lyapunov number and Lyapunov exponent are defined, 1- Give a orbit (x1,x2,,...), define a Lyapunov number L(x1)=limit(abs(f'(x1)...f'(xn))^(1/n)). I want to know if L(x1) equals L(x2)? If so, why? And how can that be proved? If not, why, and how can that be proved? 2- If L is the Lyapunov number of the orbit x1 under the map f, how can it be proved that the Lyapuonv number of the orbit x1 under the map f^k is L^k?
    Zhen Wang · Xijing University, Xi'an, china
    1 The orbit {x1,x2,x3,...} under the map f, i don't know whetherLyapunov number L(x1) equals L(x2)? If so, why? And how can that be proved? If not, why, and how can that be proved? 2 If L1 is the Lyapunov number of the orbit x1 under the map f, L2 is the Lyapunov number of the orbit x1 under the map f^k, How to prove L2=L1^k?
  • Shogo Inagaki added an answer:
    What is the most popular root for nonlinear algebraic equations that you have faced?
    According to your experience, what number is most frequently obtained as a root using any known method? You can indicate not just one number but an interval.
    Shogo Inagaki · Kyoto University
    I am rather interested in nonlinear partial differential equations not algebraic equations.
  • Shogo Inagaki added an answer:
    Do we have chaotic behaviour in two dimensional continuous dynamical systems?
    What is the minimum dimension requirement to exhibit chaotic behavior for continuous dynamical systems? I came across that it would be greater than or equal to three. But the Dixon system violates the Poincare-Bendixson theorem. My question is do we have any other systems other than Dixon system which violates Poincare-Bendixson theorem? Or is there any other two dimensional system that can exhibit chaos.
    Shogo Inagaki · Kyoto University
    Even in one spacial dimension (two dimension in phase space), chaos appears. See Konishi & Kaneko, J. of Phys. A25 (1992) 6283.
  • Thomas I. Seidman added an answer:
    If you have a time-dependent parabolic equation what is the best method to solve it other than finite difference method?
    The equation is u_t=a(t)u_xx+b(t)u_x+c(t)u+f(x,t) with initial condition and Direichet boundary conditions as follows: u(x,0)=u_0(x) and u(0,t)=u_1(t), u(1,t)=u_2(t).
    Thomas Seidman · University of Maryland, Baltimore County
    I haven't seen any consideration of what I would consider a most important aspect: where (physically) the time-variation comes from. In particular: how rapidly are the coefficients varying and are they doing so in some coordinated way (in which case an analytic change of the time variable may remove the problem)?
  • Can anyone help with a nonlinear time delay system?
    I want to approximate nonlinear time delay control system by B-splines and hybrid some of them with other functions (rather orthogonal functions). Can anyone guide or suggest me the way that this idea can be done?
    Hesham Abdelghaffar · Invensys
    I have used neural generalized predictive controller (Neural GPC) before with nonlinear non-minimum phase processes with variable dead time and was sufficient to solve the problem.
  • Mahdi Heydari.m added an answer:
    What's the basic difference between Euler-Bernoulli and Timoshenko beam theory?
    Could anyone tell me the basic difference between Euler-Bernoulli and Timoshenko beam theory? Please kindly help me by introducing good references in this respect which would present all parameters in the governing equations.
    Mahdi Heydari.m · Amirkabir University of Technology
    You Can refer to Abrate's book, Impact on composite materials,Unit 3
  • Javid Ziaei added an answer:
    Can anyone help me to compute the Lyapunov exponent?
    I'm looking for the Lyapunov exponent for a delayed system and computing it in MATLAB.
    Javid Ziaei · Urmia University of Technology
    Dear Adil Thanks for your answer. Can you give me more insight on what do you mean?
  • H. P. Salunkhe added an answer:
    How can I solve a^2 y = y^3 - y''' - y'?
    Can anyone suggest an analytical method to solve the ODE a^2 y = y^3 - y''' - y', where a is a positive constant? The solution should have y(0) = 0. Also y -> a and y' -> 0 as x -> infinity.
    H. Salunkhe · Shivaji University, Kolhapur
    As compare to analytical method Better way is to solve this problem by numerical methods
  • Adil AL-Rammahi added an answer:
    Which is the best solution for solving highly non-linear numerical problems like phase change?
    Out of the solvers available like MUMPS, PARADISO, SPOOLES, etc. which is best suited for solving highly non-linear problems
    Adil AL-Rammahi · University Of Kufa
    B-Spline method is best yet.
  • Rodolfo Reyes-Báez asked a question:
    How is robustness of a nonlinear controller based on contraction theory and incremental stability approach? Are there any physical experiments?
    I am working on the first draft of my PhD proposal about a class of nonlinear multi-physics systems. I would like to know if there are real experiments with control schemes based on contraction theory and incremental stability approach.
  • Javid Ziaei added an answer:
    Is there any relation between fractional calculus and physical concepts?
    Nowadays, fractional calculus has been introduced in many fields. However, I have not seen a relation between physics and fractional calculus. In other words, is there any natural or physical phenomenon whose dynamics may be given with fractional order differential equations?
    Javid Ziaei · Urmia University of Technology
    Thanks for all
  • Iuliu Sorin Pop added an answer:
    Why do under-compressive shocks form?
    Is there a way to see intuitively and physically why under-compressive shocks form? The formation of compressive (Lax) shocks makes perfect sense to me, but I can't see why under-compressive shocks should form at all.
    Iuliu Pop · Technische Universiteit Eindhoven
    Hi Marc (and Mark :) ), I'm not sure if I understood you right. If you mean by this that one characteristic enters the shock (this being a "shock" in what you said before) and the other one leaves it (the "rarefaction"), then, indeed, we have an undercompressive shock. For Buckley-Leverett it is possible to have solutions consisting of a rarefaction wave that ends up precisely with the value where the shock starts (actually this is the "standard" entropy one). I mean, the speed of the endpoint of the RW is exactly the speed of the shock, and the the endpoint of the RW is the (say) left state of the shock. This is not undercompressive, but a standard, compressive shock. So simply the combination RW-shock needs not to be undercompressive. Is this what you mean? Have a great weekend and hope to meet you soon, Sorin
  • David LeRoy Elliott added an answer:
    What are the current research tends in the control of nonlinear systems?
    Theses days, I am doing literature review about the new trends in control of nonlinear systems, but it looks like this is a very broad subject. What are the new research areas in the control of such systems?
    David Elliott · Institute of Electrical and Electronics Engineers
    If you are planning to write a dissertation, look for a new topic that bridges areas that are under-represented in current journals. Nano-mechanics, biological systems, discrete time systems, ...
  • Deep Shekhar Acharya asked a question:
    What are the recent advances, issues to be addressed, and scope of research in the area of stability analysis of multi-agent control systems?
    What are the recent advances, issues to be addressed, and scope of research in the area of stability analysis of multi-agent control systems during intermittent and/or permanent sensor faults or loss of observation? Please shed some light on this area. In need of help.
  • Soumya Banerjee added an answer:
    How to draw phase portrait plots for delay differential equations in matlab?
    Quiver function is being used for phase portrait plots obtained using ode. Is there a way for plotting phase portraits and vector fields for autonomous system of delay differential equations in matlab?
  • Mohammed Abu-Hilal added an answer:
    What is the best hardware for numerical integration?
    As one of my fields of study is dynamics of nonlinear systems, a lot of my work is based on numerical integration. However, I am thinking of buying a new computer, and I want it to be the best choice to make my integrations faster, so I could be more productive. Hence, regarding the processor, what I really want to know is if a greater number of cores is better, or a higher clock speed is what I'm looking for.
    Mohammed Abu-Hilal · An-Najah National University
    I prefer to use Mathematica
  • Amaechi J. Anyaegbunam added an answer:
    How can I solve a second order linear ODE with variable coefficients?
    I am trying to solve Fick's law in a radial geometry (non-axisymmetric). Using a similarity transformation, I end up with the following second-order linear ODE. x^2y''+(2x^2+1)y'-ky=0. where k is a positive constant. I am unsure as to how do I proceed from this point to obtain a general solution. Any pointers? I'm unsure if I can apply Abel's theorem to this problem?
    Amaechi Anyaegbunam · University of Nigeria
    Dear colleague and Mummin Mohammed Hussein do you know that the Homotopy perturbation method can fail to give a correct solution when applied to ODEs. See the article by Francisco M. Fernandez arXiv:0808.2078
  • Muhammad Ahsan added an answer:
    What are the new ideas in finite-time control method?
    I’m planning to use finite-time control method in a nonlinear system. I read about finite-time control method and would like some advice about recommended techniques.
    Muhammad Ahsan · National University of Science and Technology
    Here is a good book on finite time control link.springer.com/chapter/10.1007%2F0-8176-4470-9_11 www.ajc.org.tw/pages/paper/7.2PD/AC0702-P177-VS0202R.PDF
  • Mahmood Dadkhah added an answer:
    Does anyone have any suggestions for a good starting book/course in networked nonlinear systems?
    I need a good (relatively easy) book to start learning about interconnected nonlinear systems. I have taken a nonlinear systems course.
    Mahmood Dadkhah · Payame Noor University
    Hi. See for example ""Nonlinear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models Oliver Nelles"" or ""dentification of Nonlinear Systems Using Neural Networks and Polynomial Models: A Block-Oriented Approach Andrzej Janczak "" Be happy
  • Sajad Jafari added an answer:
    Does anyone have a real chaotic time series obtained from a real chaotic system (flow or map) and a valid model for that real system?
    We know that famous Lorenz system firstly developed as a model for atmospheric convection. Is there any real data which can show that model is acceptable? There are 3 parameters in Lorenz equations which can make great qualitative changes (bifurcation) in the system time series. Is there any real data which can help us to estimate these parameters? In almost all of the researches on parameter estimation of chaotic systems, there are no real data. In those researches both real system and model are exactly the same ODEs (or maps). Their difference is that the parameters are known in ODE (or map) who plays the role of real system, while the parameters supposed to be unknown in the other ODE (or map). I want to do the parameter estimation for a REAL system which has a valid model (since I don’t want to do system identification, I need a valid model which only needs parameter estimation). The only thing came to my mind was chaotic circuits, but I want something different. I am ready to collaborate on this topic if someone can provide me those data and valid model (map or flow, no difference). I think the possibility of finding discrete data which have a model in the form of map is more. Thank you
    Sajad Jafari · Amirkabir University of Technology
    Thank you very much Professor Pearson
  • Mohamed W. Mehrez added an answer:
    How can you solve NMPC objective functions with nonlinear constraint in MATLAB?
    NMPC- Nonlinear model predictive control X_dot=f(x,u) Y=C*x objective function: min J = (Y-Ys)^2+du^2+u^2 w.r.t u constraints are : 0<u<10 -0.2<du<0.2 0<Y<8 here Y is a nonlinear constraint and a vector
    Mohamed Mehrez · Memorial University of Newfoundland
    Fmincon() matlab function should allow you to do this.
  • Rohnn Sanderson added an answer:
    Can anyone help with the Lyapunov exponent from a fractional system?
    I look for a direct method for computing Lyapunov exponent from a fractional system and its code (without extracting its time series). Currently, I extract its time series and I compute the Lyapunov exponent from time series. But, because of computational and numerical errors I know that this approach is not the best.
    Rohnn Sanderson · Brescia University
    If you can do a plot, you could calculate the fractal dimension to look for the properties of the system using ImageJ and Fraclac. http://rsbweb.nih.gov/ij/plugins/fraclac/fraclac.html
  • Alfonso Bueno-Orovio added an answer:
    Does anyone have a valid MATLAB code for computing Lyapunov exponent from a time series?
    I have some time series which their models are not known. So, I try to compute Lyapunov exponent from them. I have used Rosenstein's (1993) algorithm, but I think that the results do not confirm time series. Hence, I look for a valid code to be sure that my results are true or false.
    Alfonso Bueno-Orovio · University of Oxford
    Dear Javid, I think the TISEAN package is well tested, so you could analyse your data with it and see whether their results confirm your time series. That could work as a validation of your own implementation of Resenstein's algorithm.
  • Santanu Biswas added an answer:
    Any ideas about the stability of two solutions in a nonlinear system?
    Suppose a non linear system has at least 2 positive periodic solutions in a bounded domain. If one can show that the system is globally asymptotically stable, then is it a contradiction to the previous statement? If not, what can one say about the other solution? Can anybody suggest me any relevant references?
    Santanu Biswas · Indian Statistical Institute
    Thank you sir
  • Viswanath Devan added an answer:
    How can tangent,curvature and torsion be related to the differential geometric concepts of topology, manifolds, etc?
    For example, Frenet–Serret formulas are considered as the generalization of higher dimensional Euclidean spaces. These formulas define a non-inertial coordinate system in terms of tangent,curvature and torsion. However understanding these quantities by relating them to the topological spaces, manifolds, etc is required for better visualisation of non linearities.
    Viswanath Devan · Indian Institute of Technology Guwahati
    @Rogier Brussee & Prof.James Peters: Thank you very much sirs.. @Prof.James: Sir I thanks for the copy of Theodore Shifrin..I find it to be a nice book and the way 'parametrization' has been explained in a simple way makes the book interesting. However what I am perplexed with is that: (a) The book deals with curves and surfaces and now I want to advance the concepts to higher dimensional manifolds, (b) I need a basic understanding of topological spaces the way I can relate to a curve or a surface and (c) Finally it will be really helpful if I can relate to any examples in application form. Thanks n regards.
  • Hassan Sedaghat added an answer:
    What are the sufficient conditions for the existence of chaos behavior in a linear piecewise autonomous system?
    .
    Hassan Sedaghat · Virginia Commonwealth University
    In that case, we imagine that each point of the cycle belongs to a different piece of the discontinuous or non-smooth map so the continuous or smooth theory does not apply to the cycle. In my example, the same idea extends to chaotic orbits that are distributed across two or more partitions.
  • Artur Sergyeyev added an answer:
    How can I convert a Partial Differential Equation to Coupled Map Lattice?
    In a spatially extended system, two important models for describing the nonlinear dynamics are PDE and CML. Is there a method for converting PDE to CML?
    Artur Sergyeyev · Silesian University in Opava
    The process is called discretization. For starters, see http://en.wikipedia.org/wiki/Discretization
  • Andrey L. Shilnikov added an answer:
    Is there any 2D system with a limit cycle and without any equilibrium?
    Consider a general 2D system: x' = f(x,y) y’ = g(x,y) Do you know any such system which has a limit cycle while there is no equilibrium in it? For example the famous Van Der Pol is not an answer; Because, although it has a limit cycle, it has an equilibrium in the origin (0,0) x' = y y’ = ay(1-x^2)-x
  • Viswanath Devan added an answer:
    Does state convergence necessarily imply stability?
    Stabilty
    Viswanath Devan · Indian Institute of Technology Guwahati
    Conceptually, the meaning of various types of stability (wikipedia) are : Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a distance \delta from it) remain "close enough" forever (within a distance \epsilon from it). Note that this must be true for any \epsilon that one may want to choose. Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium. Exponential stability means that solutions not only converge to the equilibrium, but in fact converge faster than or at least as fast as a particular known rate. So its not just enough that the states converge. They need to converge to the equilibrium (asymptotic or exponential stability) or atleast within a distance /epsilon from its (stability in Lyapunov sense). That justifies "State convergence does not necessarily imply Stability" since the states may converge to a point which is attractive but not anywhere close to the equilibrium to call it stable in the Lyapunov sense.

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