Nonlinear Systems

Nonlinear Systems

  • Matteo Brilli added an answer:
    Is complex network truly nonlinear?
    Personally, I take complex network (nodes can be dynamical systems) as a nonlinear system. But seems the work in this field are using linear methods extensively. From community structure detection algorithm to master stability method, they can all be boiled down to linear algebra (spectrum theory, more specifically).
    So, is complex network a nonlinear system, or just a complex linear system?
    Matteo Brilli

    What does it mean that a complex network is non-linear? What a "complex" network is? In biology, a network is a representation of a usually non-linear system, but I don't think you can say that the network is linear/non-linear without referring to the modeled system (which in case can have such a property). Complex systems are often non-linear, and networks can be their representation. In this case the system (not the network for how I read it) is non-linear. But a network can also be a representation of a linear system. Suppose to use a graph to model the water pipes of a city. Then the network will be linear? Paradoxically, the exact same graph (its structure) could represent a metabolic (non-linear) system. So in my opinion there is not such a property for networks or graph. The linearity or non-linearity is a property of the underlying system not of the mathematical object itself. Correct me of I am wrong, but I never heard of a non-linear network.

    This said, non-linearities in a system bring a lot of problems for analysis so the system is often linearised. Moreover, when dealing with experimental data to estimate the parameters of the model, it is common not to have enough observations for estimating the parameters of the full non-linear system and approximations (linearizations) has to be made. In general you take the linearization around some steady state and you study the system in a  close range, when you diverge too much from the point where you linearize the system the approximation starts to degrade. 

  • Sarah Knox added an answer:
    Why do ordinary differential equation (ODE) models of cancer suggest different behaviors for cancer cells?

    For validation part of my study, I need a comparison between my model of ductal carcinoma in situ (DCIS) and ODE models of this area. But, I’m really confused because ordinary differential equation (ODE) models of cancer have suggested different behaviors for tumor and immune cell populations. For example, the below behaviors are reported by the survey of Eftimie et al. (2011) [1]:

    • Tumor size decreases exponentially through interactions with the immune cells.
    • Tumor size decreases at first. Then, the decay of immune cells leads to an exponentially increase in it again.
    • Tumor size decays in an oscillatory manner.
    • Tumor size grows in an oscillatory manner.

    I don’t understand the reason of the difference! And, I don’t know which behavior is right. Could anyone possibly help me, please?


    [1] A Validated Mathematical Model of Tumor Growth Including Tumor–Host Interaction, Cell-Mediated Immune Response and Chemotherapy

    Sarah Knox

    I agree with Elman.  Most systems in the body are nonlinear and many (e.g. macrophages) function on the edge of criticality between order and chaos.  It is the most effective way to both be responsive to multiple needs (stimuli), yet robust enough not to be perturbed by everything that comes along.  Since the possibilities of this type of system, which functions far from thermodynamic equilibrium, also allow for bifurcations, it is not strange that you are getting the kinds of results that you are getting

  • Newton Pimentel de Ulhôa Barbosa added an answer:
    For a mixed nonlinear distribution, what is the efficient sampling method?

    We have a distribution function which is neither discrete nor continuous, but both.  The continuous portion is nonlinear as well.

    What would be an efficient method of sampling from such function, table lookup, rejection methods or ....?

    Newton Pimentel de Ulhôa Barbosa

    What kind of research is it?

  • Alejandro Fernández Villaverde added an answer:
    What is the easiest way to determine if a parameter is identifiable in a nonlinear model?

    Imagine we have a nonlinear model of the following form:

    Xdot = F(x,p,u)

    Y= h(x,p,u)

    where x is the system states, p are system parameters, u is the system input, Y is the system output and f(.) and h(.) are nonlinear mapping process and measurement functions.

    Can anyone help me with what would be the easiest way to figure out if a parameter in p set is identifiable from the outputs of the system being measured. I am looking for the fastest and easiest way possible with possible proof available.

    Alejandro Fernández Villaverde

    The papers mentioned in other replies cover a good part of the most relevant literature for solving this problem. From a structural identifiability point of view, it is worth mentioning also the COMBOS method by DiStefano's lab, which included a web implementation:  

    And maybe also the GenSSI toolbox:

    Note that, although these methods were developed for biological applications, they may be applied to nonlinear dynamic systems in general.

  • Abdalla Ahmad Tallafha added an answer:
    Is there any complete non-metrizable semi-linear uniform space?

    See the attached file.

    Abdalla Ahmad Tallafha

    Thank you Dr. Suad. Yes you are write. we should take unionwith delta.

  • Temple H. Fay added an answer:
    How can I determine whether a given point (in the phase space) lies in the basin of attraction of an attractor?
    1. For a nonlinear ODE systems, dx/dt=f(x), if one trajectory gets as close as possible to an attractor, i.e., |x(T)-x*|<eps (where x(T) is the value of the solution x(t) at the terminal time T for a given initial value x0, x* is the state vector of an attractor, and eps is a very small number), does it imply that x0  lies in the basin of attraction of an attractor?
    2. Although the definition of basin of attraction is as follows. For each such attractor, its basin of attraction is the set of initial conditions leading to long-time behavior that approaches that attractor. However, to my knowledge, nonlinear systems often display different types of attractors, such as fixed points, periodic attractors, and strange attractors. Except for the basin boundary with a smooth curve, there exist some fractal basin boundaries and riddled basins of attraction. As I understand them, for these basins with complex boundaries, if |x(T)-x*|<eps, the point x0 may not lie in the basin of attraction of that attractor. Is it true?
    3. Are there some theories or conditions that can guarantee one point definitely lies in the basin of attraction for any different type of attractors? 
    4. Thanks for your attentions and I sincerely appreciate your comments.
    Temple H. Fay

    I have only considered what might be called elementary systems and strange attractors have not arisen. I use the separatrices to determine the basin of attraction. If the system has several critical values, often there are separatrices which are determined essentially by the saddle points. These separatrices divide the phase plane into distinct regions of behavior of the system. In this sense, they become the boundary of the basin of attraction for the appropriate critical point. If this comment is to elementary or off the mark, please excuse it.

  • Roman Sznajder added an answer:
    What are good books for control of linear and nonlinear systems?

    Dear all,

                  I would like to study regarding control of linear and nonlinear systems in detail. So, please suggest me some books which can provide in-depth knowledge regarding it. Thank you.

    Roman Sznajder

    Here is a valuable edition to the collection already presented by our colleagues:

    S. Boyd, L. Ghaoui, E. Feron, and V. Balakishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, ISBN 0-89871-334-X (1994).

    There may exist an updated edition, book displays a very rich bibliography. The book shows strong connections of the subject matter with optimization, dynamical systems as well as the classical themes (Lyapunov theory).  

  • Behrouz Ahmadi-Nedushan 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?
  • Sikandar Khan added an answer:
    Can anyone help with a convergence and initial condition problem (non linear solver)?

    Actually I have a problem is running the numerical simulation.The solution did not converges and at the end give an error that the initial conditions are inconsistent. It will be highly appreciated if someone help me in this regard. I have spend weeks but didn't solved this error. Attached is the COMSOL file.

    Best Regards

    Sikandar Khan

    Sikandar Khan

    Thanks all for your kind replies. It help me a lot in solving my problems.



  • Mohammed Sabah Hussein 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).
    Mohammed Sabah Hussein

    Dear all thank you so much for all the answers that really great feedback, and I will take a huge benefit.    

  • Mariusz Buciakowski added an answer:
    How can I estimate the uncertainty of parameters?

    I design robust controller and robust observer for nonlinear system using
    norm bounded parameters uncertainty. I want to verify how controller and
    observer works. I have nominal system in typical state-space form:
    x(k+1) = Ax(k)+Bu(k) + g(x)
    y(k) = Cx(k);
    Is It possible to estimate the uncertainty of parameters (find matrix H, E1 and E2) for system in form
    x(k+1) = (A+HFE1) (kB+HFE2)u(k) + g(x)
    y(k) = Cx(k);
    base of nominal model and real measurements data form real system?


    Mariusz Buciakowski

    Thank you for your helpful answer.

  • Benar F. Svaiter added an answer:
    Does the conjugacy condition of linear conjugate gradient methods still holds for nonlinear conjugate gradient methods?

    For linear CG method the search directions must satisfy the conjugacy condition

    d_i'Hd_j=0, i not equal to j.

    It is a fact that for a nonlinear CG method the above equation does not hold, since the Hessian changes at different iterations.

    With this in mind, I am asking whether there is any modified conjugacy condition for the directions of nonlinear CG methods. If there is no such condition, then why are we using  the word  "conjugate" for nonlinear CG  methods.

    Benar F. Svaiter

    No, up to now there is no method with modified conjugacy condition.

    The name comes from the fact that the iterations of these methods (linear vs non-linear) are formally identical. In each iteration, the new search direction of the non-linear method is conjugate to the old search direction.

  • Sushant Kulkarni added an answer:
    How can I calculate the Lyapunov exponent?

    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?

    Sushant Kulkarni

    you can follow the link below

  • Sajad Jafari 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
    Sajad Jafari

    Dear Dr. Taghavi,

    Please see the attached paper.



  • Ali Taghavi added an answer:
    Is there any 2D system (flow) with more than one limit cycle?
    Consider a general 2D system:
    x' = f(x,y)
    y’ = g(x,y)
    Do you know any such system (preferably a simple one, ideally quadratic) which has more than one limit cycle? I would prefer it if there was at most one unstable equilibrium.
    Ali Taghavi

    Dear Sajad

    There  is  a  quadratic system with 4  Limit cycles  please  see

    The  method of construction of a quadratic system with 4 limit cycles is   based on the following:

    They pick a quadratic sytem with 2 hyperbolic limit cycles with disjoint interior. Existence of these 2 limit cycles  is  a consequence of Poincare-Bendixson theorem. One of thses 2 limit cycles suround a weak focus of order 2. So after perturbation we obtain two  additional  (smal) limit cycles. So we have  4 limit cycles with (3,1) distribution

    It is unknown whether there is  a quadratic  system with more than 4 limit  cycle. It is also unknown that whether there is a (2,2) distribution of limit cycles for quadratic systems.

  • Adnène Arbi added an answer:
    How can I evaluate a bifurcation diagram?

    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 ?

    Adnène Arbi

    Thanks for all giving valuable answers and suggestions 

  • Kirill Poletkin added an answer:
    How might I determine a Lyapunov Function and the corresponding region of attraction for a nonlinear system??

    When designing a Lyapunov-type controller for a nonlinear system, how should the region of attraction be determined?

    Is there any general approach to determine the Lyapunov function with respect to system dynamics?

    I was wondering if someone could help me.

    Kirill Poletkin

    There is no  generally accepted approach. However some methods  exist, you can find them in D.R. Merkin, Introduction to the Theory of Stability, Springer-Verlag, 1997. N.G. Chetayev, The Stability of Motion, Pergamon Press, 1961.

  • Suresh Kumarasamy added an answer:
    Does a chimera exist only in a network of coupled nonlinear systems and not in single nonlinear system?

    When I see the literature about the chimera state, it exists only in the network of coupled nonlinear oscillator. They define the chimera such as the existence of coherent state and in-coherent state simultaneously. My question is that, if an attractor of a system has both the coherent and incoherent state, should we call it as chimera state? Like some portion of the attractor is chaotic and rest becomes regular. 

    Suresh Kumarasamy

    Dear All, This is (see figure) what I  was talking about. You can see some portion of the attractor have regular structure and remaining of the become chaotic. 

  • Mudambi Ananthasayanam added an answer:
    What are the state of art of the estimation methods?

    I know that the title of question covers a wide area. Currently I am trying to apply Kalman Filter to estimate some parameters of a highly nonlinear system. I took a look at the papers and articles, but I could not see any obvious advances.

    If anyone can recommend me a book or a recent review article which includes and explains the state of art estimation methods, I will be glad.

    Thanks in advance.

    Mudambi Ananthasayanam

    Please see the report arXiv:1503.04313v1.pdf

  • Naresh Dixit P S added an answer:
    Which tool you prefer for non linear analysis and why?

    I just want to know which tool is used more for the fea non linear simulations. And the reason behind why we use that tool.

    Naresh Dixit P S

    I would like to suggest ANSYS 14.0 Mechanical APDL or ETABS for non linear simulations are both are robust and have wide range of options. You can get all kind of results required through it.

    + 1 more attachment

  • Nahed Mohamady added an answer:
    How can I determinate the stability region of nonlinear system of first order delay differential equations using matlab?

    I have a nonlinear system of first order delay differential equations and I'm studying its stability using a quadratic Lyapunov function and I got that

    dV[x(t)]/dt= 2x2(t){(1/t)x1(t-1)x1(t)+(3/t)x2(t-3)-3x1 ^2(t)}

    I want to determinate the stability region which means dV[x(t)]/dt<0, using matlab. can any one help me in this problem?

    Nahed Mohamady

    Thank you very much for your clarifying answer prof. Marc, I will look for these references and see how to study the stability of my system.

  • Salman Mo added an answer:
    Does anybody here have any idea how we can make an uncontrollable state dependent coefficient matrix of a nonlinear system controllable ?(SDRE)

    In SDRE survey we must first make nonlinear system into a linearized shape that matrix  A & B are state dependent matrices. The question that arises here is that I am manipulating a system in which there are lot's of zeros in matrices A & B and these zeros coerce system to violate Controllability condition. 

    My nonlinear system is :

    Xdot = V.cos(theta)

    Ydot = V.sin(theta )


    Vdot= a


    inputs are a and omega

    outputs are x,y

    my SDC form is :


    [0 0 0 cos(theta) 0]

    [0 0 0 sin(theta) 0]

    [0 0 0 (1/2)tan(phi) 0]

    [0 0 0  0  0]

    [0 0 0 0 0]

    B=[0 0;0 0; 0 0;1 0;0 1]

    Salman Mo

    thank you Daniel .this is obvious that an uncontrollable System cannot be controlled. but in the context of SDRE approach , we build a linearized form of nonlinear system while this linearized shape is not unique and by algebric manipulation every body can propose different kind of linearized form. for a multivaribale non-linear system it can be proved that there are infinite number of state dependent matrix coefficient .

    fortunately right now i could reach another form for this system which is much more better in sense of controllabilitiy . anyway thank you.

    live long and prosper .

  • Prasanna kumar Routray added an answer:
    How could I make a MATLAB code to design a PID controller?

    I'm trying to design a PID controller to control a nonlinear system, I finished all the dynamics and error (e) equations, P, I, D; but I couldn't find some useful tips, to start writing my code.

    One could found some ready code from internet, but I want to understand how one could be able to make such things; since I'm a bit new to this area!!

    Any one can suggest some tutorials, or informative resources?

    Prasanna kumar Routray

    can anyone please provide a sample that can control any physical machine like dc motor or pressure control system . not using commands but using codes. so that I can implement P,I,D,Pi,PD,PID using sampling time in matlab script and control the system in real-time. I know how to simulate in sisotool, simulink and command. but I want to control a system that has some dedicated commands and with writing some code for any controller . the dedicated functions given can be used to get current state of parameters and to start and stop the machine.

  • Javier Buceta added an answer:
    Can someone suggest how to study local stability of the nonhomogenous stationary solution of a plannar reaction diffusion system ?

    I try to linearize my plannar nonlinear system at the nonhomogeneous spatially dependent equilibrium, but the resolvent operator do not allow more explicit treatment. So I am looking for an alternative method. I would like to add that I have homogeneous Dirichlet boundary conditions and my equilibrium is spatially dependent. I do not not have explicit formula for that equilibrium but I have shown that it exists.

    Javier Buceta

    I am working in my group developing some theoretical work to address that question. As far as I know the only thing you can do is applying the Amplitude Equations formalism (may be that is enough in your case).

  • Michal Málek added an answer:
    For a DC3 dynamical system (X, f), is there an uncountable subset S of X such that any two different points of S are a DC3 pair?

    Please answer the question!

    Michal Málek

    Yes for graph maps one DC3 pair ensures existence uncountable set any two of them form a DC1 and therefore DC3 pair. (

    No for square maps. There is a paper of Forti, Paganoni and Smítal with example of triangular map with a DC1 and therefore DC3 pair and there is not other DC1,2 or 3 pair.

    No for sub shift. It is easy to construct sub shift with only one DC3 pair. This can be used in other spaces.

    • Source
      [Show abstract] [Hide abstract]
      ABSTRACT: In their famous paper from 1994, B. Schwaizer and J. Smítal, [B. Schwaizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994), pp. 737–754] fully characterized topological entropy of interval maps in terms of distribution functions of distance between trajectories. Strictly speaking, they proved that a continuous map f : [0, 1] → [0, 1] has zero topological entropy if and only if for every x, y[0, 1] the following limit exists: for every real number t except at most countable set. While many partial efforts have been made in previous years, still there is no proof that the result of Schwaizer and Smítal holds on every topological graph. Here we offer the proof of this fact, filling a gap existing in the literature of the topic.
      Dynamical Systems 09/2011; 26(3-3):273-285. DOI:10.1080/14689367.2011.588199
  • Mohammed Sadeq Al-Rawi added an answer:
    Do there exist any algorithms to transform an artificial neural network into its open equation form or its equivalent?

    I am asking the community here to get a brief overview if there exist a method or algorithm, that enables me to transform a traditional perceptron based feed forward artificial neural network into its open equation form?

    The closed for of an equation for a linear system for example is

    y =  kx + d and its open form is  R = y - kx  - d

    Any idea or hint is appreciated.

    Kind regards and thank you in advance.

  • Hussien Shafei added an answer:
    How can I construct the system of equations?

    How to construct the system of equations by susbtituting the assumed series solution in the nonlinear partial differential equation? Kindly find the attachment.  

    Hussien Shafei

    Dear Saravanan

    I read carefully your problem.

    First at all, your problem seems to be so easy, if you want to solve it Numerically. i.e. if all constants are given and the function fi(zeta) is explicit. then you can catch the values of the unknown values for a0, a1, d and b.

    Furthermore, If you want an explicit form for a1, you can let it to be a polynomial(as example) of suitable degree with unknown constants to be determined.

    Is my understanding for your problem is suitable?

  • Taher Lotfi added an answer:
    Is there any similar result as Traub's conjecture for multipoint methods?

    Traub's conjecture says that an optimal method with n-steps has order 2^n .

    Taher Lotfi

    Dear Alberto,


    First, we note that it is due to Kung (Truab's PhD student in that time) and Traub's conjecture. Moreover, it is relevant to construction optimal multipoint  iterative without memory. And, it has been proved for optimal two point without memory by Traub, I suppose. As Prof. Neta pointed out, in a special case,  ostrowski type methods, it has been established. However, for Jarrat type and Stefenssen type methods, I do not think where it has been proved. 

  • Tawfiqur Rahman added an answer:
    Solving nonlinear system with linear equality constraints which are dependent but not consistent?

    I have a nonlinear system which has a set of linear equality constraints where those are dependent but not consistent. Which solver can be used for it?

    Tawfiqur Rahman

    I have a non linear system. This system can be expressed in terms of 18 algebraic equations. But there are in total 12 variables which are constrained within specified limits.Because of these limits all these 12 variables can infact be represented through 8 variables. Out of these 8, 6 are given (fixed) and the rest two define the other 6 (out of the 12 variables). So finally I have 18 equations but only 2 variables. I suppose I should try some optimization method to find the 2 unknown variables which satisfy the 18 equations as closely as possible. But I was hoping to find some other method so that I don't have to worry about the convergence issues.

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