Nonlinear Systems

Nonlinear Systems

  • Marc R Roussel 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?

  • 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 · Isfahan University of Technology

    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 · University of Aveiro

    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 · Lehigh University

    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 · Silesian University in Opava

    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.

  • Michael Patriksson 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.

    Michael Patriksson · Chalmers University of Technology

    For quadratic problems the two algorithms can be shown to be the same, when certain strategies are used. For non-quadratic problems this no longer is the case: and as the Hessian is not constant, conjugacy in CG methods becomes lost after a series of iterations, so the method needs to be re-started regularly. The remedy in the non-quadratic case is often to actually use quasi-Newton methods instead, as they are more stable. 

  • 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.

    Mohammed Sadeq Al-Rawi · University of Coimbra

    Dear Wolframe, 

    As noted above, ANNs have linear transfer/activation functions. Your question reminded me about my paper that addressed equivalence issues among different neural network models, e.g. high order neural networks and ANN (first order networks). Unfortunately, the reviewers rejected my paper. Although my work might seem irrelevant to your question, here's the abstract:

    Equivalence studies are necessary to obtain abstracts of computational systems and thus enable better understanding of issues related their design and performance. For more than three decades, artificial neural networks have been used in many scientific applications to solve classification problems as well as other problems. Since the time of their introduction, multilayer feedforward neural network referred as Ordinary Neural Network (ONN) or ANN, that contains only summation (Sigma) activation  neurons, and multilayer feedforward High-order Neural Network (HONN), that contains Sigma neurons, and product (Pi) activation neurons, have been treated in the literature as different entities. In this work, we studied whether HONNs are mathematically equivalent to ONNs. We proved that every HONN could be converted to some equivalent ONN. In most cases, one just needs to modify the neuronal transfer function of the Pi neuron to convert it to a Sigma neuron. The theorems that we have derived clearly show that the original HONN and its corresponding equivalent ONN could give exactly the same output. We also derived equivalence theorems for several other non-standard neural networks, for example, recurrent HONNs and HONNs with translated multiplicative neurons. This work rejects the hypothesis that HONNs and ONNs are different entities.



  • 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 · Faculty of science, SVU

    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 · Islamic Azad University

    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 · Bangladesh Air Force

    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.

  • Mohamed El Naschie added an answer:
    Is it true that no weakly mixing transformation can exhibit coherent recurrence?

          One of the central themes in ergodic theory and dynamical systems is that of recurrence, which is a circle of results concerning how points in measurable dynamical systems return close to themselves under iteration. There are several types of recurrent behavior (exact recurrence, Poincaré recurrence, coherent recurrence , strictly coherent recurrence) for some classes of measurability-preserving discrete time dynamical systems. The second of these, which is the type originally introduced by Jules Henri Poincaré (published in 1890) and is still by far the most widely discussed, especially in physics, holds for all weakly mixing transformations (and so for all classes of weakly/strongly mixing dynamical systems with discrete time) by virtue of the fact that they are measure-preserving. Poincaré had shown that almost all points in a space subject to a measure-preserving transformation return over and over again to positions arbitrarily close to their original position. However, P. Johnson  and A. Sklar in  [J. Math. Anal. Appl. 54 (1976), no. 3, 752-771] regard the third type („ coherent recurrence” for measurability-preserving transformations) as being of at least equal physical significance, and this type of recurrence fails for Čebyšev polynomials. They also found that there is considerable evidence  to support a conjecture that no (strongly) mixing transformation can exhibit coherent recurrence. (This conjecture has been proved by R. E. Rice in [Aequationes Math. 17 (1978), no. 1, 104-108].)

    Mohamed El Naschie · Alexandria University Alexanderia Egypt


  • What is a Kawczynski-Strizhak chemical chaotic oscillator?

    Eminent friends at RG:

    I recently heard about Kawczynski-Strizhak chemical chaotic oscillator, but I could not get the equations describing the motion of this chemical chaotic system. It seems to be a two-dimensional chaotic oscillator. If people have worked on this chemical system, kindly give me this chemical model with the equations and the parameter values for which this system is chaotic. Phase portrait will be highly grateful!!

    Thanks a lot - kindly help me!

    With best wishes, Sundar

    Sundarapandian Vaidyanathan · Vel Tech - Technical University

    Dear Prof. Miguel A F Sanjuán:

    Thank you for providing the equations of Strizhak-Kawczynski model.

    With best wishes, Sundar

  • Bin Jiang added an answer:
    How to check whether a system is Linear or Nonlinear?
    System is a black box.
    Bin Jiang · Högskolan i Gävle

    Very stimulating question!

    I would visualize it in order to see whether it is nonlinear or not, e.g., urban growth and evolution of social media.

    Jiang B. (2015), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77, Preprint:

    Jiang B. and Miao Y. (2014), The evolution of natural cities from the perspective of location-based social media, The Professional Geographer, xx(xx), xx-xx, DOI: 10.1080/00330124.2014.968886,  Preprint:

  • Bin Jiang 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?
    Bin Jiang · Högskolan i Gävle

    Complex networks are indeed nonlinear, and we therefore must adopt nonlinear thinking and methods to develop new insights into complex networks.

    Jiang B., Duan Y., Lu F., Yang T. and Zhao J. (2014), Topological structure of urban street networks from the perspective of degree correlations, Environment and Planning B: Planning and Design, 41(5), 813-828.

    Jiang B. and Ma D. (2015), Defining least community as a homogeneous group in complex networks, Physica A: Statistical Mechanics and its Applications, 428, 154-160.

  • Diego Fasoli added an answer:
    Are you aware of examples of macroscopic differences between mean-field approximation of a system and its real (finite-size) behavior?

    For example, in Cessac 1995 "Increase in complexity in random neural networks" he proved that in the mean-field approximation the transition to chaos is very sharp, while in the real network it develops through the emergence of a limit cycle and a torus. Do you know other examples? I would like to know if there are cases when the mean-field approximation completely neglects important dynamical phenomena.Thanks in advance for any help you may provide.

    Diego Fasoli · Italian Institute of Technology (IIT)

    @Stam: thanks for your answers. I'm looking at the links you posted, it seems there is really a lot of information on the topic. Especially interesting is the part on neural networks, for what I'm currently working on.

    @ Steven: no worries, I also forgot everything about the Ising model!

    Have a nice Easter!

  • Srikant Sukumar 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.

    Srikant Sukumar · Indian Institute of Technology Bombay

    This is similar to the standard observability query for nonlinear systems. Evaluation of Lie Derivatives is a sure shot way to conclude local observability and also identifiability of parameters. In general the approach is to augment the state with the parameters, setting their derivatives to 0 (since parameters are assumed constant) and then evaluating subsequent Lie derivatives. An online search with keywords "observability, nonlinear systems" will yield several results.

  • Rodolfo Reyes-Báez added an answer:
    What are incremental stability, differential dissipativity (passivity) and the relation between them?
    Definitions and examples of physical systems which are incrementally stables and differentially dissipative (passive). Also I would like to know if there are control design methods based on differential dissipativity
    Rodolfo Reyes-Báez · University of Groningen

    Thank you again prof. Ian, Now I am working more in deep in this topic, and your previous answer has helped me.

  • Antonio Prados added an answer:
    What is the range of validity of stretched variables of the form X=(a^n) (x-vt) and T=(a^3n) t ?

    To obtain some nonlinear equation from physical models we use the stretched variables in a very limited approximation.
    For example, to obtain KdV equation which is a nonlinear equation with weak dispesrions and weak nonlinearities. We use the transformation X and T with n=1/2.

    I want to know more about this stretched variables and their domain of validity. 

    Antonio Prados · Universidad de Sevilla

    I basically agree with the answer above: you have to give more details in order to provide a more concrete answer. In some physical problems, $a$ is some parameter which may be small because it is related, for instance, to the system size $L$ that is large, ... In that case, a perturbation expansion of the exact equations may give the KdV equation over some time/length scale, which would give a concrete answer your problem.

  • Mehdi Lima 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.

    Mehdi Lima · Griffith University

    This paper may be useful:

  • Christoph Josef Backi 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
    Christoph Josef Backi · Norwegian University of Science and Technology

    Hi Chitta,

    you can check the software package ACADO, which provides a Matlab interface. This is quite user-friendly and intuitive, in my opinion.

    You can find information here:



  • Alexander Pchelintsev added an answer:
    Is it possible to have a chaotic system with one equilibrium in which the real part of all the eigenvalues are positive?
    I mean chaotic flows. That is possible for chaotic maps.
    Alexander Pchelintsev · Tambov State Technical University

    You can try to construct an example of such a system with a quadratic nonlinearities, so that all solutions have been limited (a system with dissipative). For this, I advise you to read the article of Lorenz, where he proved the limitations of solutions. But if all eigenvalues of the real part is positive, then the equilibrium will repel trajectory. These trajectories may move, for example, to limit cycle, but there is not a chaotic system.

  • Graham W Pulford 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.

    Graham W Pulford · Thales Group

    Hello again. Now that you mention your problem area, it seems that you are dealing with a system that is what I would call multi-modal. I imagine that wheel–rail contact includes different dynamics when the wheel is rolling or sliding or a bit of both. For this type of estimation problem, where it is desired to estimate the mode of the system, as well as perhaps its state, there is quite a large literature under the heading of hybrid systems. Different approaches, mostly Kalman filter based, include multiple model, GPB (generalised pseudo-Bayesian), and interacting multiple model (IMM) algorithms. As the problem of manoeuvring target tracking is a hybrid system estimation problem, these approaches have been developed quite a lot in this context. You can find some references in the survey paper below. I am sure that more has been done since then, but it is somewhere to start. Just thinking about the nonlinearity aspect - it may be that it is more important to model the modal transitions of your system than the fact that in each mode it may be nonlinear. In any case, the latter nonlinearities can be dealt with by using a mode-conditioned EKF. The IMM in particular requires you to specify the transition probabilities for the modes of the system - you can just make these uniform if you don't know them, or use average sojourn time information if you have it. BTW there should be no need to use fuzzy logic to solve a hybrid system estimation problem.

  • Vijay Bhaskar Semwal added an answer:
    Is there any formal definition of superquadratic convergence?

    The formal definitions of linear, superlinear  and quadratic convergence order are clear in the context of the convergence of the solution of nonlinear systems of equations. Recently, I have come across some works on superquadratic convergence, but I can't find the formal definition of the convergence order (superquadratic) just like we have in the rest of the existing convergence order.

    NB: I need a formal definition.  Definition like the ones given in this text, see the link below:

  • Cahit Köme added an answer:
    How to solve a highly nonlinear system of equations numerically with high efficiency? Is there a recommendation on any software package?

    I am trying to determine the equilibrium points in the astrodynamics system, but the equilibrium condition is a highly nonlinear system of equations. I have tried the 'fsolve' in Matlab, but it is very sensitive to the initial guess of the solution and is lack of robustness. So I am wondering whether there is any better solver in Matlab or any other software package.

    Cahit Köme · Nevşehir Hacı Bektaş Veli University

    Dear @Daniel_Bachrathy, have you ever use your nonlinear solver on the solution of differential equations with an implicit method?. I'm just wondering that is it really better than newton-raphson method while solving nonlinear algebraic system of equtions appearing in implicit methods?

  • Hugh Lachlan Kennedy added an answer:
    Does anyone have experience with an extended kalman filter without process noise?

    I am currently working on an extended kalman filter for a nonlinear system.

    And I am very confused.

    Assume there is a nonlinear system without process noise but with measurement noise. Then when using EKF to estimate the state of this system, how can I handle Q in EKF.

    For example:

    x(k+1)= x(k) ; or x(k+1)= x(k) + u(k)

    and observer equation
    y(k)=x²(k) + v(k)

    u(k) is the input and v(k) is the measurement noise.

    When EKF is used for the estimate of the state, should I set Q=0? 

    Hugh Lachlan Kennedy · University of South Australia

    Using a small Q matrix  (for nice smooth estimates)  is OK in simulations where your filter model (ideally) matches the process model; though you may need some process noise to help recover from linearization errors (in an EKF). In practice though, using plenty of process noise is a good idea to prevent divergence over time, because your filter models/assumptions are likely to be a poor approximation of the actual measurement and process dynamics.

  • Stam Nicolis added an answer:
    Can one identify complex Four Wave Mixing (FWM) parameters experimentally?

    The governing equation of the Manakov model with complex FWM terms is of the following form

    1 q1_t + q1_xx  + 2 (a|q1| + c |q2| + b q1q2^* + d q2 q1^*) q1 =0

    1 q2_t + q2_xx + 2 (a|q1| + c |q2| + b q1q2^* + d q2 q1^*) q2 =0

    Stam Nicolis · University of Tours

    The immediate answer is that it's impossible to answer the question, as posed, since the experimental procedure is not defined. If one asks a related question, however, what are the parameters, whose values cannot be rescaled arbitrarily, then dimensional analysis is useful tool and can lead to an answer, that is useful for numerical work, for example and experiment, also.

  • Saeed Ansari added an answer:
    How can I extract Abaqus data (U or V) from a selected result file (.sel) in Abaqus?

    How can I extract Abaqus data (U or V) from selected result file (.sel) in Abaqus? I would like to analyze these data in every time increment when analysis is running in Abaqus/Explicit.

  • Wenjun Shi added an answer:
    How to check this formula S(r,r0;t)=S(r,r0_bar:t)-(r0-r0_bar)p_s, here p_s can be seen as the initial momentum of S(r,r0_bar:t) with Symplectic maps?

    If we consider a classical action from r0 to r in terms of t(time), then for the semiclassical calculation, I can consider a small wavepacket and make an expansion around r0_bar.
    Based on classical mechanics, we can have this formula S(r,r0;t)=S(r,r0_bar:t)-(r0-r0_bar)p_s, here p_s can be seen as the initial momentum of the orbit of S(r,r0_bar:t), this is
    indeed out of any problem. But I want to check this formula with standard map or kicked top, but I find I can not make sense in this model. As the discrete characteristics, action difference is large as the two orbits should go to the same final position r in the same time. But this formula should fit for the Symplectic maps, so what is wrong with my idea?
    I am working on this seemingly very easy job for quite a few days, but now it seems hopeless to find the reason. Maybe I have a lack of some knowledge here. So I ask for your help and very glad to hear some good suggestions from you.  I give the related paper for the background with the formula(15) and (24).

    Wenjun Shi · 宁德师范学院学报

    Dear Manuel Morales, maybe you can check this formula with your idea, if you have some interest, ok? such as using the very simple model-kicked rotator. I am glad to hear your good news and I am very pleased to see that result. Best!

  • Pedro Martín Merino added an answer:
    Does the general proof of convergence for Quasi-Newton's matrix exist?

    I am wondering whether their is any general proof of the convergence of the Broyden's update matrix. Broyden (1965) discovered a new method for solving nonlinear systems of equations by replacing the Jacobian matrix in Newton's method with an approximate matrix (Broyden's matrix) that can be updated in each iteration.

    Dennis and More (1974) gives the characterization of the superlinear convergence of quasi-Newton's method based on the convergences of the sequence generated by quasi-Newton's iterates. There is an assumption for the convergence of the approximate matrix, but to the best of my ability I can't find any attempt in showing that the Broyden's matrix converges to the true jacobian at the solution.

    Please I need expert suggestions/answer to my question.

    Pedro Martín Merino · Escuela Politécnica Nacional

    Perhaps you may also want to have a look at these references

    • R.H. Byrd and J. Nocedal, "A tool for the analysis of quasi-Newton methods with application to unconstrained minimization," SIAM Journal on Numerical Analysis 26 (1989) 727-739.
    • R.H. Byrd, D.C. Liu and J. Nocedal, "On the behavior of Broyden's class of quasi-newton methods," Report No. NAM 01, Department of Electrical Engineering and Computer Science, Northwestern University (Evanston, IL, 1990).
  • How can I calculate interevent intervals (IEI) for a non-autonomous nonlinear system?

    This is used to find event rates.

    Gopalakrishnan Jayalalitha · RVS College of Engineering and Technology Dindigul-5

    using Time series Analysis or Trend Analysis for IEI

About Nonlinear Systems

Theoretic and applied results in nonlinear system theory, non-linear models and nonlinear models.

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