Nonlinear Systems

Nonlinear Systems

  • Mirjana Kocaleva added an answer:
    How can help me to draw isocline of nonlinear second order system in matlab?

    I want to use isocline  to draw phase portrait of a nonlinear system..

    Mirjana Kocaleva · Goce Delcev University of Štip

    You can see this link

  • Uche Nnolim added an answer:
    How can we design FPGA-Based sliding mode controller?

    One of the important problem to design and implement Sliding Mode controller is dynamical based FPGA-part, so what's your idea about it?

  • Mohammed Lamine Moussaoui 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?
    Mohammed Lamine Moussaoui · University of Science and Technology Houari Boumediene

    Dear Santanu Biswas, 

    To be Convergent the Solution has to be Stable and Consistent (see lax theorem). The choice of the Discretization Step(s) defines a Domain of Stability this gives you several solutions for each choice. But the solution must be unique as it is stated by Fletcher in his book.

  • Jose Gonzalez de Durana added an answer:
    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.
    Jose Gonzalez de Durana · University of the Basque Country (Spain)

    First of all I would like to know what "stability analysis of multi-agent control systems" means (I do not know any study on it).

  • Ioannis K. Dassios added an answer:
    Solving an overdetermined nonlinear system in Matlab

    Dear All,

    I have the system AF=B with Α 14nx14n F,B 14nx3n (for simplicity lets assume n=1). A, B are given and I need the solution of F.

    Once I have F I need to solve the non-linear system F=K_1X+K_2X^2

    where K_1,K_2 14x3 (are given) and X 3x3 is the unknown.

    Can anyone help me with the code? 

    Ioannis K. Dassios · The University of Edinburgh

    Dear Daniel Gaida and Micha Feigin, Thank You very much for your replies. I tried the proposed method of Daniel Gaida and combined with 'fsolve' I think it worked quiet well. I tried also to use 'fmincon' but didn't have results. I think its because this function minimizes functions subject to linear inequalities. Any other suggestions or comments are welcome!

  • James Lundeen added an answer:
    What is the recent iteration method to solve a nonlinear system of equation?
    Using some iteration method, we can solve a nonlinear system of equation. I know lots of old methods like, Newton Raphson, Gauss newton, marquardt, levenberg marquardt etc. I want to know about some upgraded methods.
    James Lundeen · Illinois State University

    Marquardt Levenberg paper published in 1961. At that time, IBM 360 memory was very limited. CPU speed slow too compared with today. That method should be put in a museum where it belongs. To use it, you must guess your final answers. This leads to non-linear least squares regression settling on local rather than global minima. Finite element analysis assures global minima least squares best fit and does not use initial guesses of anything.

  • Fernando Castanos added an answer:
    What are the methods for checking stability of zero dynamics of a nonlinear system?

    Input-Output feedback linearization method for a nonlinear system requires the check of stability of zero dynamics or internal dynamics.

    Fernando Castanos · Center for Research and Advanced Studies of the National Polytechnic Institute

    Since you ask about the zero-dynamics in particular, I would suggest you first put the complete nonlinear system in normal form (see Isidori's book). This amounts to taking the first r-1 time-derivatives of the inputs, with r being the relative degree of the system. Then find n-r additional functions to complete a diffeomorfism which will serve as coordinate transformation. In the new coordinates, the zero-dynamics are clearly decoupled from the rest. Then you can use any method at hand (Lyapunov, linear approximations...) to asses the stability of the zero-dynamics.

  • Simone Orcioni added an answer:
    How can we define Capabilities of system identification for nonlinear systems?
    What does it mean when a nonlinear system can be identified?
    What terms do we need to identify a nonlinear system?
    Simone Orcioni · Università Politecnica delle Marche

    Dear Faeze,

    I will try to answer to your third question that seems to me the clearest:

    "What terms do we need to identify a nonlinear system?"

    For a large class of nonlinear systems (those can be identified by a Volterra series) the "only" thing you need to identify the system is its answer to a Gaussian input. It exists also some methods that overcome this limitation.

    You can find some material regarding the identification of this class of nonlinear systems in the following papers (the last are presentation slides) and in the references therein:

  • Federico Zertuche added an answer:
    Is there a Hamiltonian model for the perception of music?
    The musical experience appears to contain a gap in our understanding: We have strong physical theory for the mechanics of vibration, musical theory for a range of compositional or improvisational styles, and some understanding of the processes in the brain that are stimulated by music. However, I am not aware of the mechanism by which these processes converge to the interpretation by which "music has charms to soothe a savage breast." I am wondering if the physical processes that we understand have some sort of mental mirror that "resonates" (pardon the pun) within us. Have any of you studied this?
    Federico Zertuche · Universidad Nacional Autónoma de México

    Sorry that's a stupid question

  • Qudrat Khan added an answer:
    How to test the Observability of "Discrete time" Nonlinear system?

    Is it related to the rank of the Jacobian matrix? I know in continuous time this is true, but I am not sure about discrete time. Any references and links are welcome.

    Qudrat Khan · COMSATS Institute of Information Technology

    The reachability and observability is a very famous in discrete time system. You can find your answer in the book "Linear System Theory" by Wilson J. Rugh, Second edition p. 462-472.   Having studied this chapter you will find your answer. 

  • Julien Clinton Sprott 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.
    Julien Clinton Sprott · University of Wisconsin–Madison

    Such systems exist, but they are rare. I have found such an example in a 3-D system of ODEs with quadratic nonlinearities, and a publication describing it is in preparation.

  • Adam Szewczyk added an answer:
    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.
    Adam Szewczyk · McMaster University


    How about FitzHugh-Nagumo oscillator circuit.



    or a simple pendulum will do.

  • What are lyapunov based disturbance rejection approaches for nonlinear systems?

    Just tell me their names to search on the web.

    I have searched but results were so confusing because some of the results were for linear systems.

    Jorge Luis Barahona-Avalos · Universidad Tecnológica de la Mixteca

    Hi Salman:

    Lyapunov's approach to deal with the disturbance rejection in nonlinear systems is not the only way to counteract the adverse effects which could cause such disruption in the stability and / or system performance.

    I recommend that you will revise the work done several years ago by Dr. Gao and his colleagues, whose area of ​​research is related to the active disturbance rejection control (ADRC).

    Either way, I append to this comment a related article with what you are investigating, hoping you find it useful. I'll also add some publications on active disturbance rejection which can be found here on ResearchGate.

    Best regards from Oaxaca, Mexico

  • Nikos Lazarides 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.
    Nikos Lazarides · University of Crete

    Dear Sajad,

    In case you are still interested in a simple example of a 2D flow with more than one limit cycles, you could have a look in the paper in:

    the modified van der Pol oscillator eq. (1) has from one to three limit cycles; in the latter case, two of them are stable and one unstable!



  • Dmitry Gromov added an answer:
    Do you know any results or references for asymptotic stability of linear differential equations with state-dependent delay?
    I am interested in differential equations with state dependent delay, but I don't have a book or manuscript available for study.
    Dmitry Gromov · Saint Petersburg State University

    Sounds interesting. Looking forward to seeing your results.
    Good luck. D.

  • Fotsa Mbogne David Jaures added an answer:
    Is there an existing work concerning the (global) controllabity of nonlinear systems at most affine in state (dx/dt=A(t,u)x+B(t,u))?

    But non linear with respect to the control?

    Fotsa Mbogne David Jaures · University of Ngaoundéré

    I encounter some difficulties in the exploitation of  the attached documents. Is it possible to have more precise (english) references? Thank you all.

  • Farzin Piltan added an answer:
    To control any nonlinear system; what is the difference between fuzzy sliding mode controller and sliding mode fuzzy controller?

    To design a controller for nonlinear and uncertain systems we have two choice: the first one is fuzzy sliding mode controller and the second one is sliding mode fuzzy controller. I'd like to know that which one is better and why?


    Dear Researchers;

    I think:

    Fuzzy sliding mode controller (FSMC) is a nonlinear controller based on sliding mode method when fuzzy logic methodology applied to sliding mode controller to reduce the high frequency oscillation (chattering) and compensate the dynamic model of uncertainty based on nonlinear dynamic model.

    Sliding mode fuzzy controller (SMFC) is an artificial intelligence controller based on fuzzy logic methodology when, sliding mode controller is applied to fuzzy logic controller to reduce the fuzzy rules and refine the stability of close loop system in fuzzy logic controller.
    However the SMFC has a good condition but the main drawback compared to FSMC is calculation the value of sliding surface slope coefficient pri-defined very carefully and FSMC is more suitable for implementation.


  • Narasim Ramesh added an answer:
    How do I learn Robotics, Mathematical modelling and implementation from the very basics?

    Hello everybody my respectable wishes to all,

    I am Parthiban did my Post Graduation in the area of Embedded systems.

    I have some idea about Micro-controller programming (both in Assembly and C),   Digital Electronics.

    I want to do my research in the area of  Nonlinear System Mathematical modelling, implementation and control of robots, Quadcopter using Fuzzy logic or Neural networks or Genetic algorithm,

      but I know nothing about the above mentioned area.

    So I request you all to please provide your valuable guidance to me to learn about robotics, Quadcopter modelling and implementation, Fuzzy logic, Neural networks, Genetic algorithm.

    Especially, first about Mathematical modelling of Linear control systems and then more importantly about  Mathematical modelling of Non-Linear control systems.

    Please help me, please provide useful study materials, research papers that are elaborately (easily understandable) describing the above concepts also please provide your precious suggestions in this research area.      

    Narasim Ramesh · Sri Jagadguru Chandrasekaranathaswamiji Institute of Technology

     IMHO .. you have enough background to begin work..suggest

    you identify first an application by observing problems being faced by society..

    eg 1. last kilometer safety for women..

          2. identification of pests in storage..

      3. a robot for ensuring safety of children at the gate of schools.

       4. robots to carry gas cylinders up stairs..

    5. robots for reiving outpatients for Doctors..

     6. robots to monitor crowds

    etc..then it will be clear what you need when you try to see what has already been

    done.. so development and improvements can be identifies..which will

    require specific skills that you mentioned..else ..I think there is too much material

    out  there which is distracting..

    Good Luck


  • Artur Sergyeyev added an answer:
    How can I find the integrals directly from Lie point symmetries in ODEs?
    From the vector fields, we can find the invariants by solving the characteristic equation associated with the vector field. Whether this invariant is exactly same as the first integral admitted by the given ODE?
    Artur Sergyeyev · Silesian University in Opava

    If your ODE is Lagrangian, then under certain conditions you can construct integrals from the symmetries using the Noether theorem.  

    On the other hand, you can construct first integrals for your ODE by solving the characteristic equation associated with the characteristic vector field of this ODE rather than the characteristic equation associated with the vector fields of the Lie point symmetries of your ODE. 

  • A.-J. Muñoz-Vazquez added an answer:
    What is the difference between iSMC and Fuzzy SMC?

    I'd like to design iSMC for a nonlinear system such as robot manipulator but I'd like to know about i-SMC.

    A.-J. Muñoz-Vazquez · Center for Research and Advanced Studies of the National Polytechnic Institute

    Prof Tiwari,

    I know the integral SMC, and I kwnow that it is robust to bounded disturbances, BUT this robustness is not provided by the reaching phase elimination, instead the reaching-phase avoidance is provided by the robustness, and not the converse. The robustness is associated with the permissible gain in the nonlinear control part, and it is a direct consequence that this nonlinear term appears in the derivative of an auxiliary sliding manifold, then the equivalent controller is equal to the inverse additive of the disturbance effects. 

  • Wondimu W Teka 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?
    Wondimu W Teka · University of Texas at San Antonio

    Hi Javid,

    Fractional calculus has been observed on neuronal spiking activities. For example, check the following papers.


  • Mahmood Dadkhah added an answer:
    Where can I read about an algorithm to generate colored noise?
    I need to solve a numerically stochastic differential equation with additive colored noise.
    Mahmood Dadkhah · Payame Noor University


    Visit the:

    1. ""

    2. search the site for books.


  • Does anyone know some conservative chaotic systems (3-D flows)?
    I know Sprott-A (Nose-Hoover system), and a system Heidel and Zhang investigated in “Nonchaotic and chaotic Behavior in Three-Dimensional Quadratic Systems: Five-One Conservative Cases” (which I think originally was reported by Sprott in 2000). Does anybody know another 3-D chaotic conservative flow?
    Sundarapandian Vaidyanathan · Vel Tech - Technical University

    Dear Sajad:

    There is a chapter titled "Chaos in Conservative Systems" by M. Lakshmanan et al. in their Springer book "Nonlinear Dynamics" (2003). In the introduction of this Chapter, they refer to three systems as examples of conservative chaotic flows:

     (1) Henon-Heiles system (1964)

    (2)  Conservative Duffing Oscillator under periodic forcing

    (3) Standard Map (Discrete Chaos)

    As system (4), some friends here already mentioned "Sprott-A system" (1994), which is also known as "Nose-Hoover system".

    As the Lyapunov exponents are ordered in non-increasing order, for a 3-D conservative chaotic system, we must have L1 + L2 + L3 = 0. As L2 = 0, it follows that L3 = - L1.

    My interest is about 4-D conservative chaotic systems. The Lyapunov exponents of such systems have the nice property that L2 = L3 = 0 and L4 = - L1. I am currently working on finding 4-D conservative chaotic systems. (It is not possible for hyperchaos in 4-D conservative chaotic systems).



  • Andrea Spitaleri added an answer:
    Chaos in molecular dynamics simulation?
    When using Gromacs, many times even if I use the same tpr file to run a simulation, the trajectory is tremendously different. For example, in one trajectory the protein domain stays intact, while it loses secondary structure in another run). Where does this instability come from? Is it because MD is inherently a chaotic system (which seems unlikely to me)? Or is it the result of numerical implementation in the software?
    Andrea Spitaleri · Italian Institute of Technology (IIT)

    Hi, it is not strictly correlated to your question but have look here:

    at the bottom "Exact vs binary identical continuation"


  • Shashi Kant 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.
    Shashi Kant · Delhi Technological University

    as regrads chaos in two dimensions it is mathematically not feasible. However in delayed two dimensional system, chaotic picture may be evolved.

  • Emanuel Gluskin 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?
    Emanuel Gluskin · Ben-Gurion University of the Negev
    I have performed a lot of nonlinear studies, but/and at this moment, I have a simple opinion. If your treatment naturally tends to complex analysis, -- I would vote for linearity of the system, and if you naturally tend to real-valued analysis, -- I would vote for its nonlinearity. This philosophy partly has its roots in singular (switched) systems that can be linear (LTV) or nonlinear.
    Yours, Emanuel. (main link, incomplete, but you can read the works) (unpublished technical works, several are published)
  • Djamel Teguig added an answer:
    What is the basic difference between LS and PLS?
    LS- Least Square
    PLS-Partial Least Square
    Djamel Teguig · Royal Military Academy
    Please refer to this article

    it details more what you want with numerical results
  • Steve Schneider added an answer:
    Is there any model of thinking that uses the Dynamical System approach?
    If thinking can be modeled by the Dynamical System approach, the thinking will be a self-organized system. How free will this incorporated system be? What are parameters and state variables in such a system?
    Steve Schneider · Capella University
    I seem to have come into this discussion late, but this is a great topic. First, the concept of a dynamical system is not as widely understood as some others. I also like it because, similar to systems thinking, its application was borrowed from the “hard sciences,” but is being used across disciplines. Systems, are self-organizing, but also change as the need arises so in terms of free will, I believe it is inherent in such an approach. A dynamical system of thinking is still a system. If that is true, it will adapt and structure itself. The parameters and state variables will change as the system changes (similar to people changing as they develop).

    While there is an order to dynamical systems, it is based on maturation which allows it to evolve. There are many models of thinking that use a dynamical system approach. One of the earliest is Thelen and Smith’s development of cognition and action. I found some of their work on the Internet so I assume it was OK to attach it here.
  • Carlos Eduardo Maldonado added an answer:
    What is disorder under the perspective of complex adaptive systems?
    I want to understand this concept better. Thank you for your time and effort.
    Carlos Eduardo Maldonado · Universidad del Rosario
    I would like to go a little bit farther. There is, as it happens, linear order; and there is also non-linear order.

    The concern in complexity theory can be stated thus:

    i) How does a linear system become a nonlinear one?, and
    ii) How can a linear system be changed into a nonlinear one?

    The problem is fantastic and classical science never got to this point. This is, hence, the differential of complexity science vis-à-vis traditional science.
  • Nishad T M added an answer:
    Can anyone explain Mathematics in Shakespeare's Sonnet 116?
    SONNET 116
    Let me not to the marriage of true minds
    Admit impediments. Love is not love
    Which alters when it alteration finds,
    Or bends with the remover to remove:
    O no; it is an ever-fixed mark,
    That looks on tempests, and is never shaken;
    It is the star to every wandering bark,
    Whose worth's unknown, although his height be taken.
    Love's not Time's fool, though rosy lips and cheeks
    Within his bending sickle's compass come;
    Love alters not with his brief hours and weeks,
    But bears it out even to the edge of doom.
    If this be error and upon me proved,
    I never writ, nor no man ever loved.
    Nishad T M · MES College of Engineering and Technology ,Kunnukara,Ernakulam
    I respect your argument but I will not agree with your argument.

About Nonlinear Systems

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

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