# Nonlinear Systems

Can a certain little-known 2+1 dimensional generalization of the KdV equation be solved using the inverse scattering transformation?
The system of equations u_t + (\partial_x^2 + \partial_y^2)u_x + u u_x/2 = -(p_x u_x + p_y u_y)/2 (Eq. 1) p_xx + p_yy = u_x (Eq. 2) reduces to the KdV equation if u and p are independent of y. The KdV equation can of course be solved using the inverse scattering transformation (IST). Can the system of equations (1) and (2) be solved by the IST when u and p depend on x, y and t? [These equations arise in the theory of electromigration in thin films --- see R. M. Bradley, Physica D 158, 216 (2001)].
Kamruzzaman Khan · Pabna University of Science and Technology, Bangladesh. পাবনা বিজ্ঞান ও প্রযুক্তি বিশ্ববিদ্যালয়
@ R. Bradley: Please see the following articles,( it may be helpful): 1. Kamruzzaman Khan and M. Ali Akbar. Traveling Wave Solutions of Nonlinear Evolution Equations via the Enhanced (G'/G)-expansion Method. Journal of the Egyptian Mathematical Society. doi.org/10.1016/j.joems.2013.07.009. 2. M. Wang, X. Li, J. Zhang, The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A. 372(2008) 417-423. such and such
• What are the different dynamical systems' classifications as varying from linearity or non linearity of the system?
Such as linear, weakly linear, quasi linear, weakly nonlinear and fully nonlinear and their corresponding equations.
Jefferson Portela · Fraunhofer Institute for Industrial Mathematics ITWM
Rigorously, a system is either linear (e.g., dx/dt = ax) or not. That said, in a nonlinear system there might be, say, regions on your phase-space that weakly affected by the nonlinearity if it is weak enough (e.g., if b<<1, where 'b' is the coefficient of a nonlinear term in your equations), and you might be able to describe the situation as approximately linear. Of course such approximations are problem dependent.
Solve y^3 + y''' = 0?
Can anyone suggest an analytical method to solve y^3 + y''' = 0? The solution should satisfy the boundary conditions y tends to plus or minus one as x tends to plus or minus infinity. The solution should also be bounded everywhere. The tanh method doesn't work, but perhaps some variant of it does.
Ricardo Mansilla · Universidad Nacional Autónoma de México
Is correct. I assumed trivial cases were out.
• Miguel A F Sanjuán added an answer:
Does anyone have details about William Benjamin Fite's life and work?
He is the author of one of the most important works of of the early twenty century in the qualitative theory.
Miguel Sanjuán · King Juan Carlos University
William Benjamin FITE (1869-1932) apparently was born on 23 Aug 1869 and died in 1932. He got his PhD in Cornell University in 1901. His Advisor was George Abram Miller (1863-1951) and the title of the dissertation was “On Metabelian Groups”. He was appointed Professor of Mathematics in Columbia University, New York, in 1911. He was the Treasurer of the American Mathematical Society during the period (1921-1929). Besides writing a book on Metabelian Groups, he wrote other books in College Algebra among others. He is the author of the paper: W.B. Fite. Concerning the zeros of solutions of certain differential equations. Trans. Amer. Math. Soc., 19 (1917), pp. 341–352. I am sure that if you search carefully with the right keywords in google you will get more details.
• Robert J. Low added an answer:
How do I calculate the fundamental group of fibre bundles?
I am looking for a way to compute the fundamental group of fiber bundles in terms of the fundamental group of the base, fibers and whatever else is needed.
Robert Low · Coventry University
If your fibre bundle is S^3, thought of as the Hopf fibration with fibre S^1 and base S^2, then the fundamental group is 0, since S^3 is simply connected. On the other hand, if your fibre bundle is just R^2 x S^1, the fundamental group is Z, since S^1 is a deformation retract of R^2 x S^1. But in each case the fibre has fundamental group Z and the base has fundamental group 0. This suggests to me that the 'whatever else' is going to be quite complicated. You might get a more helpful answer by asking your question in the topic Algebraic Topology.
When do two nontrivial solutions to a nonlinear ODE add to give a new nontrivial solution?
I learned to my surprise from an earlier question that there are special cases in which two nontrivial solutions to a nonlinear ODE add to give a new nontrivial solution. (See "Does this ODE have a solution?") Does it show something special about the ODE when this happens? Is there a general property of nonlinear ODEs that tells us when this is possible? Are there other nonlinear ODEs that have this strange property? Note added --- I'm interested in the case in which the two solutions that are added to give a new one are not simply proportional to one another --- see my comment below.
K. Kassner · Otto-von-Guericke-Universität Magdeburg
Well, there is a somewhat trivial answer to that question. Suppose, your equation can be written in the general form L[y] + N[y] = 0, where L is some linear differential operator and N a non-linear differential operator. Then a sufficient condition for z = y_1+y_2 being a solution, provided y_1 and y_2 are solutions themselves is N[y_1] + N[y_2] = N[y_1+y_2]. (A) If we allow N to contain an additive free constant, then it is even sufficient, if we have N[y_1] + N[y_2] = N[y_1+y_2] +c, (B) where c is some new constant. This is obviously the case realized in the equation considered by you previously y''' - y'^2 + x y - c = 0 (1) where N[y] = - y'^2 - c. One solution was y1 = -6/x, with c = -6, the other y2 = x^3/9, with c = 2/3. Then we have N[y_1] + N[y_2] = -36/x^2 + 6 - x^4/9 - 2/3 = -36/x^2 - x^4/9 + 16/3 (*) and N[y_1+y_2] = -(6/x^2+x^2/3)^2 - c = -36/x^2 - 4 - x^4/9 - c (**) (*) and (**) are identical if c = -28/3. As you have observed yourself, this is due to the fact that the mixed term in N[y_1+y_2] becomes a constant. I doubt that one can get more specific than (A) or (B) without additional restrictions on the form of the equations to be considered. [(A) holds for the original form of your equation, obtained by differentiating (1) once. Then N[y] = - 2 y' y''.]
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.
Dear Sorin, Thanks very much for the very helpful answer. The specific problem we're trying to understand is the one studied by Bertozzi et al. in their 1998 PRL [PRL 81, 5189 (1998)]. Their equation of motion is h_t + (h^2 - h^3)_x = -\eps (h^3 h_xxx). I understand the mathematical viewpoint you describe in your answer above. However, it is unclear to me from a physical standpoint why the UC shocks form, and why these solutions seem to be attractive. I imagine that the formation of the UC shocks has something to do with balancing gravity and the Marangoni stress, but is it possible to be more precise? BTW, could you give me the reference for your Eq. (4)? Thanks and best regards, Mark
• Abdallah Daddi Moussa Ider asked a question:
How can a plan Couette flow be embedded in a Ra-Bénard situation with differential heating across the plates or a Ta-Couette flow with a narrow gap?
How can a given flow be embedded in a family of other flows in the aim of visualizing the 3D stationary states?
• Brian G Higgins added an answer:
How does one implement wetting and drying? Is it just by ensuring positive height or by incorporating some terms in the model equations?
I am learning to develop algorithm for flooding and drying. And am using the 2D shallow water model.
Brian Higgins · University of California, Davis
Note that the 2D-shallow water model excludes viscous effects; thus it inherently cannot describe the dynamics of dewetting that would lead to dry spots. Further, the shallow water model cannot describe the flow of thin liquid films near a boundary, the dynamics of which are dominated by viscous effects. I think the approach you will have to take is one based on a singular perturbation analysis, in which the outer solution is described by the shallow water wave model and the inner solution ( when the layer thickness is sufficiently thin) described by a lubrication-like flow dominated by viscosity and surface tension. There are many examples in the literature that discuss deweting of thin films using a lubrication model. But your challenge will be to define the appropriate scalings such that you have the correct limits as the layer becomes vanishingly small in thickness, and have the correct matching conditions. Perhaps others on the forum know if this has been done before- I am not familiar with the literature of shallow water waves....
Does this ODE have a solution?
Does the nonautonomous nonlinear ODE [y''' - (y')^2 + xy]' = 0 have a nonzero solution that is bounded everywhere? If there is a solution, can it be found in analytical form? Note that there is of course the first integral y''' - (y')^2 + xy = const. This equation has the solution y = -6/x with the constant equal to -6, but it isn't bounded.
K. Kassner · Otto-von-Guericke-Universität Magdeburg
Sorry, equation (1) should of course have a minus sign in front of the nonlinear term and the nonlinearity is therefore always negative. But I have not analysed the wrong equation, it is just a typo (otherwise the local behaviour b would be different). Also when I am saying there are "no bounded solutions", it is to be understood as "no nontrivial bounded solutions". The solution y=0 for c=0 is of course bounded, but it may be the only globally bounded solution.
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.
Amin Amani · Delft University Of Technology
there are some new analytical methods to solve this problem, like "homotopy perturbation method", "energy balance method", "Variational iteartion method, " homotopy analysis method", "variational approach method", "exp-function method",
Linearization around an equilibrium point versus linearization around a trajectory - can anyone help?
What is the fundamental difference, in calculation, between linearizing a nonlinear system around an equilibrium point and that around a trajectory?
Muhammad Aftab · Sultan Qaboos University
Thanks Douglas and Marco for providing the information.
• Walter Edgardo Legnani added an answer:
Is the concept of signal complexity a closed scientific question?
I am writing a review on signal complexity and a peer told me that is a closed discussion in signal processing. Is this the understand of the scientific community?
Walter Legnani · National University of Technology
Dear Jean, could you send me the complete reference to the work of Bravi, I can´t found in the electronic library in Argentina. Thanks in advance.
Which text is the best as an introduction to nonlinear ordinary differential equations?
Which text is the best as an introduction to nonlinear ordinary differential equations? I am having a really hard time learning this on my own.
Rodolfo Reyes-Báez · Center for Research and Advanced Studies of the National Polytechnic Institute
Do you need a pure theoretical approach?, I can recomend you a book that engineers and applied math professional uses a lot in the context of control systems. The book es called "Nonlinear Systems" of professor Hassan Khalil. The first part is about an introduction to qualitative theory of nonlinear dynamical systems represented by a vectorial nonlinear ordinary differential equations.
Is this nonlinear diffusion equation exactly solvable?
Can anyone suggest a method to solve the nonlinear diffusion equation u_t = -u_xxxx + (u^2)_xx subject to the initial condition u(x,0)=delta(x) and the boundary conditions u, u_x -> 0 as x -> +/- infinity? Note that if the term -u_xxxx were absent, this would be the porous medium equation.
@Aref: The term means the second partial of u squared with respect to x. There are traveling wave solutions that can be obtained by solving a fourth order nonlinear ODE, but these will not match the initial condition. A better approach is to seek a similarity solution. This gives a nasty fourth order nonlinear ODE as well.
Is there any repository of large sparse systems of nonlinear equations associated to mathematical models?
The goal of my work is a structural study of large systems of equations, with linear and nonlinear equations within, for improving the efficiency of their resolution. I work with incidence matrices and graph theory for the structural reorganization of systems of equations. It would be interesting to find large systems that arise in different disciplines.
Afaq Ahmad · Sultan Qaboos University
Dear Eduardo Xamena, Check the following link for its usefulness. http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/71101/6/Ueda_03.pdf
Topics in nonlinear system identification?
I plan to do my PhD thesis in nonlinear vibration. After reviewing some papers, I have finalized the topic as nonlinear system identification. Can anyone give me some ideas about this topic?
Huajiang Ouyang · University of Liverpool
I suggest that study a simple system with one mass, one viscous damper, one spring and one cubic srping under a harmonic excitation. You can get the motion from numerical integration. With this simulated motion try to work out the coefficient of the cubic spring using a nonlinear identification method you choose. You will find how good this nonlinear identification method.
• Juan B Gutierrez added an answer:
Robust stability of uncertain polynomial systems?
I have the following problem. Consider a system \dot x1 = \mu f1(x1,x2,x3), \dot x2=\mu f2(x1,x2,x3), \dot x3 = f3(x1,x2,x3), where 1 \le \mu \le M and f_i's are polynomial. If I find a Lyapunov function in the vertex \mu = 1 and a Lyapunov function in the vertex \mu = M, can I conclude that the system is asymptotically stable for all 1 \le \mu \le M?
Juan Gutierrez · University of Georgia
It depends. The reason is this: The fundamental assumption is that your system \dot{X} = f(X) where X represents the variables x1, x2 and x3, has a fixed point X*. Your Lyapunov function is a continuously differentiable real-valued function V(X) such that: (i) V(X)>0 for all X \neq X* and V(X*)=0, and (ii) \dot{V} < 0 for all X \neq X*; only when these two conditions are met, the fixed point X* is globally asymptotically stable. Hence the "depends" in the opening sentence; your parameter \mu could change the stability of your system. Therefore, provided that there are no bifurcations of f(X) for 1 \le \mu \le M, then the system is globally asymptotically stable at a fixed point X*. A final comment: you can claim that your system is asymptotically stable around a given fixed point; this precision is important because you could have multiple fixed points, and not all are necessarily asymptotically stable even if you find a Lyapunov function that works in one of them.
• Miguel A F Sanjuán 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.
Miguel Sanjuán · King Juan Carlos University
The general problem of finding the number of limit cycles for a specific dynamical system is related to the Hilbert’s 16th problem. Among the dynamical systems with several limit cycles you can find Lienard systems, being the van der Pol system a particular case. Some references and examples can be found in: M. A. F. Sanjuan, Lienard Systems, Limit Cycles, and Melnikov Theory”, Physical Review E57(1), 340-344 (1998)
What is dynamic stability definition in human movement? We say a state is stable, what is dynamic stability?
Equilibrium means moving without acceleration. Balance is the control of equilibrium. Stability is the resistance to perturbation.
G. Filligoi · Sapienza University of Rome
Dynamic movement is a series of short-time intervals of quasi stationary states. The length of these intervals depends on the rapidity with which the movements are carried out.
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
DaHui Wang · Beijing Normal University
it is impossible. According to index theory,(http://www.scholarpedia.org/article/Periodic_orbit), one limit cycle must surround one or more saddle or focus.
How can I determine the stability of periodic solutions in a non-autonomous set of differential equations?
In a autonomous system, a limit cycle has its stability determined by the linearization around the internal fixed point. In a non-autonomous system (like a damped forced pendulum), there is no fixed point inner the limit cycles. There is some way to analytically determine the stability of a period-p solution in this kind of system? Considering a linear piecewise term in the differential equations, is this possible yet?
Marco Storace · Università degli Studi di Genova
Dear Tiego, to find an analytical solution to your problem, you should carry out local analysis using approximate Poincare' maps (see, e.e., Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Springer-Verlag, New York, 2004). For numerical analysis with smooth flows you can apply continuation methods (e.g., with the software packages MATCONT or AUTO). Hope this helps!
What sense have derived fractional to control theory?
We know that the derivative represents the slope straight to a curve at a point, thenwhat fractional derivatives representing. In fractional control where this utility derived fractional out. I hope someone can answer me, I am new in this, but I is enough interest, a reference would also be quite useful.
Emmanuel Gonzalez · De La Salle University
Let's talk about fractional calculus applied in control theory. I will not be talking about what fractional calculus is. Control systems are usually represented using integer-order differentiation equations or transfer functions. You could have a simple first-order model for a DC motor that is contorolled by a simple PID controller. The combination of your controller and DC motor could be situation in a simple unity-feedback system and we have learned in various textbooks that we could derive its frequency and time responses as we may. Fractional calculus gives a twist in this area. Since fractional calculus means "non-integer-order", the you could have 4 possibilities: 1.) Integer-order controller + integer-order plant (the typical one) 2.) Integer-order controller + fractoinal-order plant 3.) Fractional-order controller + integer-order plant 4.) Fractional-order controller + fractional-order plant So what is the sense of having, for example, a fractional-order controller? If you have a fractional-order controller, then the entire control system will have dynamics following fractional-order differential equations. It has been found from various researchers that there are many advantages of incorporating fractional-order dynamics in a system: 1.) You can produce iso-damping property in your system. Iso-damping means the overshoot of the system doesn't change even if there are gain variations in your plant. Thus, your system becomes robusts in gain changes which could be achieved only with fractional dynamics. 2.) You can satisfy more than 3 robustness criteria using fractional-order PID controller (FOPID). An FOPID has 5 degrees of freedom which you can tune as necessary. Therefore, you can optimize your feedback control system and satisfy 5 parameters of interest (usually 5 robustness criteria) by just tuning your 5 parameters. Classical PID controller on has 3, i.e. Kp, Ki, and Kd, while FOPID has five, i.e. Kp, Ki, Kd, diff_order, integ_order. Above are just examples. So, does it make any sense of having fractional calculus in control theory? The answer is a big yes :D Hope this helps. Emm
What is the dynamic type of a continuous nonlinear system with the following Lyapunov exponents: LE1= 6.9857, LE2= -0.0091 and LE3 = -6.9948?
Is it chaotic or conservative chaotic? The LE values are calculated with Lyapunov Exponent Toolbox (LET), which is based on "A. Wolf et al., Determining Lyapunov exponents from a time series", Physica D, Vol 16, pp 285-317, 1985.
Hamed Ghane · Amirkabir University of Technology
Dear Prof. J. C. Sprott, Many thanks for your helpful answer.
How does numerical dispersion behave in time?
I am using finite difference of central in space type, so I should get dispersion error due to truncation. Also my solution is oscillatory. So how do I say that the resulting oscillation is my solution, not the numerical dispersion? Is there any trend of the numerical dispersion that I can recognize that. Any reference material is welcome.
N.C Markatos · National Technical University of Athens
I have no idea of the numbers, but if Peclet=0.015 obviously it is Not the problem
Vorticity and potential vorticity what is the best way to understand the difference?
With linear shallow water theory (LSW) we find del(omega)/del(t) = 0, that is vorticity is preserved, whereas if we consider non-linear shallow water theory (NLSW) it is the potential vorticity that is conserved following each fluid particle. In oceanographic context potential vorticity is a very important parameter. I wish to distinguish the vorticity and potential vorticity terms physically? How can I do this? By the word 'physically' I mean some real life examples by which this distinctions seems obvious.
James Andrews · University of Birmingham
The attached document distinguishing the vorticity and potential vorticity terms physically is attached here. Sorry for its absence in the previous post.
Is there any practical system in which the output is a nonlinear function of states?
Usually in practical systems, output is simply some states or maybe a sum of them. I'm searching for the case that output is a nonlinear function of states and off-course this non-linearity is not the only non-linearity present in the system model.
Prakash Rajendran · Indian Institute of Technology Madras
I can say few examples like discharge of water from water tank , stress exceeds yield limit in forging, forming operation etc.. these are live examples of nonlinear functions of states.