# What is Nonlinear systems: Definition and 24 Discussions

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one.
In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.
As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.
Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others:

Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.

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1. ### I Number of Solutions of a Nonlinear System

I developed a program to solve the nonlinear system below through the method of successive approximations and was only able to find one solution, namely ##x_1 = 0.93377## and ##x_2 =0.88417##, even though I tried many different starting points. I was wondering if there's a way to determine if...
2. ### Solving a set of nonlinear quadratic equations

I would like to solve this system, which is a sets of non linear quadratic equations, the system needed to be solved can be expressed in general as follow: ϒϒ'C – ϒα = B Where ϒ=(ϒ1,ϒ2,...ϒn)’ is a column vector and ϒ’ its transpose C=(c1,c2,…,cn)’ and B=(b1,b2,…bn)’ are a columns vector And...
3. ### Conserved quantities in the Korteweg-de Vries equation

Homework Statement Consider the Kortweg-de Vires Equation in the form $$\frac{\partial \psi}{\partial t}+\frac{\partial^3 \psi}{\partial x^3}+6\psi\frac{\partial \psi}{\partial x}=0$$ Find the relation between the coefficients ##c## and ##d## , such that the following quantity is conserved...
4. ### Problems with Unscented Kalman Filter

Hi guys, I have problems understanding UKF and its environment. I will ask a couple of questions and see if you can help me. I have a nonlinear system (SYS) with one input called P (scalar) and one output called IP (also scalar). I would like to estimate the future output (the IP), so in my...
5. ### Fixed point method for nonlinear systems - complex roots

Homework Statement I've been asked to graphically verify that the system of equations F (that I've uploaded) has exactly 4 roots. And so I did, using the ContourPlot function in Mathematica and also calculated them using FindRoot. Now, I've to approximate the zeros of F using the fixed point...
6. ### Equilibria in Nonlinear Systems

So I'm reading the Example on page 161 of Differential Equations, Dynamical Systems and an Introduction to Chaos by Hirsh, Smale, and Devaney. I'm not understanding everything. So given the system x' = x + y^2 y' = -y we see this is non-linear. I get it that near the origin, y^2 tends to...
7. ### MHB Solving Linear & Nonlinear System w/ Reparametrization Function

I have a problem on which I am stuck and would like help on how to proceed. The problem and my work is fairly lengthy, so please bear with me. **Problem:** A model for transport of a solute (moles of salt) and solvent (volume of water) across a permeable membrane has the form...
8. ### Perturbation Techniques and Theory for Nonlinear Systems

Homework Statement Given the equation \ddot{\theta}=\Omega^2\sin{\theta}\cos{\theta}-\frac{g}{R}\sin{\theta} Determine a first-order uniform expansion for small but finite theta. Homework Equations Other than the equation above, none so far as I am aware. The Attempt at a...
9. ### Time dependent forcing and nonlinear systems

Hi, I'm trying to find a toy (i.e. analytic) example of a nonlinear system that has very different behavior for two different types of forcing: 1) \frac{\partial u(x,t)}{\partial t}+ N(u(x,t)) = F(x) where u(x,t) is the dependent variable, N represents some nonlinear operator with only...
10. ### Software to solve Nonlinear Systems (ineq and eq)

Hi everyone, I've got an optimisation/computing question. I have a system of nonlinear equalities and inequalities, which I've written below for reference. It's the conditions for a minimiser of a Karush-Kuhn-Tucker problem. Would anyone be kind enough to explain how I could use software to...
11. ### Nonlinear Systems & Weighted Sum of Impulses

Hello, my question is that almost all textbooks say that a linear system will give the output to a weighted sum of impulses which equals the superposition of scaled responses to each of the shifted impulses. But if we apply the same input which is a weighted sum of impulses to a non linear time...
12. ### Schools Preparation for grad school - nonlinear systems

Lately I have been reading about nonlinear systems and chaos. It's fascinating and I would like to know more about how I could prepare myself to possibly study this in grad school. I would be interested in looking at biological systems such as neural networks or even animal populations. I'm a...
13. ### What branch of math deals with nonlinear systems?

Like linear algebra goes in depth about linear systems, what should I look for to learn about the extension of linear algebra to nonlinear systems? Is there a name of the field of study? If I go into a book store to buy books about it, what should I be looking for? Abstract Algebra? Complex...
14. ### Roots of polynomials as nonlinear systems of equations

Ok, to start off I have been examining the structure of polynomials. For instance, consider the general polynomial P(x)=\sum^{n}_{k=0}a_{k}x^{k} (1) Given some polynomial, the coefficients are known. Without the loss of generality...
15. ### A quick and clever method for solving nonlinear systems of equations

if we are given a polynomial s^6+as^5+bs^4+cs^3+ds^2+es+k (if a,b,c,d are known) what is a clever method to solving for the value k if we are given the following: the above polynomial is equal to the following(zeta is given as some constant, say 1 for simplicity)...
16. ### Interested in Chaos Theory, Complex Systems, Nonlinear Systems

As the thread title says I'm interested in Chaos Theory, Complex Systems, and Nonlinear Systems. If I can help it, I'd like to study these at graduate level. My question is what kind and how much mathematics I'm supposed to know if I'm to study these?
17. ### Nonlinear systems of differential equations

The complete question I've been given: The Rossler equations are formally deﬁned as dx/dt=−y−z dy/dt=x+ay dz/dt=b+z(x−c). Let us suppose that a=0.2, b=0.2, c=5.7, x(0)=y(0)=z(0)=0, t∈[0,400]. Let v1(t) be the solution to the given initial value problem, and let v2(t) be the solution of the...
18. ### Nonlinear systems of differential equations

I think I posted in the wrong section and will repost in the textbook/coursework section but don't know how to delete this. Although if you want to answer feel free. The complete question I've been given: The Rossler equations are formally deﬁned as dx/dt=−y−z dy/dt=x+ay dz/dt=b+z(x−c). Let us...
19. ### Finding Equilibrium Points of Nonlinear Systems

Hi, So I keep making mistakes trying to find all of the equilibrium points of different simple nonlinear systems. These problems aren't difficult, it's just that I keep taking different approaches to finding the equilibrium points. Is there a methodological way to know that I have found...
20. ### Solving Nonlinear System: 3x2+2y2=35, 4x2-3y2=24

Homework Statement solve the system of 3x^{2}+2y^{2}=35 and 4x^{2}-3y^{2}=24 Homework Equations The Attempt at a Solution I re arranged for y^2 and got 1\frac{1}{3}x^{2}-16=y^{2} I keep getting x to equal \pm 2.473 this is clearly wrong, the answers...
21. ### Nonlinear System Solution Strategies

Homework Statement Solve the nonlinear system. x^2-2y^2=16 x^2+y^2=25 Homework Equations n/a? The Attempt at a Solution ive tried subtracting 1 equation from the other... (x^2-2y^2+9)=25 -(x^2+y^2)=25 -3y^2+9=25 -3y^2=16 y^2=-16/3 y=root of -16/3... im not even sure...
22. ### Exploring Phasor Analysis for Nonlinear Systems

Hello, What happens when phasor analysis is applied to nonlinear systems? could someone explain me how it would work? :confused: Thanks
23. ### Solving nonlinear systems efficiently

so I'm doing these lagrangian multipliers in calc class and it involves nonlinear systems and apparently the techniques I'm used to applying to linear systems aren't appropriate because i keep losing solutions, getting the wrong ones etc. so yea what are some efficient algorithms for this
24. ### Nonlinear Systems of Equations Question

Ive been working on a bunch of non linear systems problems for homework and can't figure this one out: y=xcubed x-y=0 I get to the substitution stage and get x-xcubed=0, but you can't factor that so I am wondering what you do next to get the answers for x, and y. Thanks in advance for...