# Questions tagged [nonlinear-system]

In mathematics, a nonlinear system of equations 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.

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### An first integral of nonlinear differential equation as like forced pendulum nonlinear diff. eq.

I'm trying to face this nonlinear differential equation: $$y''(x)+\omega^2\sin\,y(x)=a\,x \,\;(1)$$ and I'm interested to found the solution of $y'(x)$ (an first integral) The homogeneous part of ...
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### Proving that a pair of non-linear equations has a unique solution

Assuming that we have a pair of equations with two unknowns $a$ and $b$: \begin{cases} f(a,b) = 0 \\ g(a,b) = 0 \end{cases} What kind of conditions are needed to show that the solution to this pair is ...
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### Numerical solution to a non-linear PDE

I have this Non-linear PDE $$\frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2}$$ Where C is a function of (x,t) It comes from the ...
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### Show whether the following function is radially bounded or radially unbounded: [closed]

my function is V(x) = (Sin(x))^2 Please give me an appropriate answer
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### Non linear system

Anyone can help me with this system? $$(I)\ 2x\sin(\theta)\cos(\theta) - 4x\sin(\theta) + 10\sin(\theta) = 0$$ $$(II)\ 10x\cos(\theta) - 2x^2\cos(\theta) + x^2(\cos^2(\theta) - \sin^2(\theta)) = 0$$ ...
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### nonlinear differential equations

I am interested in solving coupled differential equations with the following shape: \begin{equation} \ddot y_p(x) =\sum_{lmn=1}^N C(l,m,n,p)y_l (x)y_m (x)y_n(x) \end{equation} where $p=1\cdots N$. ...
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### Phase trajectory must always enclose a fixed point

I found this problem in strogatz nonlinear dynamics . The theorem says, A closed phase space trahectory must enclose a fixed point . The question is asked as, is this true for phase surfaces other ...
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### Jacobian of a Fixed Point Method

I am trying to determine the Jacobian of a multi-dimensional fixed point method with the following form $\begin{equation} x = -c(x)Ax + Bx, \end{equation}$ where $A$ and $B$ are square matrices (not ...
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### Four equations with five unknown and the Wolframalpha can solve them

$$a+b+c=1\tag 1$$ $$a*k_1+b*k_2+c=1\tag2$$ $$2a*k_1^2+2b*k_2^2+2c=1\tag3$$ $$3a*k_1^3+3b*k_2^3+3c=1\tag4$$ the Wolframalpha gives the solution as follow: How did this happen?????
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### Linearization of Partial Differential Equation

I have this beautiful Non-linear PDE $$\frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2}$$ Where C is a function of (x,t) It comes from ...
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### Conditions for the solution of the ODE $\dot{x}(t) = f(x(t))$ continuous on $t \in [0, +\infty)$?

Given the ODE: \begin{align} \dot{x}(t) = f(x(t)) \end{align} It is known that the solution $x(t)$ is continuously differentiable on a compact interval $t \in [0, t_{1}]$ when $f(x(t))$ is ...
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### Is there any way to solve the following system of non linear equations

$$a\,(k_1-1)+b\,(k_2-1)+c\,(k_3-1)=-1/2$$ $$a\,(k_1-k_1^2)+b\,(k_2-k_2^2)+c\,(k_3-k_3^2)=1/6$$ $$a\,(k_1-k_1^3)+b\,(k_2-k_2^3)+c\,(k_3-k_3^3)=2/8$$ $$a\,(k_1-k_1^4)+b\,(k_2-k_2^4)+c\,(k_3-k_3^4)=3/10$$...
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### How can I prove that $10=2^{a}*3^{b}*7^{c}$ has infinite solutions?

Both in a unrescrited case and with the following restriction: $a+b+c=1$
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### Proving a property of a solution to a set of nonlinear polynomial equations

Consider the following system of equations for $R_{i}(\lambda)$ \begin{align} R_1 &= \frac{\lambda}{4}(1 + R_3 + 2 R_2 R_1) \tag{1.1}\\ R_2 &= \lambda \left[q + \left(\frac{1}{2} - q\right)...
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### Plotting an iterative system of nonlinear equations using MATLAB

Consider the following coupled system: Let $x(n) = \left[ \begin{array}{c} x_1(n)\\ . \\ .\\ .\\ x_{32}(n) \end{array} \right]$, and the system of $32$ first order nonlinear ...
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### Test if a function given as a non-integrable ode set is Bijective

Given that state space trajectories of an autonomous system do not cross, can I deduce that a mapping function f:(x,y)→(x',y') given by a solution of an ODE of an autonomous system is bijective? ...
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### Revisit “example of an unstable fixed point for which the linearized dynamics are stable”

I am reading the following discussion: example of an unstable fixed point for which the linearized dynamics are stable The above discussion is for the vector field (continuous time). Is there an ...
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### stability of $(0,0)$ for $\dot{\theta} = y$ and $\dot{y} = -\sin\theta$

Given the system $\dot{\theta} = y$ and $\dot{y} = -\sin\theta$. For the fixed point $(0,0)$ I can see through linearisation that the flow corresponds to a centre which moves anticlockwise Why is ...
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### how to find form of $h(x)$ in reduction to centre manifold

given the system $\dot{x} = y - x - x^2$ and $\dot{y} = x - y - y^2$ I can find that the centre subspace is spanned by $E^c = [1,1]^T$ It says the centre manifold can be expressed as \$y=h(x) = x + ...