2D Nonlinear Systems of Differential Equations

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\vec{F}(x,y)=\left[\begin{array}{c}f(x,y)\\ g(x,y)\end{array}\right]$$

Equilibria

Find the equilibria by solving for all $(x,y)$ that satisfy $\vec{F}=\vec{0}$

Linearization

Find the Linearization of the system.

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{c}f(x,y)\\ g(x,y)\end{array}\right]⟹\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\vec{F}(\text{EP})+J_{\vec{F}(\text{EP})}\left[\begin{array}{c}x-\text{EP.x}\\ y-\text{EP.y}\end{array}\right]$$

where $J_{\vec{F}(EP)}$ is the Jacobian matrix at a particular equilibrium point

where EP is each Equilibrium Point, where EP.x and EP.y are the x and y components

By definition at an equilibrium point, the function is equal to $\vec{0}$. See Equilibria.

Therefore the linearization becomes this:

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=J_{\vec{F}(\text{EP})}\left[\begin{array}{c}x-\text{EP.x}\\ y-\text{EP.y}\end{array}\right]$$ $$J_{\vec{F}(\text{EP})}=\left[\begin{array}{cc}{f}_x& f_y\\ g_x& g_y\end{array}\right]@\text{EP}$$

The final expression for the linearization looks like this.

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left(\left[\begin{array}{cc}{f}_x& f_y\\ g_x& g_y\end{array}\right]@\text{EP}\right)\left[\begin{array}{c}x-\text{EP.x}\\ y-\text{EP.y}\end{array}\right]$$

This is a Shifted Systems of Linear Differential Equations

Linearization in 1D with Complex Polynomials

$$\text{Let }{y}^{\prime }=f(y)=-\frac{1}{16}(y+1{)}^3(y-2{)}^{2}y(y-3)(y-3)$$

The equilibrium points of this equation are $y=-2,-1,0,2,3$ The linearization at an equilibrium point is

$$y^{\prime }=J_{f(\text{EP})}(y-\text{EP})=f^{\prime }(\text{EP})(y-\text{EP})$$ $$y^{\prime }=J_f(y-\text{EP})=f^{\prime }(y-\text{EP})$$

In short, if you have a polynomial of order 1, the linearization is just f, but you drop the EP term. If the order is greater than 1, then the linearization is 0.

$$\begin{array}{c}\text{Case: }y=-2\\ f(y)=-\frac{1}{16}\cdot (y+1{)}^3\cdot (y-2{)}^2\cdot y\cdot (y-3)\cdot (y+2)\\ \text{Define a function with every factor, besides the -2 term.}\\ \text{Let }G(y)=-\frac{1}{16}\cdot (y+1{)}^3\cdot (y-2{)}^2\cdot y\cdot (y-3)\\ f(y)=G(y)\cdot (y+2)\\ f^{\prime }(y)=G^{\prime }(y)\cdot (y+2)+G(y)\\ f^{\prime }(-2)=G^{\prime }(-2)\cdot (-2+2)+G(-2)=G(-2)\\ y^{\prime }=f^{\prime }(-2)(y+2)=G(-2)(y+2)=10(y+2)\\ \text{Linearization: }\boxed{y^{\prime }=10(y+2)}\\ \\ \text{Case: }y=-1\\ f(y)=-\frac{1}{16}\cdot (y+1{)}^3\cdot (y-2{)}^2\cdot y\cdot (y-3)\cdot (y+2)\\ \text{Define a function with every factor, besides the -1 term.}\\ \text{Let }H(y)=-\frac{1}{16}\cdot (y-2{)}^2\cdot y\cdot (y-3)\cdot (y+2)\\ f(y)=H(y)\cdot (y+1{)}^3\\ f^{\prime }(y)=H^{\prime }(y)\cdot (y+1{)}^3+H(y)\cdot 3(y+1{)}^2\\ f^{\prime }(-1)=H^{\prime }(-1)\cdot (-1+1{)}^3+H(-1)\cdot 3(-1+1{)}^2=0\\ y^{\prime }=f^{\prime }(-1)(y+1)=0\\ \text{Linearization: }\boxed{y^{\prime }=0}\end{array}$$

Perterbation

Given

$${\vec{v}}^{\prime }=A\vec{v}$$

Which is the exact form of the homogenous part of our Linearization, except we have

$$\vec{v}=J_{\vec{F}(\text{EP})}(\vec{v}-\text{EP})$$

Neutral Eigenvalues ($\mathrm{Re}\lambda =0$) lead to structural instability, e.g. a perterbation leads to signficant change.

Approximating Dynamics

Given

$$\vec{v}=J_{\vec{F}(\text{EP})}(\vec{v}-\text{EP})$$

We approximate the dynamics of the original system, by finding the dynamics of the Linearization

• TLDR: We find the Eigenstuff of the Jacobian, and use that to construct our Phase Portrait
• If the eigenvalues of the Jacobian lead to Unstable or Asymptotically Stable, then the approximation is sufficient to determine the dynamics of the original system
• If Stable, we consider the approximation “degenerate”, and conclude that the approximation is insufficient.

Lotka-Volterra Competition Model

$$\frac{d🐺}{dt}=g_{🐺}🐺\left(1-\frac{🐺}{K_{🐺}}-{\alpha }_{🐺🐯}\frac{🐯}{K_{🐺}}\right)$$ $$\frac{d🐯}{dt}=r_{🐯}🐯\left(1-\frac{🐯}{K_{🐯}}-{\alpha }_{🐯🐺}\frac{🐺}{K_{🐯}}\right)$$

where $🐺$, $🐯$ are population sizes

where $g_{🐺}$, $r_{🐯}$ are intrinsic per capita growth rates

where $K_{🐺}$, $K_{🐯}$ are carrying capacities.

where ${\alpha }_{🐺🐯}$, ${\alpha }_{🐯🐺}$ are competition coefficients

• where ${\alpha }_{🐺🐯}$ measures the negative effect tigers have on wolves
• where ${\alpha }_{🐯🐺}$ measures the native effect wolves have on tigers

Lotka-Volterra Predator-Prey Model

$$\frac{d🐰}{dt}=\alpha 🐰-\beta 🐰🐺$$ $$\frac{d🐺}{dt}=\delta 🐰🐺-\gamma 🐺$$

where🐰 is the prey population size

where 🐺 is the predator population size

where $\alpha$ is the prey growth rate (birth rate)

where $\beta$ is the predation rate/interaction rate

where $\gamma$ is the predator death rate

where $\delta$ is the conversion efficiency of prey into new predators.