ch14: nonlinear systems, linearization, Lotka-Volterra
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% =============================================================================
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% ch14_nonlinear_systems.tex
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% Chapter 14: Nonlinear Systems
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% =============================================================================
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\section{Nonlinear Systems}
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\section{Nonlinear Systems}
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\label{ch:nonlinear_systems}
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\label{ch:nonlinear_systems}
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\subsection{Equilibrium Points}
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Linear systems admit a powerful theory built on superposition and eigenvalue analysis
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\label{sec:ch14_equilibrium}
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(\cref{ch:systems}). When the equations become nonlinear, the principle of superposition
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breaks down: the sum of two solutions is generally \emph{not} a solution. No universal
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closed-form solution formula exists for nonlinear systems. Instead, we rely on
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\textbf{qualitative methods} --- studying the structure of solutions without finding
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explicit formulas.
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% Content goes here
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\subsection{Nonlinear Autonomous Systems}
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\label{sec:ch14_autonomous}
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A \textbf{nonlinear autonomous system} in two variables takes the form
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\begin{equation}
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\label{eq:nonlinear_autonomous}
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\frac{\mathrm{d}x}{\mathrm{d}t} = f(x,y), \qquad
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\frac{\mathrm{d}y}{\mathrm{d}t} = g(x,y),
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\end{equation}
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where $f$ and $g$ are nonlinear functions. The system is \emph{autonomous} because
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$f$ and $g$ do not depend explicitly on $t$.
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\paragraph{Why analytical methods fail.}
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For linear systems $\mathbf{x}' = A\mathbf{x}$, we can construct the general solution
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from eigenvalues and eigenvectors. For nonlinear systems:
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\begin{itemize}
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\item \textbf{No superposition:} If $\mathbf{x}_1(t)$ and $\mathbf{x}_2(t)$ are solutions,
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$c_1\mathbf{x}_1(t) + c_2\mathbf{x}_2(t)$ is \emph{not} generally a solution.
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\item \textbf{No general formula:} There is no algorithm that produces a closed-form
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solution for an arbitrary nonlinear system.
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\item \textbf{Rich behavior:} Nonlinear systems exhibit phenomena absent in linear
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systems, including multiple equilibria, limit cycles, bifurcations, and chaos.
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\end{itemize}
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The strategy is to understand the system's \textbf{phase portrait} --- a qualitative
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map of all possible solution trajectories in the $(x,y)$-plane --- by combining
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local linear analysis near equilibria with global theorems.
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\paragraph{Equilibrium points.}
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As in \cref{sec:ch03_autonomous} and \cref{sec:ch08_phase_plane}, equilibrium
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(singular) points are constant solutions where the system comes to rest.
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\begin{definition}[Equilibrium point]
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An \textbf{equilibrium point} (or \textbf{fixed point}, or \textbf{critical point})
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of \cref{eq:nonlinear_autonomous} is a point $(x^*,y^*)$ satisfying
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\[
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f(x^*,y^*) = 0 \quad\text{and}\quad g(x^*,y^*) = 0.
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\]
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At an equilibrium, both derivatives vanish and the solution is stationary:
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$x(t) \equiv x^*$, $y(t) \equiv y^*$.
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\end{definition}
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Finding equilibria is an algebraic problem: solve the system of two nonlinear
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equations $f = 0$, $g = 0$. Unlike the one-dimensional case (\cref{ch:qualitative}),
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there may be zero, one, or many equilibrium points.
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\begin{workedexample}
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\textbf{Find all equilibrium points of the system}
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\[
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\frac{\mathrm{d}x}{\mathrm{d}t} = x - x^2 - xy, \qquad
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\frac{\mathrm{d}y}{\mathrm{d}t} = y - y^2 - xy.
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\]
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\textbf{Solution.} Set both equations to zero simultaneously:
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\begin{align}
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x(1 - x - y) &= 0, \label{eq:ex_eq1} \\
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y(1 - x - y) &= 0. \label{eq:ex_eq2}
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\end{align}
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From \cref{eq:ex_eq1}, either $x = 0$ or $1 - x - y = 0$.
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\textit{Case 1:} $x = 0$. Substituting into \cref{eq:ex_eq2}: $y(1 - 0 - y) = 0$,
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so $y = 0$ or $y = 1$. This gives equilibria $(0,0)$ and $(0,1)$.
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\textit{Case 2:} $1 - x - y = 0$, i.e.\ $y = 1 - x$. Substituting into
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\cref{eq:ex_eq2}: $y(0) = 0$, which is satisfied for any $y$. But we must
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also have $x = 1 - y$ from the same relation. Setting $x = 0$ gives $y = 1$
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(already found), and setting $y = 0$ gives $x = 1$, yielding the equilibrium $(1,0)$.
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\textit{Verification:} Check $(1,0)$: $f(1,0) = 1 - 1 - 0 = 0$, $g(1,0) = 0 - 0 - 0 = 0$. $\checkmark$
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The system has three equilibrium points: $(0,0)$, $(0,1)$, and $(1,0)$.
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\end{workedexample}
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\subsection{Jacobian Linearization}
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\subsection{Jacobian Linearization}
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\label{sec:ch14_jacobian}
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\label{sec:ch14_jacobian}
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% Content goes here
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The central technique for analyzing nonlinear systems near an equilibrium is
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\textbf{linearization}. Just as in \cref{sec:ch03_stability}, we approximate the
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nonlinear functions by their first-order Taylor expansions near the equilibrium.
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This reduces the nonlinear system to a linear one, whose behavior we understand
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completely from \cref{ch:systems}.
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\paragraph{Derivation via Taylor expansion.}
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Let $(x^*,y^*)$ be an equilibrium. Introduce deviation variables
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$u = x - x^*$ and $v = y - y^*$. Expanding $f$ and $g$ in a
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two-variable Taylor series about $(x^*,y^*)$:
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\begin{align*}
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f(x,y) &= f(x^*,y^*) + \pd{f}{x}\Big|_{(x^*,y^*)} (x-x^*)
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+ \pd{f}{y}\Big|_{(x^*,y^*)} (y-y^*) + O\!\bigl(|u|^2 + |v|^2\bigr), \\
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g(x,y) &= g(x^*,y^*) + \pd{g}{x}\Big|_{(x^*,y^*)} (x-x^*)
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+ \pd{g}{y}\Big|_{(x^*,y^*)} (y-y^*) + O\!\bigl(|u|^2 + |v|^2\bigr).
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\end{align*}
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Since $(x^*,y^*)$ is an equilibrium, $f(x^*,y^*) = g(x^*,y^*) = 0$.
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Discarding the higher-order terms gives the \textbf{linearized system}
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\begin{equation}
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\label{eq:linearized_system}
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\begin{pmatrix} u' \\ v' \end{pmatrix}
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=
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\underbrace{
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\begin{pmatrix}
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\pd{f}{x}\big|_{(x^*,y^*)} & \pd{f}{y}\big|_{(x^*,y^*)} \\[6pt]
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\pd{g}{x}\big|_{(x^*,y^*)} & \pd{g}{y}\big|_{(x^*,y^*)}
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\end{pmatrix}
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}_{\displaystyle = \; J}
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\begin{pmatrix} u \\ v \end{pmatrix}.
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\end{equation}
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\begin{keyresult}
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\textbf{Jacobian matrix and linearization.}
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The \textbf{Jacobian matrix} of the system $x' = f(x,y)$, $y' = g(x,y)$ is
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\[
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J(x,y) =
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\begin{pmatrix}
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\pd{f}{x} & \pd{f}{y} \\[6pt]
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\pd{g}{x} & \pd{g}{y}
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\end{pmatrix}.
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\]
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Evaluated at an equilibrium $(x^*,y^*)$, the linearized system is
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$\begin{pmatrix} u' \\ v' \end{pmatrix} = J(x^*,y^*) \begin{pmatrix} u \\ v \end{pmatrix}$,
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where $u = x-x^*$, $v = y-y^*$. Classify $(x^*,y^*)$ by the eigenvalues
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of $J(x^*,y^*)$ using the phase plane classification from \cref{ch:systems}.
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\end{keyresult}
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\begin{keyresult}
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\textbf{Eigenvalue classification at an equilibrium of a nonlinear system.}
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Let $\lambda_1, \lambda_2$ be the eigenvalues of $J(x^*,y^*)$.
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\begin{center}
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\begin{tabular}{l l p{5cm}}
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\toprule
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\textbf{Eigenvalues} & \textbf{Type} & \textbf{Stability} \\
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\midrule
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Real, both $> 0$ & Unstable node (source) & Unstable \\[4pt]
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Real, both $< 0$ & Stable node (sink) & Asymptotically stable \\[4pt]
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Real, opposite signs & Saddle point & Unstable \\[4pt]
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Complex $\alpha \pm i\beta$, $\alpha > 0$ & Unstable spiral & Unstable \\[4pt]
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Complex $\alpha \pm i\beta$, $\alpha < 0$ & Stable spiral & Asymptotically stable \\[4pt]
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Purely imaginary $\pm i\beta$ & Center (inconclusive) & \textit{Linearization inconclusive} \\
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\bottomrule
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\end{tabular}
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\end{center}
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The center case is the only one where the linearization does \emph{not}
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determine the true nonlinear behavior. See \cref{sec:ch14_hartman_grobman} for details.
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\end{keyresult}
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\begin{workedexample}
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\textbf{Linearize and classify the equilibria of}
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\[
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\frac{\mathrm{d}x}{\mathrm{d}t} = y, \qquad
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\frac{\mathrm{d}y}{\mathrm{d}t} = x - x^2.
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\]
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\textit{(This is the undamped nonlinear pendulum in phase-plane form.)}
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\textbf{Step 1: Find equilibria.}
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Set $y = 0$ and $x - x^2 = 0 \implies x(1-x) = 0$.
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\[
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\text{Equilibria: } (0,0) \text{ and } (1,0).
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\]
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\textbf{Step 2: Jacobian.}
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With $f(x,y) = y$ and $g(x,y) = x - x^2$:
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\[
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\pd{f}{x} = 0, \quad \pd{f}{y} = 1, \quad
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\pd{g}{x} = 1 - 2x, \quad \pd{g}{y} = 0.
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\]
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\[
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J(x,y) =
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\begin{pmatrix}
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0 & 1 \\
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1 - 2x & 0
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\end{pmatrix}.
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\]
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\textbf{Step 3: Classify $(0,0)$.}
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\[
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J(0,0) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.
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\]
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Characteristic equation: $\lambda^2 - 1 = 0 \implies \lambda = \pm 1$.
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Opposite signs $\implies$ \textbf{saddle point} (unstable).
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\textbf{Step 4: Classify $(1,0)$.}
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\[
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J(1,0) = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.
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\]
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Characteristic equation: $\lambda^2 + 1 = 0 \implies \lambda = \pm i$.
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Purely imaginary eigenvalues $\implies$ \textbf{center by linearization}.
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The linearization predicts closed orbits, but the nonlinear behavior
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requires further analysis (confirmed below to be a true center).
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\end{workedexample}
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\begin{workedexample}
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\textbf{Linearize and classify the equilibria of the competing species model}
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\[
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\frac{\mathrm{d}x}{\mathrm{d}t} = 2x - x^2 - xy, \qquad
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\frac{\mathrm{d}y}{\mathrm{d}t} = y - y^2 - xy.
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\]
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\textbf{Step 1: Find equilibria.}
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\begin{align}
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x(2 - x - y) &= 0, \label{eq:comp1} \\
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y(1 - x - y) &= 0. \label{eq:comp2}
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\end{align}
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From \cref{eq:comp1}, either $x = 0$ or $x + y = 2$.
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\textit{Case 1:} $x = 0$. From \cref{eq:comp2}: $y(1-y) = 0$, giving $y = 0$ or $y = 1$.
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Equilibria: $(0,0)$ and $(0,1)$.
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\textit{Case 2:} $x + y = 2$, so $y = 2 - x$. From \cref{eq:comp2}: $y(1 - x - y) = y(1 - 2) = -y = 0$,
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so $y = 0$, which gives $x = 2$. Equilibrium: $(2,0)$.
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The three equilibria are $(0,0)$, $(0,1)$, and $(2,0)$.
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\textbf{Step 2: Jacobian.}
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With $f(x,y) = 2x - x^2 - xy$ and $g(x,y) = y - y^2 - xy$:
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\[
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J(x,y) =
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\begin{pmatrix}
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2 - 2x - y & -x \\[4pt]
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-y & 1 - 2y - x
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\end{pmatrix}.
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\]
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\textbf{Step 3: Classify $(0,0)$.}
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\[
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J(0,0) = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix},
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\quad \lambda_1 = 2,\; \lambda_2 = 1.
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\]
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Both positive $\implies$ \textbf{unstable node (source)}. Both species die out is unstable:
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a small introduction of either population grows.
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\textbf{Step 4: Classify $(0,1)$.}
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\[
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J(0,1) = \begin{pmatrix} 2-1 & 0 \\ -1 & 1-2 \end{pmatrix}
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= \begin{pmatrix} 1 & 0 \\ -1 & -1 \end{pmatrix}.
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\]
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Since $J$ is lower triangular, eigenvalues are the diagonal entries:
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$\lambda_1 = 1$, $\lambda_2 = -1$.
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Opposite signs $\implies$ \textbf{saddle point} (unstable).
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Species~$y$ alone at carrying capacity is unstable to invasion by species~$x$.
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\textbf{Step 5: Classify $(2,0)$.}
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\[
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J(2,0) = \begin{pmatrix} 2-4 & -2 \\ 0 & 1-2 \end{pmatrix}
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= \begin{pmatrix} -2 & -2 \\ 0 & -1 \end{pmatrix}.
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\]
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Upper triangular: $\lambda_1 = -2$, $\lambda_2 = -1$.
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Both negative $\implies$ \textbf{stable node (sink)}.
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Species~$x$ alone at carrying capacity is stable; species~$y$ cannot invade.
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\end{workedexample}
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\subsection{Trace-Determinant Classification}
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\subsection{Trace-Determinant Classification}
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\label{sec:ch14_trace_determinant}
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\label{sec:ch14_trace_determinant}
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% Content goes here
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Just as for linear systems (\cref{sec:ch08_trace_det}), the trace and determinant
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of the Jacobian at an equilibrium provide a quick classification without computing
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eigenvalues explicitly.
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For $J = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ at an equilibrium:
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\[
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\tau = \tr(J) = a + d, \qquad
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\Delta = \det(J) = ad - bc.
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\]
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The eigenvalues satisfy $\lambda^2 - \tau\lambda + \Delta = 0$ with
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discriminant $D = \tau^2 - 4\Delta$.
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\begin{keyresult}
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\textbf{Trace-determinant classification for nonlinear equilibria.}
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Evaluate $\tau = \tr(J)$ and $\Delta = \det(J)$ at the equilibrium point.
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\begin{center}
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\begin{tabular}{l l l}
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\toprule
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\textbf{Region} & \textbf{Conditions} & \textbf{Classification} \\
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\midrule
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$\Delta < 0$ & Below $\tau$-axis & Saddle (unstable) \\[4pt]
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$\Delta > 0$, $\tau > 0$, $D > 0$ & Right, above parabola & Unstable node \\[4pt]
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$\Delta > 0$, $\tau < 0$, $D > 0$ & Left, above parabola & Stable node \\[4pt]
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$\Delta > 0$, $\tau > 0$, $D < 0$ & Right, below parabola & Unstable spiral \\[4pt]
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$\Delta > 0$, $\tau < 0$, $D < 0$ & Left, below parabola & Stable spiral \\[4pt]
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$\Delta > 0$, $\tau = 0$ & On positive $\Delta$-axis & Center (inconclusive) \\
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\bottomrule
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\end{tabular}
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\end{center}
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\end{keyresult}
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\paragraph{The trace-determinant plane.}
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\begin{center}
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\begin{tikzpicture}[scale=0.85]
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% Axes
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\draw[->, thick] (-5,0) -- (5,0) node[right] {$\tau$ (trace)};
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\draw[->, thick] (0,-1) -- (0,5) node[above] {$\Delta$ (det)};
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% Parabola \Delta = \tau^2 / 4
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\draw[thick, dashed, red!80] plot[domain=-4.5:4.5, samples=100, smooth, variable=\t]
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({\t}, {\t*\t/4});
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\node[red!80, font=\footnotesize, anchor=west] at (3.8, 4.3) {$\Delta = \tau^2/4$};
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% Fill saddle region (below \tau-axis)
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\fill[pattern=north west lines, pattern color=red!25] (-5,0) rectangle (5,-0.45);
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||||||
|
% Stability boundary (\tau = 0 line)
|
||||||
|
\draw[dashed, thick, blue!60] (0,0) -- (0,5);
|
||||||
|
|
||||||
|
% Labels for regions
|
||||||
|
\node[font=\footnotesize, align=center] at (0,-0.25) {\textbf{Saddle}\\$(\Delta < 0)$};
|
||||||
|
|
||||||
|
\node[font=\footnotesize, align=center] at (-3,3.5) {\textbf{Stable node}\\$(\tau < 0,\; D > 0)$};
|
||||||
|
\node[font=\footnotesize, align=center] at (3,3.5) {\textbf{Unstable node}\\$(\tau > 0,\; D > 0)$};
|
||||||
|
|
||||||
|
\node[font=\footnotesize, align=center] at (-2,1.1) {\textbf{Stable spiral}\\$(\tau < 0,\; D < 0)$};
|
||||||
|
\node[font=\footnotesize, align=center] at (2,1.1) {\textbf{Unstable spiral}\\$(\tau > 0,\; D < 0)$};
|
||||||
|
|
||||||
|
\node[font=\footnotesize, align=center] at (0,4.5) {\textbf{Center}\\$(\tau = 0,\; \Delta > 0)$};
|
||||||
|
|
||||||
|
% Origin
|
||||||
|
\filldraw[black] (0,0) circle (1.5pt);
|
||||||
|
\node[font=\scriptsize, below left] at (0,0) {$(0,0)$};
|
||||||
|
|
||||||
|
% Parabola tip label
|
||||||
|
\node[font=\scriptsize, red!80, align=center] at (0,0.15) {parabola};
|
||||||
|
\end{tikzpicture}
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
The parabola $\Delta = \tau^2/4$ separates real from complex eigenvalues.
|
||||||
|
The $\Delta$-axis ($\tau = 0$) separates stable from unstable systems.
|
||||||
|
The $\tau$-axis ($\Delta = 0$) separates saddles from nodes/spirals/centers.
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Classify the equilibria of the competing species model from
|
||||||
|
the previous section using the trace-determinant plane.}
|
||||||
|
|
||||||
|
Recall the Jacobian:
|
||||||
|
\[
|
||||||
|
J(x,y) =
|
||||||
|
\begin{pmatrix}
|
||||||
|
2 - 2x - y & -x \\[4pt]
|
||||||
|
-y & 1 - 2y - x
|
||||||
|
\end{pmatrix}.
|
||||||
|
\]
|
||||||
|
|
||||||
|
\textbf{At $(0,0)$:}
|
||||||
|
\[
|
||||||
|
\tau = 2 + 1 = 3 > 0, \qquad \Delta = (2)(1) - (0) = 2 > 0.
|
||||||
|
\]
|
||||||
|
Discriminant: $D = 9 - 8 = 1 > 0$.
|
||||||
|
Region: $\tau > 0$, $\Delta > 0$, $D > 0$ $\implies$ \textbf{unstable node}. Consistent with $\lambda = 2, 1$.
|
||||||
|
|
||||||
|
\textbf{At $(0,1)$:}
|
||||||
|
\[
|
||||||
|
\tau = 1 + (-1) = 0, \qquad \Delta = (1)(-1) - (0) = -1 < 0.
|
||||||
|
\]
|
||||||
|
Region: $\Delta < 0$ $\implies$ \textbf{saddle point}. Consistent with $\lambda = \pm 1$.
|
||||||
|
|
||||||
|
\textbf{At $(2,0)$:}
|
||||||
|
\[
|
||||||
|
\tau = (-2) + (-1) = -3 < 0, \qquad \Delta = (-2)(-1) - (0) = 2 > 0.
|
||||||
|
\]
|
||||||
|
Discriminant: $D = 9 - 8 = 1 > 0$.
|
||||||
|
Region: $\tau < 0$, $\Delta > 0$, $D > 0$ $\implies$ \textbf{stable node}. Consistent with $\lambda = -2, -1$.
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
\subsection{Hartman--Grobman Theorem}
|
\subsection{Hartman--Grobman Theorem}
|
||||||
\label{sec:ch14_hartman_grobman}
|
\label{sec:ch14_hartman_grobman}
|
||||||
|
|
||||||
% Content goes here
|
The linearization tells us about the behavior of the nonlinear system in a
|
||||||
|
\emph{small neighborhood} of the equilibrium. The Hartman--Grobman theorem
|
||||||
|
provides the rigorous justification for this connection.
|
||||||
|
|
||||||
|
\begin{theorem}[Hartman--Grobman]
|
||||||
|
\label{thm:hartman_grobman}
|
||||||
|
Let $(x^*,y^*)$ be an equilibrium point of the autonomous system
|
||||||
|
$x' = f(x,y)$, $y' = g(x,y)$, where $f$ and $g$ are $C^1$ (continuously differentiable).
|
||||||
|
Let $J$ be the Jacobian matrix evaluated at $(x^*,y^*)$.
|
||||||
|
|
||||||
|
If $J$ has \textbf{no eigenvalues with zero real part} (i.e., the equilibrium is
|
||||||
|
\textbf{hyperbolic}), then there exists a neighborhood of $(x^*,y^*)$ in which
|
||||||
|
the nonlinear system is \textbf{topologically conjugate} to its linearization
|
||||||
|
$u' = a u + b v$, $v' = c u + d v$.
|
||||||
|
|
||||||
|
In particular, the local phase portrait of the nonlinear system near $(x^*,y^*)$
|
||||||
|
has the same topological structure as that of the linearized system.
|
||||||
|
\end{theorem}
|
||||||
|
|
||||||
|
\paragraph{What topological conjugacy means.}
|
||||||
|
Two systems are topologically conjugate if there exists a continuous, invertible
|
||||||
|
change of coordinates (a homeomorphism) that maps the trajectories of one system
|
||||||
|
onto the trajectories of the other, preserving the direction of time. Practically,
|
||||||
|
this means:
|
||||||
|
\begin{itemize}
|
||||||
|
\item A stable node of the linearization corresponds to a stable node of the nonlinear system.
|
||||||
|
\item A saddle of the linearization corresponds to a saddle of the nonlinear system.
|
||||||
|
\item The qualitative flow pattern (arrows, attraction, repulsion) is preserved.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
The actual shapes of trajectories may differ --- a nonlinear trajectory near
|
||||||
|
a stable spiral may not be a perfect logarithmic spiral --- but the
|
||||||
|
essential features (spiraling inward, counterclockwise direction, etc.) are identical.
|
||||||
|
|
||||||
|
\paragraph{Limitations: non-hyperbolic equilibria.}
|
||||||
|
The theorem \textbf{does not apply} when the Jacobian has eigenvalues with zero real part:
|
||||||
|
\begin{itemize}
|
||||||
|
\item \textbf{Centers} ($\lambda = \pm i\beta$): The nonlinear system near a center
|
||||||
|
can be a center, a stable spiral, or an unstable spiral. The linearization
|
||||||
|
alone cannot distinguish between these cases. Higher-order terms in the
|
||||||
|
Taylor expansion must be examined.
|
||||||
|
\item \textbf{Non-hyperbolic points} with real zero eigenvalues ($\lambda = 0$):
|
||||||
|
The dynamics can be extremely sensitive to nonlinear terms.
|
||||||
|
Examples include semi-stable equilibria and bifurcation points.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Hartman--Grobman applies: stable spiral.}
|
||||||
|
|
||||||
|
Consider $x' = -x + y + x(x^2+y^2)$, $y' = -x - y + y(x^2+y^2)$.
|
||||||
|
|
||||||
|
\textit{Equilibrium:} $(0,0)$ (the only one easily found).
|
||||||
|
|
||||||
|
\textit{Jacobian at $(0,0)$:}
|
||||||
|
\[
|
||||||
|
J(0,0) = \begin{pmatrix} -1 & 1 \\ -1 & -1 \end{pmatrix},
|
||||||
|
\quad \tau = -2,\; \Delta = 2,\; D = 4-8 = -4 < 0.
|
||||||
|
\]
|
||||||
|
Eigenvalues: $\lambda = -1 \pm i$. Both have nonzero real part
|
||||||
|
($\Re(\lambda) = -1 \neq 0$).
|
||||||
|
|
||||||
|
\textbf{Conclusion:} The equilibrium is hyperbolic. By the Hartman--Grobman
|
||||||
|
theorem, the nonlinear system near $(0,0)$ is a \textbf{stable spiral},
|
||||||
|
topologically equivalent to its linearization.
|
||||||
|
|
||||||
|
(In fact, the nonlinear terms $x(x^2+y^2)$ and $y(x^2+y^2)$ cause
|
||||||
|
outward spiraling for large radii, creating a stable limit cycle ---
|
||||||
|
but near the origin, the local portrait is exactly a stable spiral.)
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Hartman--Grobman does not apply: center vs.\ spiral.}
|
||||||
|
|
||||||
|
Consider the two systems:
|
||||||
|
\begin{align}
|
||||||
|
\text{(A)} \quad x' &= y, \quad y' = -x, \\
|
||||||
|
\text{(B)} \quad x' &= y + x(x^2+y^2), \quad y' = -x + y(x^2+y^2).
|
||||||
|
\end{align}
|
||||||
|
|
||||||
|
\textit{Jacobian at $(0,0)$ for both systems:}
|
||||||
|
\[
|
||||||
|
J(0,0) = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix},
|
||||||
|
\quad \lambda = \pm i.
|
||||||
|
\]
|
||||||
|
Purely imaginary eigenvalues $\implies$ the equilibrium is \textbf{not hyperbolic}.
|
||||||
|
Hartman--Grobman \textbf{does not apply}.
|
||||||
|
|
||||||
|
\textit{System (A):} The linear system. Trajectories are circles $x^2 + y^2 = C$.
|
||||||
|
This is a true \textbf{center}.
|
||||||
|
|
||||||
|
\textit{System (B):} Convert to polar coordinates ($x = r\cos\theta$, $y = r\sin\theta$):
|
||||||
|
\[
|
||||||
|
r' = r^3, \quad \theta' = -1.
|
||||||
|
\]
|
||||||
|
Since $r' = r^3 > 0$ for $r > 0$, the radius strictly increases.
|
||||||
|
The origin is actually an \textbf{unstable spiral}, \emph{not} a center,
|
||||||
|
despite the linearization suggesting a center.
|
||||||
|
|
||||||
|
\textbf{Lesson:} When $\lambda = \pm i\beta$, the nonlinear terms determine
|
||||||
|
the true behavior. Always use additional tools (Lyapunov functions,
|
||||||
|
polar coordinates, first integrals) to resolve non-hyperbolic equilibria.
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
\subsection{Limit Cycles}
|
\subsection{Limit Cycles}
|
||||||
\label{sec:ch14_limit_cycles}
|
\label{sec:ch14_limit_cycles}
|
||||||
|
|
||||||
% Content goes here
|
\begin{definition}[Limit cycle]
|
||||||
|
A \textbf{limit cycle} is a \textbf{closed, isolated periodic orbit}
|
||||||
|
of an autonomous system in the phase plane.
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
\subsection{Lotka--Volterra Predator-Prey}
|
The key word is \emph{isolated}: a limit cycle is a single closed trajectory
|
||||||
|
that is not part of a continuous family of closed orbits. Nearby trajectories
|
||||||
|
either spiral toward the limit cycle (stable limit cycle) or spiral away from
|
||||||
|
it (unstable limit cycle).
|
||||||
|
|
||||||
|
This contrasts with the center in linear systems, where every trajectory near
|
||||||
|
the equilibrium is a closed orbit forming a continuous family.
|
||||||
|
|
||||||
|
\begin{theorem}[Poincar\'{e}--Bendixson]
|
||||||
|
\label{thm:poincare_bendixson}
|
||||||
|
Let $R$ be a closed, bounded (compact) region in the plane containing no
|
||||||
|
equilibrium points, and let a trajectory enter $R$ and remain there for
|
||||||
|
all future time $t > 0$. Then the trajectory either:
|
||||||
|
\begin{enumerate}
|
||||||
|
\item approaches a periodic orbit (a closed trajectory) as $t \to \infty$, or
|
||||||
|
\item is itself a periodic orbit.
|
||||||
|
\end{enumerate}
|
||||||
|
\end{theorem}
|
||||||
|
|
||||||
|
\paragraph{Implications.}
|
||||||
|
The Poincar\'{e}--Bendixson theorem is a powerful tool for proving the
|
||||||
|
existence of limit cycles in two-dimensional systems:
|
||||||
|
\begin{itemize}
|
||||||
|
\item If you can construct a trapping region $R$ that contains no equilibria,
|
||||||
|
any trajectory entering $R$ must approach a periodic orbit.
|
||||||
|
\item In the plane, the only possible long-term behaviors of bounded
|
||||||
|
trajectories are: equilibrium points, periodic orbits (limit cycles),
|
||||||
|
or trajectories that approach limit cycles.
|
||||||
|
\item \textbf{No chaos in 2D autonomous systems:} Strange attractors and
|
||||||
|
chaotic behavior require at least three dimensions.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\paragraph{Physical examples.}
|
||||||
|
Limit cycles model sustained oscillations in physical systems:
|
||||||
|
\begin{itemize}
|
||||||
|
\item \textbf{Electrical circuits:} The van der Pol oscillator models self-sustained
|
||||||
|
oscillations in vacuum tube circuits.
|
||||||
|
\item \textbf{Chemical reactions:} The Belousov--Zhabotinsky reaction exhibits
|
||||||
|
periodic color changes due to a chemical limit cycle.
|
||||||
|
\item \textbf{Biological rhythms:} Heartbeat, neural firing, and circadian
|
||||||
|
rhythms can be modeled as limit cycles.
|
||||||
|
\item \textbf{Mechanical oscillators:} A clock pendulum with a periodic
|
||||||
|
driving force (the escapement mechanism) exhibits a limit cycle.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\subsection{Lotka--Volterra Predator--Prey Model}
|
||||||
\label{sec:ch14_lotka_volterra}
|
\label{sec:ch14_lotka_volterra}
|
||||||
|
|
||||||
% Content goes here
|
The Lotka--Volterra model is the classic example of a nonlinear system with
|
||||||
|
closed orbits, first introduced in the 1920s to model predator--prey
|
||||||
|
population dynamics.
|
||||||
|
|
||||||
|
\paragraph{Biological motivation and model derivation.}
|
||||||
|
Consider two interacting populations:
|
||||||
|
\begin{itemize}
|
||||||
|
\item $x(t)$: number of \textbf{prey} (e.g., rabbits).
|
||||||
|
\item $y(t)$: number of \textbf{predators} (e.g., foxes).
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
We make four biological assumptions:
|
||||||
|
\begin{enumerate}
|
||||||
|
\item In the absence of predators, prey grow exponentially at rate $\alpha$:
|
||||||
|
$x' = \alpha x$.
|
||||||
|
\item Predators eat prey at a rate proportional to encounters $\alpha xy$.
|
||||||
|
Combined with assumption~1: $x' = \alpha x - \beta xy$.
|
||||||
|
\item In the absence of prey, predators die off at rate $\gamma$:
|
||||||
|
$y' = -\gamma y$.
|
||||||
|
\item Predators reproduce at a rate proportional to food consumption $\beta xy$.
|
||||||
|
Combined with assumption~3: $y' = \delta xy - \gamma y$.
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
This yields the \textbf{Lotka--Volterra predator--prey equations}:
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:lotka_volterra}
|
||||||
|
\boxed{
|
||||||
|
\frac{\mathrm{d}x}{\mathrm{d}t} = \alpha x - \beta xy, \qquad
|
||||||
|
\frac{\mathrm{d}y}{\mathrm{d}t} = \delta xy - \gamma y,
|
||||||
|
}
|
||||||
|
\end{equation}
|
||||||
|
where $\alpha, \beta, \gamma, \delta > 0$ are parameters.
|
||||||
|
|
||||||
|
\begin{definition}[Lotka--Volterra model]
|
||||||
|
The Lotka--Volterra predator--prey system is the autonomous nonlinear
|
||||||
|
system \cref{eq:lotka_volterra} with $\alpha, \beta, \gamma, \delta > 0$.
|
||||||
|
It has two equilibrium points:
|
||||||
|
\[
|
||||||
|
(0,0) \text{ (extinction)}, \qquad
|
||||||
|
\left(\frac{\gamma}{\delta},\, \frac{\alpha}{\beta}\right) \text{ (coexistence)}.
|
||||||
|
\]
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
|
\paragraph{Equilibrium analysis.}
|
||||||
|
Setting the right-hand sides to zero:
|
||||||
|
\begin{align}
|
||||||
|
x(\alpha - \beta y) &= 0, \label{eq:lv_eq1} \\
|
||||||
|
y(\delta x - \gamma) &= 0. \label{eq:lv_eq2}
|
||||||
|
\end{align}
|
||||||
|
From \cref{eq:lv_eq1}, $x = 0$ or $y = \alpha/\beta$.
|
||||||
|
|
||||||
|
\textit{Case 1:} $x = 0$. From \cref{eq:lv_eq2}, $y(\,-\gamma) = 0 \implies y = 0$.
|
||||||
|
Equilibrium: $(0,0)$.
|
||||||
|
|
||||||
|
\textit{Case 2:} $y = \alpha/\beta$. From \cref{eq:lv_eq2},
|
||||||
|
$\delta x - \gamma = 0 \implies x = \gamma/\delta$.
|
||||||
|
Equilibrium: $(\gamma/\delta, \alpha/\beta)$.
|
||||||
|
|
||||||
|
\paragraph{Jacobian and classification.}
|
||||||
|
The Jacobian of the system is
|
||||||
|
\[
|
||||||
|
J(x,y) =
|
||||||
|
\begin{pmatrix}
|
||||||
|
\alpha - \beta y & -\beta x \\[6pt]
|
||||||
|
\delta y & \delta x - \gamma
|
||||||
|
\end{pmatrix}.
|
||||||
|
\]
|
||||||
|
|
||||||
|
\textbf{At $(0,0)$:}
|
||||||
|
\[
|
||||||
|
J(0,0) = \begin{pmatrix} \alpha & 0 \\ 0 & -\gamma \end{pmatrix},
|
||||||
|
\quad \lambda_1 = \alpha > 0,\; \lambda_2 = -\gamma < 0.
|
||||||
|
\]
|
||||||
|
Opposite signs $\implies$ \textbf{saddle point} (unstable).
|
||||||
|
Both populations at zero is unstable: a small number of prey grows,
|
||||||
|
triggering predator response.
|
||||||
|
|
||||||
|
\textbf{At $(\gamma/\delta, \alpha/\beta)$:}
|
||||||
|
\[
|
||||||
|
J\!\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right)
|
||||||
|
=
|
||||||
|
\begin{pmatrix}
|
||||||
|
0 & -\beta\gamma/\delta \\[6pt]
|
||||||
|
\delta\alpha/\beta & 0
|
||||||
|
\end{pmatrix}
|
||||||
|
=
|
||||||
|
\begin{pmatrix}
|
||||||
|
0 & -\dfrac{\beta\gamma}{\delta} \\[8pt]
|
||||||
|
\dfrac{\delta\alpha}{\beta} & 0
|
||||||
|
\end{pmatrix}.
|
||||||
|
\]
|
||||||
|
Trace: $\tau = 0$. Determinant: $\Delta = (0) - (-\beta\gamma/\delta)(\delta\alpha/\beta) = \alpha\gamma > 0$.
|
||||||
|
Characteristic equation: $\lambda^2 + \alpha\gamma = 0$, so
|
||||||
|
\[
|
||||||
|
\lambda = \pm i\sqrt{\alpha\gamma}.
|
||||||
|
\]
|
||||||
|
Purely imaginary eigenvalues $\implies$ \textbf{center} by linearization.
|
||||||
|
Hartman--Grobman does not apply. We must analyze the nonlinear system.
|
||||||
|
|
||||||
|
\paragraph{First integral (conservation law).}
|
||||||
|
The Lotka--Volterra system admits an exact first integral, proving that
|
||||||
|
the orbits are indeed closed. Divide the two equations:
|
||||||
|
\[
|
||||||
|
\frac{\mathrm{d}y}{\mathrm{d}x}
|
||||||
|
= \frac{\delta xy - \gamma y}{\alpha x - \beta xy}
|
||||||
|
= \frac{y(\delta x - \gamma)}{x(\alpha - \beta y)}.
|
||||||
|
\]
|
||||||
|
Separate variables:
|
||||||
|
\[
|
||||||
|
\frac{\alpha - \beta y}{y}\,\mathrm{d}y
|
||||||
|
= \frac{\delta x - \gamma}{x}\,\mathrm{d}x.
|
||||||
|
\]
|
||||||
|
Rewrite and integrate:
|
||||||
|
\[
|
||||||
|
\left(\frac{\alpha}{y} - \beta\right)\mathrm{d}y
|
||||||
|
= \left(\delta - \frac{\gamma}{x}\right)\mathrm{d}x.
|
||||||
|
\]
|
||||||
|
\[
|
||||||
|
\alpha \ln y - \beta y = \delta x - \gamma \ln x + C.
|
||||||
|
\]
|
||||||
|
Rearranging gives the \textbf{first integral}:
|
||||||
|
\begin{equation}
|
||||||
|
\label{eq:lv_first_integral}
|
||||||
|
V(x,y) = \delta x - \gamma \ln x + \beta y - \alpha \ln y = \text{constant}.
|
||||||
|
\end{equation}
|
||||||
|
Each value of $C$ defines a closed orbit in the phase plane.
|
||||||
|
The function $V(x,y)$ has a strict global minimum at the coexistence equilibrium
|
||||||
|
$(\gamma/\delta, \alpha/\beta)$, and level curves $V(x,y) = C$ for
|
||||||
|
$C > V_{\min}$ are closed curves surrounding this point.
|
||||||
|
|
||||||
|
\begin{keyresult}
|
||||||
|
\textbf{Lotka--Volterra closed orbits.}
|
||||||
|
The coexistence equilibrium $(\gamma/\delta, \alpha/\beta)$ is a true
|
||||||
|
center for the nonlinear system. Every solution starting in the
|
||||||
|
first quadrant $(x > 0, y > 0)$ traces a closed orbit around
|
||||||
|
the coexistence point, determined by the initial conditions through
|
||||||
|
the first integral \cref{eq:lv_first_integral}.
|
||||||
|
\end{keyresult}
|
||||||
|
|
||||||
|
\paragraph{Nullclines.}
|
||||||
|
The \textbf{nullclines} are curves in the phase plane where one of the
|
||||||
|
derivatives vanishes:
|
||||||
|
\begin{itemize}
|
||||||
|
\item \textbf{$x$-nullcline} ($x' = 0$): $x = 0$ or $y = \alpha/\beta$.
|
||||||
|
On $x = 0$, the prey population is zero.
|
||||||
|
On $y = \alpha/\beta$ (horizontal line), the prey population
|
||||||
|
is momentarily stationary.
|
||||||
|
\item \textbf{$y$-nullcline} ($y' = 0$): $y = 0$ or $x = \gamma/\delta$.
|
||||||
|
On $y = 0$, the predator population is zero.
|
||||||
|
On $x = \gamma/\delta$ (vertical line), the predator population
|
||||||
|
is momentarily stationary.
|
||||||
|
\end{itemize}
|
||||||
|
The two nontrivial nullclines intersect at the coexistence equilibrium
|
||||||
|
$(\gamma/\delta, \alpha/\beta)$. The nullclines divide the first quadrant
|
||||||
|
into four regions, each with a characteristic direction of motion:
|
||||||
|
\begin{itemize}
|
||||||
|
\item Region I ($x > \gamma/\delta, y < \alpha/\beta$): $x' > 0$, $y' > 0$
|
||||||
|
$\implies$ motion up and right.
|
||||||
|
\item Region II ($x < \gamma/\delta, y < \alpha/\beta$): $x' > 0$, $y' < 0$
|
||||||
|
$\implies$ motion down and right.
|
||||||
|
\item Region III ($x < \gamma/\delta, y > \alpha/\beta$): $x' < 0$, $y' < 0$
|
||||||
|
$\implies$ motion down and left.
|
||||||
|
\item Region IV ($x > \gamma/\delta, y > \alpha/\beta$): $x' < 0$, $y' > 0$
|
||||||
|
$\implies$ motion up and left.
|
||||||
|
\end{itemize}
|
||||||
|
The resulting flow is counterclockwise around the equilibrium.
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Full analysis of the Lotka--Volterra system with parameters}
|
||||||
|
$\alpha = 1.5$, $\beta = 1$, $\delta = 3$, $\gamma = 1$.
|
||||||
|
|
||||||
|
\textbf{System:}
|
||||||
|
\[
|
||||||
|
\frac{\mathrm{d}x}{\mathrm{d}t} = 1.5\,x - x y, \qquad
|
||||||
|
\frac{\mathrm{d}y}{\mathrm{d}t} = 3\,x y - y.
|
||||||
|
\]
|
||||||
|
|
||||||
|
\textbf{Equilibria:}
|
||||||
|
\[
|
||||||
|
(0,0) \text{ and } \left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right)
|
||||||
|
= \left(\frac{1}{3}, \frac{1.5}{1}\right) = \left(\frac{1}{3}, 1.5\right).
|
||||||
|
\]
|
||||||
|
|
||||||
|
\textbf{Jacobian:}
|
||||||
|
\[
|
||||||
|
J(x,y) =
|
||||||
|
\begin{pmatrix}
|
||||||
|
1.5 - y & -x \\[6pt]
|
||||||
|
3y & 3x - 1
|
||||||
|
\end{pmatrix}.
|
||||||
|
\]
|
||||||
|
|
||||||
|
\textbf{At $(0,0)$:}
|
||||||
|
\[
|
||||||
|
J(0,0) = \begin{pmatrix} 1.5 & 0 \\ 0 & -1 \end{pmatrix},
|
||||||
|
\quad \lambda_1 = 1.5 > 0,\; \lambda_2 = -1 < 0.
|
||||||
|
\]
|
||||||
|
\textbf{Saddle point} (unstable).
|
||||||
|
|
||||||
|
\textbf{At $(1/3, 1.5)$:}
|
||||||
|
\[
|
||||||
|
J\!\left(\tfrac{1}{3}, 1.5\right)
|
||||||
|
= \begin{pmatrix} 0 & -1/3 \\ 4.5 & 0 \end{pmatrix}.
|
||||||
|
\]
|
||||||
|
\[
|
||||||
|
\tau = 0, \quad \Delta = (0) - (-1/3)(4.5) = 1.5.
|
||||||
|
\]
|
||||||
|
\[
|
||||||
|
\lambda = \pm i\sqrt{1.5} \approx \pm 1.225\,i.
|
||||||
|
\]
|
||||||
|
Purely imaginary eigenvalues. The coexistence point is a \textbf{center}
|
||||||
|
(confirmed by the first integral, not just linearization).
|
||||||
|
|
||||||
|
\textbf{Nullclines:}
|
||||||
|
\begin{itemize}
|
||||||
|
\item $x$-nullcline: $x = 0$ or $y = 1.5$ (horizontal line).
|
||||||
|
\item $y$-nullcline: $y = 0$ or $x = 1/3$ (vertical line).
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\textbf{First integral:}
|
||||||
|
\[
|
||||||
|
V(x,y) = 3x - \ln x + y - 1.5\,\ln y = \text{constant}.
|
||||||
|
\]
|
||||||
|
|
||||||
|
\textbf{Period of oscillation:} Near the equilibrium, the period is
|
||||||
|
approximately $T \approx 2\pi/\sqrt{\alpha\gamma} = 2\pi/\sqrt{1.5}
|
||||||
|
\approx 5.13$ time units.
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
\paragraph{Phase portrait.}
|
||||||
|
|
||||||
|
\begin{center}
|
||||||
|
\begin{tikzpicture}[scale=0.9]
|
||||||
|
% Axes
|
||||||
|
\draw[->, thick] (-0.3,0) -- (5,0) node[right] {$x$ (prey)};
|
||||||
|
\draw[->, thick] (0,-0.3) -- (0,5) node[above] {$y$ (predator)};
|
||||||
|
|
||||||
|
% Nullclines
|
||||||
|
\draw[dashed, thick, blue!60] (1,0) -- (1,4.5) node[above, font=\scriptsize, blue!60] {$x=1/3$};
|
||||||
|
\draw[dashed, thick, red!60] (0,1.875) -- (4.8,1.875) node[right, font=\scriptsize, red!60] {$y=1.5$};
|
||||||
|
|
||||||
|
% Closed orbits (ellipses centered at (1/3, 1.5))
|
||||||
|
% Inner orbit
|
||||||
|
\draw[thick, purple!80] (1,1.875) ellipse (0.6 and 0.6);
|
||||||
|
% Middle orbit
|
||||||
|
\draw[thick, purple!80] (1,1.875) ellipse (1.2 and 1.2);
|
||||||
|
% Outer orbit
|
||||||
|
\draw[thick, purple!80] (1,1.875) ellipse (1.9 and 1.8);
|
||||||
|
|
||||||
|
% Arrows on orbits (counterclockwise)
|
||||||
|
% Inner orbit arrows
|
||||||
|
\draw[thick, purple!80, ->] (1.6,1.875) -- (1.65,2.05);
|
||||||
|
\draw[thick, purple!80, ->] (1,2.475) -- (0.83,2.45);
|
||||||
|
\draw[thick, purple!80, ->] (0.4,1.875) -- (0.4,1.65);
|
||||||
|
\draw[thick, purple!80, ->] (1,1.275) -- (1.17,1.295);
|
||||||
|
|
||||||
|
% Middle orbit arrows
|
||||||
|
\draw[thick, purple!80, ->] (2.2,1.875) -- (2.28,2.12);
|
||||||
|
\draw[thick, purple!80, ->] (1,3.075) -- (0.76,3.05);
|
||||||
|
\draw[thick, purple!80, ->] (-0.2,1.875) -- (-0.2,1.55);
|
||||||
|
\draw[thick, purple!80, ->] (1,0.675) -- (1.24,0.70);
|
||||||
|
|
||||||
|
% Equilibrium points
|
||||||
|
\filldraw[black] (0,0) circle (2.5pt) node[below left=1pt] {$(0,0)$};
|
||||||
|
\filldraw[black] (1,1.875) circle (2.5pt) node[below right=1pt] {$(\gamma/\delta,\,\alpha/\beta)$};
|
||||||
|
|
||||||
|
% Region labels
|
||||||
|
\node[font=\scriptsize, align=center, blue!60] at (2.5,2.8) {Region I\\($x' > 0,\; y' > 0$)};
|
||||||
|
\node[font=\scriptsize, align=center, blue!60] at (0.4,0.9) {Region II\\($x' > 0,\; y' < 0$)};
|
||||||
|
\node[font=\scriptsize, align=center, blue!60] at (-0.5,2.5) {Region III\\($x' < 0,\; y' < 0$)};
|
||||||
|
\node[font=\scriptsize, align=center, blue!60] at (2.5,3.5) {Region IV\\($x' < 0,\; y' > 0$)};
|
||||||
|
|
||||||
|
% Legend
|
||||||
|
\node[font=\footnotesize, align=left, anchor=west] at (2.8,4.2) {
|
||||||
|
\textcolor{blue!60}{\rule{6pt}{1pt}} $x$-nullcline \\
|
||||||
|
\textcolor{red!60}{\rule{6pt}{1pt}} $y$-nullcline \\
|
||||||
|
\textcolor{purple!80}{\rule{6pt}{1pt}} Closed orbits
|
||||||
|
};
|
||||||
|
\end{tikzpicture}
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
\paragraph{Interpretation of the oscillations.}
|
||||||
|
The closed orbits correspond to periodic predator--prey cycles:
|
||||||
|
\begin{enumerate}
|
||||||
|
\item Prey population grows (few predators) $\to$ predators have abundant food.
|
||||||
|
\item Predator population grows $\to$ predation pressure increases.
|
||||||
|
\item Prey population declines $\to$ predators face food shortage.
|
||||||
|
\item Predator population declines $\to$ prey can recover.
|
||||||
|
\item Cycle repeats.
|
||||||
|
\end{enumerate}
|
||||||
|
The amplitude of the oscillation is determined by the initial conditions:
|
||||||
|
the further the initial point from the equilibrium, the larger the orbit.
|
||||||
|
|
||||||
|
\paragraph{Limitations of the model.}
|
||||||
|
The Lotka--Volterra model has several simplifications:
|
||||||
|
\begin{itemize}
|
||||||
|
\item Unlimited prey growth in the absence of predators (no carrying capacity).
|
||||||
|
\item Linear functional response (predation rate proportional to $xy$).
|
||||||
|
\item No time delays in predator reproduction.
|
||||||
|
\item Homogeneous mixing (well-mixed populations).
|
||||||
|
\end{itemize}
|
||||||
|
Realistic extensions include logistic prey growth, Holling-type functional
|
||||||
|
responses, and more complex age-structured models. These modifications can
|
||||||
|
create stable limit cycles (rather than neutral centers) and richer dynamics.
|
||||||
|
|
||||||
\subsection{Summary}
|
\subsection{Summary}
|
||||||
\label{sec:ch14_summary}
|
\label{sec:ch14_summary}
|
||||||
|
|
||||||
% Summary table at end of chapter
|
Nonlinear systems cannot be solved by superposition, but their qualitative
|
||||||
|
behavior near equilibria is accessible through linearization. The Jacobian
|
||||||
|
matrix, combined with the trace-determinant classification, provides a
|
||||||
|
systematic framework for understanding local dynamics. The Hartman--Grobman
|
||||||
|
theorem guarantees that this linear picture is topologically correct for
|
||||||
|
hyperbolic equilibria. Global phenomena like limit cycles require additional
|
||||||
|
tools: the Poincar\'{e}--Bendixson theorem, first integrals, and Lyapunov
|
||||||
|
functions.
|
||||||
|
|
||||||
\begin{table}[htbp]
|
\begin{table}[htbp]
|
||||||
\centering
|
\centering
|
||||||
\caption{Chapter Summary}
|
\caption{Chapter 14 Summary: Nonlinear Systems}
|
||||||
\label{tab:ch14_summary}
|
\label{tab:ch14_summary}
|
||||||
\begin{tabular}{l l}
|
\begin{tabular}{l p{8.5cm}}
|
||||||
\toprule
|
\toprule
|
||||||
\textbf{Concept} & \textbf{Key Formula/Method} \\
|
\textbf{Concept} & \textbf{Key Formula/Method} \\
|
||||||
\midrule
|
\midrule
|
||||||
TBD & TBD \\
|
Nonlinear autonomous system & $\displaystyle x' = f(x,y), \;\; y' = g(x,y)$, no superposition \\
|
||||||
|
Equilibrium point & Solve $f(x^*,y^*) = 0$, $g(x^*,y^*) = 0$ \\
|
||||||
|
Jacobian matrix & $J = \begin{pmatrix} \pd{f}{x} & \pd{f}{y} \\[4pt] \pd{g}{x} & \pd{g}{y} \end{pmatrix}$ \\
|
||||||
|
Linearized system & $\begin{pmatrix} u' \\ v' \end{pmatrix} = J(x^*,y^*) \begin{pmatrix} u \\ v \end{pmatrix}$, $u=x-x^*$, $v=y-y^*$ \\
|
||||||
|
Trace-determinant & $\tau = \tr(J)$, $\Delta = \det(J)$, $D = \tau^2 - 4\Delta$ \\
|
||||||
|
Hartman--Grobman theorem & Hyperbolic equilibria ($\Re(\lambda) \neq 0$): nonlinear $\cong$ linearization \\
|
||||||
|
Non-hyperbolic case & $\Re(\lambda) = 0$: linearization inconclusive; use first integrals, Lyapunov functions \\
|
||||||
|
Limit cycle & Closed, isolated periodic orbit \\
|
||||||
|
Poincar\'{e}--Bendixson theorem & Bounded trajectory in 2D with no equilibria $\implies$ periodic orbit \\
|
||||||
|
Lotka--Volterra model & $x' = \alpha x - \beta xy$, $y' = \delta xy - \gamma y$ \\
|
||||||
|
LV equilibria & $(0,0)$ saddle; $(\gamma/\delta, \alpha/\beta)$ center (closed orbits) \\
|
||||||
|
LV first integral & $V(x,y) = \delta x - \gamma\ln x + \beta y - \alpha\ln y = C$ \\
|
||||||
|
LV nullclines & $x$-nullcline: $y = \alpha/\beta$; $y$-nullcline: $x = \gamma/\delta$ \\
|
||||||
\bottomrule
|
\bottomrule
|
||||||
\end{tabular}
|
\end{tabular}
|
||||||
\end{table}
|
\end{table}
|
||||||
|
|
||||||
|
\begin{hintbox}
|
||||||
|
\textbf{Problem-solving workflow for nonlinear autonomous systems.}
|
||||||
|
\begin{enumerate}
|
||||||
|
\item Find all equilibrium points by solving $f(x,y) = 0$, $g(x,y) = 0$.
|
||||||
|
\item Compute the Jacobian matrix $J(x,y)$ and evaluate it at each equilibrium.
|
||||||
|
\item Classify each equilibrium using eigenvalues or the trace-determinant plane.
|
||||||
|
\item Apply the Hartman--Grobman theorem: if the equilibrium is hyperbolic,
|
||||||
|
the local phase portrait matches the linear classification.
|
||||||
|
\item For non-hyperbolic equilibria ($\Re(\lambda) = 0$), use additional tools:
|
||||||
|
first integrals, Lyapunov functions, or polar coordinates.
|
||||||
|
\item Draw nullclines to understand the flow direction in each region.
|
||||||
|
\item Look for limit cycles using the Poincar\'{e}--Bendixson theorem
|
||||||
|
(requires constructing a trapping region).
|
||||||
|
\item Sketch the full phase portrait combining local and global information.
|
||||||
|
\end{enumerate}
|
||||||
|
\end{hintbox}
|
||||||
|
|||||||
Reference in New Issue
Block a user