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In a two-variable calcium-wave model, the fast front scale can reduce to the displayed dynamical system. Its equilibria are intersections of the kinetic nullcline $V=g(U)$ and a line of slope $-1/\gamma$. At a simple intersection with $f_V<0$, the Jacobian determinant is $f_V(1+\gamma g^{\prime })$; a negative determinant identifies a saddle point. Matching a rising front requires a heteroclinic orbit from one saddle to another, selected by a shooting method in the speed $s$. A falling connection must use the same speed. Between fronts the slow equation is $s(1+\gamma g^{\prime })U^{\prime }=\mu -U$. Recovery on the two outer rising branches requires the resting value $\mu$ to lie between the low landing value and the high takeoff value. The nullcline shape supplies this geometry but does not by itself prove existence of both connections.

For the travelling wave coordinate $\xi =x+st$, the chain rule givesFast-slow matching for a calcium pulse joins two fast transitions, $B\to C$ and $D\to A$, joined by slow portions $C\to D$ and $A\to B$. Write the nullcline as $V=g(U)$. The resting point has $U_B=\mu$, $V_B=g(\mu )$ in the leading matched description.

Put $\xi =\epsilon X$. Multiplying the first equation by $\epsilon$ gives $U_{XX}=s(U_X+\gamma V_X)+\epsilon (U-\mu )$. At leading order, integrate and use $U_X\to 0$, $(U,V)\to (U_B,V_B)$ at the start of the front. Together with the second equation this givesThe fast equilibria are intersections of $V=g(U)$ and $V=V_B-(U-U_B)/\gamma$. As the slope $-1/\gamma$ changes, there may be one or three simple intersections, with two distinct intersections at a tangency. At a simple point of the nullcline, $f_U=-f_V{g}^{\prime }$, and the sign convention above/below it gives $f_V<0$. The Jacobian matrix of the fast system hasFor the usual three-intersection configuration, the two outer points have $1+\gamma g^{\prime }>0$, hence negative determinant and one positive and one negative eigenvalue: they are saddle points. The middle point has positive determinant; it is linearly stable if $s^2<-f_V$ and unstable if $s^2>-f_V$. Zero trace is a possible oscillatory stability threshold, with a nonlinear classification requiring further information about $f$.

To select the front speed, follow the appropriate branch of the unstable manifold of $B$ and vary $s$ until it meets the stable manifold of $C$. This is a shooting method for a heteroclinic orbit. Follow the high-$U$ slow branch from $C$ to $D$; the falling front obeys the same fast equations with the integration constant anchored at $D$. Choose $D$ so that its descending saddle connection reaches a low-$U$ point $A$ at that same speed. The remaining slow branch returns toward the resting point, completing the matched spike. The construction requires the indicated connections; a nullcline sketch alone does not specify their numerical speed or prove their existence for every possible kinetic function $f$.

On the slow portions, $f(U,V)=0$ at leading order, so $V=g(U)$ and $V^{\prime }=g^{\prime }(U)U^{\prime }$. Neglecting the small diffusion term yieldsOn the outer rising branches the coefficient on the left is positive. Recovery from $C$ toward $D$ requires decreasing $U$, hence $\mu <U_D$; the low branch from $A$ back toward rest requires increasing $U$, hence $U_A<\mu$. Therefore $\boxed{U_A<\mu <U_D}$. These inequalities distinguish the recovery branches from a trajectory that would move away from the required matching points.

Introduce $v=u^{\prime }$ and solve the initial-value systemThe unknown initial slope $s$ is the shooting method parameter. Integrate to one and solve the scalar residual equation $R(s)=u(1;s)-\beta =0$, using a secant iteration or a bracketed root finder when suitable data are available.

For a Newton update, integrate the sensitivity equations along the current shot:Then $R^{\prime }(s)=w(1)$ and the update is $s\leftarrow s-R(s)/w(1)$ when this derivative is nonzero. After a root has been found, the corresponding shot satisfies both endpoint conditions. This formulation does not presume that a nonlinear boundary-value problem necessarily has a unique solution.

For the second-order boundary value problem, introduce an unknown initial slope $s$ and solve the initial value problem $u^{\prime }=v$, $v^{\prime }=g(x,u,v)$ with $u(a)=\alpha$, $v(a)=s$. Its endpoint mismatch is $F(s)=u(b;s)-\beta$. The shooting method seeks a root $F(s)=0$; in general the dependence of the flow on $s$ is nonlinear. A sign-changing bracket for a continuous $F$ allows the bisection method, provided the intervening initial value solutions exist up to $b$.

For Newton's method, solve the parameter sensitivity equation alongside the trajectory. With $z={\partial }_{s}u$ and $w={\partial }_{s}v$,Then $F^{\prime }(s)=z(b)$ and the update is $\boxed{s_{\mathrm{new}}=s-F(s)/z(b)}$ whenever $z(b)\ne 0$.

For the third-order problem, shooting from $a$ leaves both $u^{\prime }(a)$ and $u^{\prime \prime }(a)$ unknown, requiring two shooting parameters to satisfy two right-endpoint conditions. Shooting backwards from $b$ already knows $u(b)=\beta$ and $u^{\prime }(b)=\gamma$; only $u^{\prime \prime }(b)$ is unknown, with the single residual $u(a)-\alpha$. Backward shooting reduces the nonlinear solve from two scalar unknowns to one. This parameter-count advantage does not by itself guarantee numerical stability of backward integration.