More on Riccati equations and fractional linear transformations

In the last post, we saw that solutions of the non-autonomous Riccati equation

\displaystyle  \dot{q} = p_t(q) = a(t) q^2 + b(t) q + c(t) \ \ \ \ \ (1)

are given by fractional linear transformations — that is, if {\Phi(t)} is the map taking the initial condition {q=q_0} to the solution {q(t)} at time {t}, then

\displaystyle  \Phi(t)(q) = \frac{A(t) q + B(t)}{C(t) q + D(t)} \ \ \ \ \ (2)

for some functions {A,B,C,D\colon {\mathbb R}\rightarrow {\mathbb R}} with {AD-BC=1}.

We also saw that if (1) is autonomous — the polynomial {p=p_t} is independent of {t} — then the fixed points of {\Phi(t)} are the roots of {p}, and in particular, the classification of {\Phi(t)} as hyperbolic, parabolic, or elliptic can be deduced from knowledge of the roots of {p}.

Is a similar qualitative analysis available in the non-autonomous case? What if we know that {p_t} always has two real roots, but those roots may move around? Can we still deduce that {\Phi(t)} is a hyperbolic FLT? This question (along with some more sophisticated variants) arose recently in an example of a dynamical system, which is part of a joint project with Dmitry Dolgopyat and Yakov Pesin, for which we would like to understand the evolution of tangent vectors under a particular flow. This evolution turns out to be governed by a non-autonomous Riccati equation where the roots of {p_t} may vary, and one would like to nevertheless have information on the map {\Phi(t)} — in particular, to know where its fixed points are.

By differentiating (2) and using (1), one could obtain ODEs for the coefficients {A,B,C,D}. The problem is that these end up being no easier to solve explicitly than (1), and moreover the information we are most interested in — the presence and location of fixed points of {\Phi(t)} — cannot be read directly from these coefficients. The solution is to derive ODEs that deal directly with the fixed points themselves. This is Proposition 2 below — first it’s worth formulating the qualitative result that can eventually be obtained. The second paragraph of this result contains some technical conditions that can be improved with a little more work.

Proposition 1 Let {p_t(q)} be a time-dependent quadratic polynomial in {q} for {t\in[0,T]}, and let {I_1,I_2\subset {\mathbb R}} be disjoint closed intervals (independent of {t}) such that {p_t} has one root in {I_1} and one root in {I_2} for each {t\in [0,T]}. Suppose that {I_1 < I_2} and let {\Delta} be the length of the interval between them. Suppose also that each {p_t} is decreasing on {I_1} and increasing on {I_2}. (There is an analogous result for concave {p_t}.)

Furthermore, suppose that {J\subset (0,\infty)} is such that {p_t'(q)\in J} for every {q\in I_2} and {p_t'(q)\in -J} for every {q\in I_1}. Let {m<0<M} be the minimum and maximum values, respectively, of {p_t(q)} taken over {t\in[0,T]} and {q\in I_1 \cup I_2}. Suppose that {2m > - \Delta \inf J}, and let {J'=J + [2m/\Delta, 2M/\Delta] \supset J}. (Note that {J'\subset (0,\infty)}.)

Under the above conditions, for every {t\in[0,T]}, the solution map {\Phi(t)} of (1) is a hyperbolic fractional linear transformation with fixed points {w_1\in I_1} and {w_2\in I_2}. Moreover, {D_q(\Phi(t))(w_2) \in e^{tJ'}} and {D_q(\Phi(t))(w_1) \in e^{-tJ'}} for every {t\in[0,T]}.

Observe that a fractional linear transformation is uniquely specified by its two fixed points {w_1,w_2\in \hat{\mathbb C}} and by the derivative {z_j := D_q(\Phi(t))(w_j)} at either one of those fixed points. We will keep track of the four quantities {w_1}, {w_2}, {z_1}, {z_2}. We use the notational convention that {w_j} denotes either {w_1} or {w_2}, and {w_i} denotes the other one of the two.

Proposition 2 The fixed points {w_j} of the FLT {\Phi(t)} evolve according to the ODE

\displaystyle   \dot{w}_j = \frac{p_t(w_j)}{1-z_j}, \ \ \ \ \ (3)

and the derivatives {z_j} at those fixed points evolve according to

\displaystyle   \dot{z}_j = z_j \left( p_t'(w_j) + \frac{2p_t(w_j)}{w_i-w_j}\right), \ \ \ \ \ (4)

whenever the right-hand sides are well-defined.

This will be proved momentarily. First we observe that Proposition 2 is enough to solve autonomous Riccati equations, and give a sketch of how it can be used to establish Proposition 1.

In the autonomous case, we have {p_t(w_j)=0} for all {t}, and so all that remains is to solve (4) in the form {\dot{z}_j = z_j p'(w_j)}, and we get {z_j(t) = e^{tp'(w_j)}}, which is enough information to compute the FLT {\Phi(t)}.

For Proposition 1, the idea is to observe that (4) together with the technical condition in the second paragraph of the proposition guarantees that {\frac{d}{dt} (\log z_2)\in J' \subset (0,\infty)}, and similarly {\frac d{dt} (\log z_1) \in -J'}, as long as {w_j\in I_j}. As long as {z_1 < 1 < z_2}, (3) guarantees that {w_j} moves in the direction of the corresponding root of {p_t}, and hence cannot leave {I_j}.

We end with the proof of Proposition 2.

Proof: The fixed points {w_j\in\hat{\mathbb C}} are the roots of {q = \frac{Aq+B}{Cq+D}}, or equivalently of the quadratic equation

\displaystyle  Cq^2 + (D-A)q - B = 0. \ \ \ \ \ (5)

We will write {\phi(t,q) = \Phi(t)(q)} and {D_t,D_q} for the partial derivatives of {\phi}. Assuming {z_j(t_0) = D_q\phi(t_0,w_j(t_0)) \neq 1}, the fixed points for {t\approx t_0} can be found by observing that

\displaystyle  \begin{aligned} w_j(t) - w_j(t_0)&= \phi(t,w_j(t)) - \phi(t_0,w_j(t_0)) \\ &= (t-t_0)D_t\phi(t_0,w_j(t_0)) + (w_j(t)-w_j(t_0)) D_q\phi(t_0,w_j(t_0)) \\ &\qquad\qquad + o(t-t_0) + o(w_j(t)-w_j(t_0)), \end{aligned}

so that dividing by {t-t_0} and taking the limit gives

\displaystyle  \dot{w}_j = D_t\phi(t_0,w_j(t_0)) + \dot{w}_j \cdot D_q\phi(t_0,w_j(t_0)).

Recalling the ODE (1) and the definition of {z_j} gives (3).

The derivation of the equation for the evolution of {z_j} is a little more involved. First we differentiate {z_j = (D_q\phi)(t,w_j(t))} to get

\displaystyle  \dot{z}_j = (D_{tq}\phi)(t,w_j(t)) + \dot{w}_j(t) \cdot (D_{qq}\phi)(t,w_j(t)). \ \ \ \ \ (6)

The first term is given by

\displaystyle  D_{tq}\phi(t,w_j(t)) = D_q D_t\phi(t,w_j(t)) = D_q p_t(\phi(t,w_j(t))) = D_q\phi(t,w_j(t)) \cdot p_t'(\phi(t,w_j(t))) = z_j p_t'(w_j).

In the second term, {\dot{w}_j} is given by (3), while for {D_{qq}\phi} we need to recall that the derivative of a fractional linear transformation is

\displaystyle  D_q\Phi = D_q\left(\frac{Aq+B}{Cq+D}\right) = \frac 1{(Cq+D)^2},

thanks to the normalisation {AD-BC=1}. Differentiating again gives

\displaystyle  D_{qq}\Phi = -2C(Cq+d)^{-3} = -2C (D_q\Phi)^{3/2},

and so {D_{qq}\phi(t,w_j(t)) = -2C(t) z_j^{3/2}}. Thus we must solve for {C(t)}. Recall that the fixed points are roots of (5), and so their sum and product are given in terms of the coefficients of that equation as

\displaystyle  A-D = C(w_1 + w_2), \qquad -B = Cw_1 w_2. \ \ \ \ \ (7)

Moreover, we have {AD-BC=1} and {z_j^{-1/2} = Cw_j + D}. Substituting {D=z_j^{-1/2} - Cw_j} in the first equation of (7) gives

\displaystyle  A = z_1^{-1/2} + C w_2 = z_2^{-1/2} + C w_1.

Eliminating {B} in the equation for the determinant gives

\displaystyle  AD + C^2w_1w_2 = 1,

and using the above expressions for {A,D}, we have

\displaystyle  (z_1^{-1/2} + Cw_2)(z_1^{-1/2} - Cw_1) + C^2 w_1w_2 = 1,

which simplifies to

\displaystyle  z_1^{-1} + C(w_2 - w_1)z_1^{-1/2} =1,

so that {C} is given in terms of {z_j,w_j} as

\displaystyle  C = \frac{1-z_1^{-1}}{(w_2-w_1)z_1^{-1/2}},

and the same equation holds with the indices reversed. We will write this state of affairs as

\displaystyle  C = \frac{1-z_j^{-1}}{(w_i-w_j)z_j^{-1/2}},

with the understanding that {i} and {j} are the two indices {1} and {2}, in either order. Now returning to (6), we have

\displaystyle  \begin{aligned} \dot{z}_j &= z_j p_t'(w_j) + \frac{p_t(w_j)}{1-z_j}(-2C(t) z_j^{3/2}) \\ &= z_j p_t'(w_j) -2\frac{p_t(w_j)}{1-z_j} z_j^{3/2}\frac{1-z_j^{-1}}{(w_i-w_j)z_j^{-1/2}} \\ &= z_j p_t'(w_j) + \frac{2p_t(w_j)z_j}{w_i-w_j}. \end{aligned}

Thus {z_j} evolves according to the equation (4). \Box

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About Vaughn Climenhaga

I'm an assistant professor of mathematics at the University of Houston. I'm interested in dynamical systems, ergodic theory, thermodynamic formalism, dimension theory, multifractal analysis, non-uniform hyperbolicity, and things along those lines.
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