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

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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