---
title: Ermakov Mapping in Differential Systems
url: https://www.emergentmind.com/topics/ermakov-mapping
type: topic
---

# Ermakov Mapping in Differential Systems

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Ermakov mapping denotes a class of structurally invariant transformations that relate linear differential equations, Riccati- or Schwarzian-type equations, and nonlinear Ermakov, Ermakov–Pinney, or Painlevé–Ermakov systems. In the literature surveyed here, the term appears in several technically distinct but closely related senses: as the passage from a linear oscillator to an inverse-cubic amplitude equation; as a projective or Schwarzian construction generating Ermakov and Painlevé XXV–Ermakov solutions from a Schwarzian field; as a time-rescaling and coordinate-scaling transformation sending nonautonomous systems to autonomous Hamiltonian ones; and as a reduction of quantum, hydrodynamic, or PDE models to finite-dimensional Ermakov dynamics [2201.02267], [1302.1316], [2109.06276], [2602.00507], [1208.4666]. A common theme is that the nonlinear variable is not introduced ad hoc: it is produced by an invariant structure, typically an Ermakov–Lewis invariant, a Schwarzian derivative, a Liouville normalization, or a Sundman/Arnold-type transformation.

## 1. Core concept and canonical forms

At its most classical, the relevant nonlinear equation is the Ermakov–Pinney equation
\[
u''(z)=B(z)u(z)+\frac{I}{u(z)^3},
\tag{27}
\]
or, equivalently,
\[
\ddot{\rho}+\omega^2(t)\rho=\frac{k}{\rho^3}.
\]
This equation is paired with a linear oscillator,
\[
\eta''(z)=B(z)\eta(z),
\tag{4}
\]
or
\[
\ddot{u}(t)+\omega^2(t)u(t)=0.
\]
The standard nonlinear superposition principle expresses the Ermakov variable as a quadratic form in two linearly independent linear solutions. In one formulation,
\[
\rho^2(q)=A y_1^2(q)+B y_2^2(q)+2D y_1(q)y_2(q),\qquad AB-D^2=\frac{k}{W^2},
\tag{5.1}
\]
where \(W\) is the constant Wronskian of \(y_1,y_2\) [2602.00507]. In another,
\[
x(t)=\Big(Au^2(t)+2B\,u(t)v(t)+Cv^2(t)\Big)^{1/2},
\tag{3}
\]
with
\[
AC-B^2=\frac{L^2}{(\mathrm{Wr}[u,v])^2},
\tag{7}
\]
solves
\[
\ddot{x}(t)+\omega^2(t)x(t)=\frac{L^2}{x^3(t)}.
\tag{1}
\]
These formulas are the prototypical Ermakov mappings from linear solution spaces to nonlinear inverse-cubic dynamics [1609.00248].

The associated invariant is the Ermakov–Lewis invariant. In one standard form,
\[
J(\rho,\dot{\rho},x,\dot{x})
=\frac12\left(\rho\dot{x}-\dot{\rho}x\right)^2
+\frac12\left(\frac{x}{\rho}\right)^2,
\tag{3}
\]
while in separated stationary Bohm–Madelung sectors it becomes
\[
I_i=\frac{1}{2}\left[\big(\rho_i y_i'-\rho_i' y_i\big)^2+\frac{k_i\,y_i^2}{\rho_i^2}\right],
\tag{2.6}
\]
with \(k_i=C_i^2/\hbar^2\) [2205.14577], [2602.00507]. In multidimensional generalized Ermakov systems the invariant assumes Ray–Reid form, for example
\[
I_0=\frac{1}{2}(x\dot{y}-y\dot{x})^2+\int^{y/x}\Big[uF(u)-u^{-3}G(u)\Big]\,du.
\tag{6}
\]

A concise way to describe the mapping principle is therefore: one starts from a linear problem, constructs an amplitude, ratio, or scaled coordinate obeying an inverse-cubic equation, and obtains a conserved quantity that organizes the nonlinear dynamics.

## 2. Linearization, time rescaling, and autonomous reformulations

One major class of Ermakov mappings is built from changes of variables that convert nonautonomous equations into autonomous systems. For the two-dimensional generalized Ermakov system
\[
\ddot{x}=-\omega^2(t)x+x^{-3}F\!\left(\frac{y}{x}\right),
\tag{4}
\]
\[
\ddot{y}=-\omega^2(t)y+y^{-3}G\!\left(\frac{y}{x}\right),
\tag{5}
\]
the transformation
\[
T=\int \rho^{-2}(t)\,dt,\qquad X=\rho^{-1}(t)x,\qquad Y=\rho^{-1}(t)y,
\tag{7}
\]
with \(\rho\) solving
\[
\ddot{\rho}+\omega^2(t)\rho=0,
\tag{8}
\]
maps the original nonautonomous system into the autonomous equations
\[
X''=X^{-3}F\!\left(\frac{Y}{X}\right),
\tag{9}
\qquad
Y''=Y^{-3}G\!\left(\frac{Y}{X}\right).
\tag{10}
\]
The invariant becomes time-independent:
\[
I_0=\frac{1}{2}(XY'-YX')^2+\int^{Y/X}\Big[uF(u)-u^{-3}G(u)\Big]\,du.
\tag{11}
\]
This is the central nonautonomous-to-autonomous Ermakov mapping in the geometric treatment of generalized conservative systems [2109.06276].

Within the conservative subclass, the autonomous system is identified with a Hamiltonian system possessing
\[
H=\frac12(X'^2+Y'^2)+\frac{N(u)}{X^2},\qquad u=\frac{Y}{X},
\tag{17}
\]
provided
\[
F(u)=2N(u)+u\frac{dN}{du},\qquad G(u)=-u^3\frac{dN}{du}.
\tag{16}
\]
This embeds the generalized Ermakov equations into the known class of integrable two-dimensional autonomous conservative systems with Euclidean kinetic metric [2109.06276].

A distinct, but structurally similar, mapping arises in the Arnold framework. For a linear second-order ODE
\[
\ddot{x}(t)+\dot{f}(t)\dot{x}(t)+\omega^2(t)x(t)=\Lambda(t),
\tag{1}
\]
the Classical Arnold Transformation sends it to the free equation \(\ddot{\kappa}(\tau)=0\) by
\[
\tau=\frac{u_1(t)}{u_2(t)},\qquad \kappa=\frac{x-u_p(t)}{u_2(t)},
\tag{4}
\]
where \(u_1,u_2\) are canonical homogeneous solutions [1302.1316]. Composing two such maps yields the Arnold–Ermakov–Pinney transformation
\[
x_1=\frac{x_2}{b(t_2)},\qquad W_1(t_1)\,dt_1=\frac{W_2(t_2)}{b(t_2)^2}\,dt_2,
\tag{14}
\]
with \(b(t_2)\) satisfying the generalized Ermakov–Pinney equation
\[
\ddot{b}+\dot{f}_2\dot{b}+\omega_2^2(t_2)b
=\frac{W_2^2(t_2)}{b^3}\,\omega_0^2
\tag{17}
\]
when the target system is a constant-frequency oscillator [1302.1316]. Here the mapping function itself is the Ermakov variable.

## 3. Schwarzian and projective formulations

A projective formulation of Ermakov mapping is developed through the Schwarzian derivative
\[
\{f,z\}:=\frac{f'''(z)}{f'(z)}-\frac{3}{2}\left(\frac{f''(z)}{f'(z)}\right)^2.
\tag{3}
\]
If \(\eta_1,\eta_2\) are linearly independent solutions of
\[
\eta''(z)=B(z)\eta(z),
\tag{4}
\]
then their quotient \(\Omega=\eta_1/\eta_2\) satisfies
\[
\{\Omega,z\}=-2B(z).
\tag{5}
\]
This linear–Schwarzian correspondence is the backbone of the mapping structure in the Schwarzian treatment of Ermakov and Painlevé XXV–Ermakov equations [2201.02267].

From a solution \(\Omega\) of the Schwarzian equation, one obtains in closed form:
\[
w(z)=\frac{\Omega(z)}{\Omega'(z)},
\tag{98}
\]
which solves the third-order linear equation
\[
w'''-4Bw'-2B'w=0,
\tag{107}
\]
\[
u(z)=2r\,\frac{\Omega(z)}{\Omega'(z)},
\tag{99}
\]
which solves the Ermakov equation
\[
u''=B(z)u-\frac{r^2}{u^3},
\tag{109}
\]
and
\[
y(z)=-\frac{3}{2}\frac{\Omega'(z)}{\Omega(z)},
\tag{100}
\]
which solves the \(A=0\) Painlevé XXV–Ermakov equation
\[
y''y-\frac{5}{4}y'^2+2y^3+4B(z)y=0.
\tag{111}
\]
This identifies the Schwarzian equation as a master equation for linear, Ermakov, and Painlevé XXV–Ermakov dynamics [2201.02267].

The Möbius invariance of the Schwarzian,
\[
\left\{\frac{Af+B}{Cf+D},x\right\}=\{f,x\},\qquad AD-BC\neq 0,
\]
lifts to full four-parameter solution families:
\[
w(z)=\frac{(a\Omega+b)(c\Omega+d)}{(ad-bc)\,\Omega'},
\tag{108}
\]
\[
u(z)=2r\,\frac{(a\Omega+b)(c\Omega+d)}{(ad-bc)\,\Omega'},
\tag{110}
\]
\[
y(z)=-\frac{3}{2}\frac{(ad-bc)\,\Omega'}{(a\Omega+b)(c\Omega+d)}.
\tag{112}
\]
Accordingly, the mapping is many-to-one at the level of \(\Omega\), but one-to-one at the level of Möbius-equivalence classes [2201.02267].

The same work also provides a direct linear-to-Painlevé XXV–Ermakov map:
\[
y(z)=\frac{3w''}{w}-\frac{3}{2}\left(\frac{w'}{w}\right)^2-6B(z),
\tag{25}
\]
when \(w\) solves
\[
w'''-4Bw'-2B'w+\frac{A}{3}w=0.
\tag{24}
\]
When \(A=0\), the first integral
\[
ww''-(w')^2-2B(z)w^2=2I
\tag{26}
\]
and the substitution \(w=u^2\) recover the classical Ermakov equation
\[
u''=B(z)u+\frac{I}{u^3}.
\tag{27}
\]
This is the paper’s explicit formulation of the classical Ermakov mapping [2201.02267].

## 4. Geometric, multidimensional, and Riemannian extensions

In higher dimensions, Ermakov mappings are closely tied to homothetic geometry and \(SL(2,\mathbb{R})\) symmetry. For the \(n\)-dimensional Hamiltonian Ermakov system,
\[
L=\frac12\Big[\dot{\rho}^{2}+\rho^{2}(\dot{\theta}_1^{2}+\sin^{2}\theta_1\,\dot{\theta}_2^{2}+\cdots)\Big]
-\frac{1}{\rho^{2}}V(\theta_1,\dots,\theta_{n-1}),
\tag{45}
\]
the potential has the characteristic form \(\rho^{-2}V(\text{angles})\), and the system always admits an \(SL(2,\mathbb{R})\) algebra of Noether symmetries generated by the homothetic structure of flat space [2205.14577]. The corresponding Ermakov invariant is obtained as a Casimir-like combination of the Noether integrals; in two dimensions,
\[
J(\rho,\dot{\rho})=\frac14 I_3^2 H-I_2^2
=\frac12\dot{\rho}^{2}+\frac12\rho^{-2}.
\tag{24--25}
\]

The classification of generalized \(n\)-dimensional Hamiltonian Ermakov systems with additional conservation laws is expressed in Cartesian form by potential families such as
\[
V(x_1,\dots,x_n)=\sum_i V_i x_i^{-2},
\tag{47}
\]
along with inverse-square hyperplane generalizations of the type described in equations (48)–(49) of the same work [2205.14577]. A central conclusion is that Ermakov invariants and additional linear or quadratic first integrals are determined by Killing vectors, Killing tensors, and the homothetic vector of the background Euclidean metric.

A further generalization replaces Euclidean space by a Riemannian manifold admitting a gradient homothetic vector. In adapted coordinates the metric is
\[
ds^2=du^2+u^2 h_{AB}(y)\,dy^A dy^B,
\]
and the autonomous Hamiltonian Riemannian Kepler–Ermakov Lagrangian is
\[
L=\frac12\left(u'^2+u^2 h_{AB}y'^A y'^B\right)-V(u,y),
\]
with
\[
V(u,y)=\frac{1}{u^2}V'(y)
\]
for the polynomial \(sl(2,\mathbb{R})\) representation, or
\[
V(u,y)=-\frac{\mu^2}{2}u^2+\frac{1}{u^2}V'(y)
\]
for the exponential one [1205.4114]. The Riemannian Ermakov invariant is
\[
J=u^4 h_{AB}y'^A y'^B + 2V'(y),
\]
and the radial equation becomes an Ermakov–Pinney equation in \(u\). In this setting, the “mapping” is from a dynamical system on a manifold with gradient homothety to a radial inverse-cubic equation plus angular dynamics encoded by \(h_{AB}\) and \(V'(y)\).

This geometric picture is used to embed specific cosmological models into Riemannian Kepler–Ermakov systems. In particular, a scalar-field cosmology with exponential potential and stiff fluid, and the \(f(R)\) model
\[
f(R)=(R-2\Lambda)^{3/2},
\]
are shown to reduce to autonomous Hamiltonian Riemannian Kepler–Ermakov systems that are Liouville integrable via Noether integrals [1205.4114]. A plausible implication is that Ermakov mapping, in this geometric sense, functions as a symmetry-adapted coordinate reduction for minisuperspace dynamics.

## 5. Quantum-mechanical realizations

In quantum mechanics, Ermakov mapping appears both in time-dependent and stationary settings. For the one-dimensional harmonic oscillator
\[
i\hbar\partial_t\psi
=\left(-\frac{\hbar^2}{2m}\partial_x^2+\frac{m\omega_0^2x^2}{2}\right)\psi,
\tag{1}
\]
writing \(\psi=\rho e^{iS/\hbar}\) and imposing a Gaussian density leads to the width equation
\[
m\ddot{\sigma}+m\omega_0^2\sigma=\frac{\hbar^2}{4m\sigma^3}.
\tag{5}
\]
This reduces the full Schrödinger dynamics of Gaussian states to a single Ermakov equation for the width \(\sigma(t)\), with \(\sigma(t)\) interpreted as the rms displacement or wave-packet width [1002.2545]. Dissipative and thermal extensions preserve the inverse-cubic quantum term while adding friction, thermal, or radiative contributions, for example
\[
m\ddot{\sigma}+b\dot{\sigma}+m\omega_0^2\sigma=\frac{\hbar^2}{4m\sigma^3},
\tag{7}
\]
\[
m\ddot{\sigma}+b\dot{\sigma}+m\omega_0^2\sigma=\frac{k_BT}{\sigma}+\frac{\hbar^2}{4m\sigma^3},
\tag{11}
\]
and
\[
m\ddot{\sigma}-r\dddot{\sigma}+m\omega_0^2\sigma=\frac{\hbar^2}{4\sigma^3}.
\tag{12}
\]
Here the mapping is from a PDE for \(\psi(x,t)\) to a nonlinear ODE for a single amplitude variable [1002.2545].

A stationary analogue appears in Bohm–Madelung quantum mechanics for diagonal, separable Hamiltonians. After separation and Liouville normalization, each sector satisfies a linear normal-form equation
\[
y_i''(q_i)+\Omega_i^2(q_i)\,y_i(q_i)=0,
\tag{2.5}
\]
while the stationary continuity constraint implies the nonlinear companion
\[
\rho_i''(q_i)+\Omega_i^2(q_i)\rho_i(q_i)=\frac{k_i}{\rho_i^3(q_i)},
\tag{2.4}
\]
with \(k_i=C_i^2/\hbar^2\) [2602.00507]. The corresponding Ermakov–Lewis invariant is coordinate-constant:
\[
I_i=\frac{1}{2}\left[\big(\rho_i y_i'-\rho_i' y_i\big)^2+\frac{k_i\,y_i^2}{\rho_i^2}\right].
\tag{2.6}
\]
The mapping here is from a stationary Schrödinger sector to a linear Sturm–Liouville problem plus a nonlinear Ermakov–Pinney amplitude equation.

The Quantum Arnold Transformation provides another quantum Ermakov mapping. The Quantum Arnold Transformation
\[
\varphi(\kappa,\tau)
=
A^*\!\left[
\sqrt{u_2(t)}\,
\exp\!\left(-\frac{i}{2}\frac{m}{\hbar}\frac{1}{W(t)}\frac{\dot{u}_2(t)}{u_2(t)}x^2\right)\phi(x,t)
\right]
\tag{10}
\]
maps solutions of a time-dependent quadratic Schrödinger equation to free-particle solutions, while the Quantum Arnold–Ermakov–Pinney transformation
\[
\varphi(x_1,t_1)
=
E^*\!\left[
\sqrt{b(t_2)}\,
\exp\!\left(-\frac{i}{2}\frac{m}{\hbar}\frac{1}{W_2(t_2)}\frac{\dot{b}(t_2)}{b(t_2)}x_2^2\right)
\phi(x_2,t_2)
\right]
\tag{18}
\]
maps between two quadratic systems, with \(b\) solving a generalized Ermakov–Pinney equation [1302.1316].

The same theme appears in time-dependent oscillator invariants. If
\[
w(t)=\xi(t)e^{-i\theta(t)},\qquad 2\xi^2\dot{\theta}=1,
\tag{11--12}
\]
then \(\xi\) satisfies
\[
\ddot{\xi}+\omega^2\xi=\frac{1}{4\xi^3},
\tag{13}
\]
and the Lewis–Riesenfeld invariant is
\[
\hat I_0(t)=\frac{1}{2}\big(\xi p-\dot{\xi}q\big)^2+\frac{1}{2}\left(\frac{q}{\xi}\right)^2.
\tag{15}
\]
In this formulation, choosing \(\xi(t)\) and then reconstructing \(\omega^2(t)\) yields an inverse Ermakov mapping from amplitude profiles to exactly solvable time-dependent oscillators [1609.00248].

A photonic realization implements the same structure experimentally. In a semi-infinite waveguide array, the lattice Hamiltonian is mapped to
\[
\hat H=
-\left[
\frac{1}{2M(z)}\hat p^2+\frac12 M(z)\Omega^2(z)\hat q^2
+\sqrt{2}\,\alpha_1(z)\hat q-\frac{\alpha_0(z)}{2}
\right],
\]
with
\[
M(z)=\frac{1}{\alpha_0(z)-2\alpha_2(z)},
\qquad
\Omega^2(z)=\alpha_0^2(z)-4\alpha_2^2(z).
\]
After displacement and squeezing transformations, the dynamics are governed by the Ermakov–Lewis invariant and the generalized Ermakov equation for \(\rho(t)\) [1403.1498]. This is an optical realization of an Ermakov-mapped time-dependent harmonic oscillator.

## 6. PDE reductions, Painlevé hybridization, and nonlocal transforms

Ermakov mapping also operates at the level of nonlinear PDEs and nonlocal transformations. In one direction, generalized Sundman transformations
\[
X(T)=F(t,x),\qquad dT=G(t,x)\,dt,
\tag{3.3}
\]
are used to map generalized Liénard-type equations
\[
\ddot{x}+A_2(t;x)\dot{x}^2+A_1(t;x)\dot{x}+A_0(t;x)=0
\tag{3.1}
\]
to the linear oscillator
\[
X''(T)+\omega^2X(T)=0.
\tag{3.2}
\]
The pullback of the oscillator energy gives the invariant
\[
I(t,x,\dot{x})
=
\left(\frac{F_t+F_x\dot{x}}{G}\right)^2+\omega^2F^2.
\tag{3.8}
\]
Specializations of the resulting mapped equations include dissipative generalized Ermakov–Pinney systems such as
\[
\ddot{x}
+\frac{\dot{a}}{a}\dot{x}
-2\left(\frac{\dot{a}}{a}\right)^2x
+\frac{\omega}{a x^3}=0,
\tag{3.36}
\]
with invariant
\[
I_{\mathrm{gEP}}(t,x,\dot{x})
=
\frac{\dot{x}^2}{a}
+\frac{\dot{a}}{a^2}x\dot{x}
+\omega^2(ax)^2.
\tag{3.36}
\]
Coupled Liénard systems are similarly mapped to coupled dissipative Ermakov–Milne–Pinney equations with coupled Ermakov-type invariants [1905.00610].

In integrable PDE theory, Ermakov–Painlevé reductions provide another extension. The temporally modulated extended mKdV equation
\[
u_t-6u^2u_x+u_{xxx}+\lambda (t+a)^\mu u^{-4}u_x=0
\tag{2.1}
\]
admits the similarity ansatz
\[
u(x,t)=(t+a)^m\Psi(\xi),\qquad \xi=\frac{x}{(t+a)^n},
\tag{2.2}
\]
and for
\[
m=-\frac13,\qquad n=\frac13,\qquad \mu=-2,
\]
the reduction becomes
\[
\Psi''-2\Psi^3-\frac13\xi\Psi-\frac{\lambda}{3}\Psi^{-3}=\zeta.
\tag{2.5}
\]
With \(\zeta=0\) and an additional scaling, this is brought to the canonical Ermakov–Painlevé II equation
\[
w^*_{zz}=2w^{*3}+zw^*+\frac{\delta}{w^{*3}}.
\tag{2.8}
\]
This PDE-to-ODE reduction is explicitly described as an Ermakov mapping in the moving-boundary analysis of extended mKdV equations [2511.03356].

A closely related \(2+1\)-dimensional construction starts from the temporally modulated modified Kadomtsev–Petviashvili-type equation
\[
U_t + U_{xxx} - 3U^2U_x - 3U_x\partial_x^{-1}U_y
+ \delta^* U \partial_x^{-1}U_{yy}
+ \lambda (t+a)^\mu U^{-4}U_x = 0,
\tag{2}
\]
and, under
\[
U(x,y,t)=(t+a)^{-1/3}\Psi\!\left(\frac{x+\alpha^*y}{(t+a)^{1/3}}\right),
\]
reduces it to the same Ermakov–Painlevé II structure [2603.16520]. In that setting, involutory transformations
\[
dt^*=\rho^{-2}(t)\,dt,\qquad
U^*(x^*,y^*,t^*)=\frac{U(x,y,t)}{\rho(t)},\qquad
\rho^*(t^*)=\frac{1}{\rho(t)}
\tag{25}
\]
with \(\rho^*\) satisfying the classical Ermakov equation
\[
\rho^*_{t^*t^*}+\omega(t^*)\rho^*
=
\frac{\mathcal E}{\rho^{*3}}
\tag{28}
\]
generate a broad class of temporally modulated PDEs that preserve the Ermakov–Painlevé II reduction.

A hydrodynamic example is supplied by the \(2+1\)-dimensional anisotropic non-isothermal magnetogasdynamic system. Under an elliptic vortex ansatz for the density and a linear ansatz for the velocity, the PDEs reduce to a finite-dimensional dynamical subsystem; after the introduction of a scaling variable \(\Omega\), the semi-axes \(\Phi,\Psi\) of the elliptical cross-section satisfy an Ermakov–Ray–Reid system [1208.4666]. The Hamiltonian of the semi-axis dynamics,
\[
H=\frac{1}{2}(\dot{\Phi}^2+\dot{\Psi}^2)
-\frac{1}{2(\Phi^2+\Psi^2)}
\left[
Z^2-\frac{f^2}{4}(\Phi^2+\Psi^2)^2+\frac{k}{4}
\right],
\tag{5.2}
\]
is the relevant Ermakov-type invariant. In this context, the mapping is from a \(2+1\)-dimensional plasma model to a Hamiltonian Ermakov–Ray–Reid subsystem.

## 7. Invariants, symmetries, and the role of solution generation

Across these formulations, the central organizing objects are invariants and symmetry groups. In classical Ermakov systems the invariant is quadratic in a Wronskian-like combination; in generalized multidimensional systems it is tied to \(SL(2,\mathbb{R})\) or to homothetic algebras; in the Schwarzian setting it is tied to Möbius invariance; in quantum mechanics it becomes a Lewis–Riesenfeld invariant operator; and in Painlevé hybridizations it is converted into a Painlevé parameter [2201.02267], [2205.14577], [1302.1316], [2602.00507], [1703.02282].

Bäcklund transformations are especially important in the Schwarzian and Painlevé-based settings. In the Schwarzian approach, one family is Wronskian-based: if \(w_1,w_2\) solve
\[
w'''-4Bw'-2B'w=0,
\tag{57}
\]
then their Wronskian
\[
w=W(w_1,w_2)=w_1w_2'-w_2w_1'
\tag{58}
\]
is also a solution, and this induces Bäcklund transformations for corresponding Ermakov equations through \(w=u^2\) [2201.02267]. A second family is Möbius/Schwarzian-based: if
\[
\Omega(f(z))=\frac{a\Omega(z)+b}{c\Omega(z)+d},
\tag{69}
\]
then
\[
u_1(z)=\frac{u_0(f(z))}{f'(z)}
\tag{70}
\]
maps one Ermakov solution to another.

In Ermakov–Painlevé IV systems, the invariant
\[
\mathcal I=\frac12(\Phi_1\Phi_2'-\Phi_2\Phi_1')^2
+\left(\frac{\Phi_1^2+\Phi_2^2}{\Phi_1^2}\right)J\!\left(\frac{\Phi_2}{\Phi_1}\right)
\tag{7}
\]
enters the scalar reduction
\[
\Sigma''=\frac{1}{2\Sigma}(\Sigma')^2+\frac{3}{2}\Sigma^3+4z\Sigma^2+2(z^2-\alpha)\Sigma+\frac{\beta}{\Sigma},
\tag{10}
\]
through the identification \(\beta=4\mathcal I\) [1703.02282]. The map
\[
(\Phi_1,\Phi_2)\mapsto \Sigma=\Phi_1^2+\Phi_2^2
\]
is therefore an Ermakov mapping from a coupled Ermakov system to Painlevé IV, and the Bäcklund transformations of Painlevé IV generate new coupled Ermakov solutions.

Taken together, these constructions show that Ermakov mapping is less a single transformation than a family of structurally related correspondences. What persists across all variants is the passage from a linear or symmetry-reduced description to a nonlinear amplitude equation with inverse-power structure, together with an invariant that survives the transformation and controls reconstruction of the original variables.

Source: https://www.emergentmind.com/topics/ermakov-mapping