# Dynamics of a generalized Rayleigh system

Maíra Duran Baldissera∗ ∗Departamento de Matemática, Instituto de Ciências
Matemáticas e Computação, Universidade de São Paulo, Avenida Trabalhador São
Carlense, 400, 13566-590, São Carlos, SP, Brazil.
maira.baldissera<EMAIL_ADDRESS>,  Jaume Llibre∗∗ ∗∗Departament
de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra,
Barcelona, Catalonia, Spain<EMAIL_ADDRESS>and  Regilene Oliveira∗

###### Abstract.

Consider the first order differential system given by

$\begin{array}[]{l}\dot{x}=y,\qquad\dot{y}=-x+a(1-y^{2n})y,\end{array}$

where $a$ is a real parameter and the dots denote derivatives with respect to
the time $t$. Such system is known as the generalized Rayleigh system and it
appears, for instance, in the modeling of diabetic chemical processes through
a constant area duct, where the effect of adding or rejecting heat is
considered. In this paper we characterize the global dynamics of this
generalized Rayleigh system. In particular we prove the existence of a unique
limit cycle when the parameter $a\neq 0$.

###### Key words and phrases:

Rayleigh system; limit cycles; averaging theory; Poincaré compactification.

###### 2010 Mathematics Subject Classification:

Primary: 34C07, 34C05, 37G15

The differential system here studied modelizes a diabetic chemical processes
through a constant area duct, where the effect of adding or rejecting heat is
considered. Always is important to know if the model exhibits or not periodic
motions. This model depens on a paramater, when this parameter is sufficiently
small but non-zero it is knwon that the system has a limit cycle (an isolated
periodic orbit), but it is not knwon its uniqueness, its kind of stability and
all the values of the parameteer for which such a limit cycle exist. Here we
solve all these questions. Moreover we describe all the global dynamics of the
model controlling the orbits which come from or escape at infinity.

##  1\. Introduction

Consider the non-linear second order differential equation

$\stackrel{{\scriptstyle..}}{{x}}+x=a(1-\dot{x}^{2n})\dot{x},$ (1)

where $n$ is a positive integer and $a\in\mathbb{R}$ and the dots denote
derivatives with respect to the time $t$. This equation is equivalent to the
following first order differential system

$\begin{array}[]{l}\dot{x}=y,\\\ \dot{y}=-x+a(1-y^{2n})y,\end{array}$ (2)

that is known as the generalized Rayleigh differential equation (the Rayleigh
equation was introduced by Lorde Rayleigh in [5], for more details about the
differential system (1) see [3]). Such system appears, for instance, in the
modeling of diabetic chemical processes through a constant area duct, where
the effect of adding or rejecting heat is considered.

The aim of this study is to investigate the global dynamics of system (1) in
the Poincaré disc, and in particular, investigate the existence and uniqueness
of the limit cycles of the differential system (1) for diferent values of
$a\in\mathbb{R}$. Applying the change of variables $(x,y,t)\to(y,x,-t)$,
system (2) becomes

$\begin{array}[]{l}\dot{x}=y+a(x^{2n}-1)x,\\\ \dot{y}=-x,\end{array}$ (3)

that is a $1$-parameter Liénard differential system already studied by Lins-
Neto, Melo and Pugh in [4] and by López and Martínez in [3]. The main results
of this paper are the following.

######  Theorem 1.

The generalized Rayleigh differential system (2) has a unique limit cycle if
$a\neq 0$. When the limit cycle exists it is stable if $a<0$ and unstable if
$a>0$. For $a=0$ the system is a linear center.

######  Theorem 2.

For the generalized Rayleigh differential system (2) we obtain three non-
topologically equivalent phase portraits on the Poincaré disc, they are
described in Figure 1. The bifurcation set of the phase portraits in the
parameter space is $a=0$.

Figure 1. Phase portraits of the generalized Rayleigh systems in the Poincaré
disc.

##  2\. Preliminary results

###  2.1. Basic concepts

Let $\mathbb{R}[x,y]$ be the set of all polynomials in the variables $x$ and
$y$ with real coefficients. Associated to the polynomial differential system

$\begin{array}[]{cc}\dot{x}=P(x,y),&\\\ \dot{y}=Q(x,y),&\end{array}$ (4)

where $P,Q\in\mathbb{R}[x,y]$, it is the vector field
$\mathcal{X}(x,y)=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$
or, simply $\mathcal{X}=(P,Q)$ in $\mathbb{R}^{2}$. The degree of the vector
field $\mathcal{X}$ is $n=\max(\deg{P},\deg{Q})$.

The orbit $\gamma_{p}$ of the vector field $\mathcal{X}$ through the point $p$
is the image of the maximal solution
$\phi_{p}:I_{p}\in\mathbb{R}\to\mathbb{R}^{2}$ endowed with an orientation,
moreover, only one of the following statements holds.

  1. (1)

$\phi_{p}$ is a injection onto its image (i.e. the orbit is in correspondence
with an interval of $\mathbb{R}$;

  2. (2)

$I_{p}=\mathbb{R}$ and $\phi_{p}$ is constant (i.e. the orbit is a point);

  3. (3)

$I_{p}=\mathbb{R}$ and $\phi_{p}$ is a periodic function, i.e, there is a
constant $\tau$ such that $\phi_{p}(t+\tau)=\phi_{p}(t)$, for any $t$ and
$\phi_{p}$ is non constant (in this case the orbit is diffeomorphic to a
circle).

######  Definition 1.

1\. A singular point (or a singularity) of the vector field $X=(P,Q)$ is a
point $p\in\mathbb{R}^{2}$ such that $\phi_{p}$ is constant i.e.
$P(p)=Q(p)=0$. The points of $\mathbb{R}^{2}$ which are not singular are
called regular points.

2\. A periodic orbit of the vector field $X=(P,Q)$ is an orbit diffeomorphic
to a circle.

An isolated periodic orbit in the set of all periodic orbits of a planar
differential system is a limit cycle. In recent years a variety of methods
were used to investigate the existence of limit cycles. In this paper we shall
see that the Poincaré-Bendixson Theorem could be an useful tool in such
investigation.

Let $p$ be a singular point of the vector field $\mathcal{X}=(P,Q)$. Denote by
$\Delta(p)=P_{x}(p)Q_{y}(p)-P_{y}(p)Q_{x}(p)$ and $\Gamma=P_{x}(p)+Q_{y}(p)$,
where $F_{a}$ denotes the derivative of the function $F(a,b)$,
$a,b\in\mathbb{R}$, with respect to the variable $a$.

######  Definition 2.

An isolated singular point $p$ is said to be non-degenerate if $\Delta(p)\neq
0$.

It is known that $p$ is a saddle if $\Delta<0$, a node if
$\Gamma^{2}-4\Delta>0$ (stable if $\Gamma<0$, unstable if $\Gamma>0$), a focus
if $\Gamma^{2}-4\Delta<0$ (stable if $\Gamma<0$, unstable if $\Gamma>0$), and
either a weak focus or a center if $\Gamma=0$ and $\Delta>0$ (see Theorem 2.15
of [2] for more details about non-degenerate singular points).

######  Definition 3.

An isolated singular point $p$ of $\mathcal{X}=(P,Q)$ is a hyperbolic singular
point if the eigenvalues of the Jacobian matrix $J\mathcal{X}(p)$, defined by
the first derivative of $P$ and $Q$ with respect to the variables $x$ and $y$,
have both nonzero real part.

We note that the hyperbolic singular points are non-degenerate singular points
except by the weak foci and centers.

######  Definition 4.

If $p$ is an isolated singular point of the vector field $\mathcal{X}$ such
that $\Delta(p)=0$ and $\Gamma\neq 0$ then $p$ is called a semi-hyperbolic
singular point.

The next result describes the local phase portraits of the semi-hyperbolic
singular points. For more details about the next proposition see Theorem 2.19
of [2].

######  Proposition 3.

Let $(0,0)$ be an isolated singular point of the vector field $\mathcal{X}$
given by

$\begin{array}[]{l}\dot{x}=A(x,y),\\\ \dot{y}=\lambda y+B(x,y),\end{array}$

where $A$ and $B$ are analytic in a neighborhood of the origin starting with,
at least, degree $2$, in the variables $x$ and $y$. Let $y=f(x)$ be the
solution of the equation $\lambda y+B(x,y)=0$ in a neighborhood of the point
$(0,0)$, and suppose that the function $g(x)=A(x,f(x))$ has the expression
$g(x)=ax^{\alpha}+o(x^{\alpha})$, where $\alpha\geq 2$ and $a\neq 0$. So, when
$\alpha$ is odd, $(0,0)$ is either an unstable node or saddle, depending if
$a>0$ or $a<0$, respectively. In the saddle’s case, the separatices are
tangent to the $x$-axis. If $\alpha$ is even, the $(0,0)$ is a saddle-node,
i.e., the singular point is formed by the union of two hyperbolic sectors with
one parabolic sector. The stable separatrix is tangent to the positive
(respectively, the negative) $x$-axis at $(0,0)$, according to $a<0$
(respectively, $a>0$). The two unstable separatices are tangent to the
$y$-axis at $(0,0)$.

###  2.2. Limit sets

Let $\phi_{p}(t)=(x(t),y(t))$ be the maximal solution of a planar differential
system defined on the interval $(\alpha_{p},\omega_{p})$ such that
$\phi_{p}(0)=p$. If $\omega_{p}=\infty$ we define the $\omega$–limit set of
$p$ as

$\omega(p)=\\{q\in\mathbb{R}^{2}:\exists\\{t_{n}\\}\mbox{ with
}t_{n}\to\infty\mbox{ and }\phi_{p}(t_{n})\to q\mbox{ when }n\to\infty\\}.$

Analogously, if $\alpha_{p}=-\infty$ we define the $\alpha$–limit set of $p$
as

$\alpha(p)=\\{q\in\mathbb{R}^{2}:\exists\\{t_{n}\\}\mbox{ with
}t_{n}\to-\infty\mbox{ and }\phi_{p}(t_{n})\to q\mbox{ when }n\to\infty\\}.$

In what follows we denote by $\gamma_{p}^{+}$ the positive semi-orbit of a
differential vector field $\mathcal{X}$ passing through the point $p$. The
next result is known as Poincaré-Bendixson Theorem and it can be used to
guarantee the existence of periodic orbits in compact sets, for a proof see
Theorem 1.25 of [2].

######  Theorem 4.

Let $\phi_{p}(t)$ be the solution of the vector field $\mathcal{X}$ defined
for all $t\geq 0$ and such that $\gamma_{p}^{+}=\\{\phi_{p}(t):t\geq 0\\}$ is
contained in a compact set $K$. Assume that the vector field $\mathcal{X}$ has
at most a finite number of singular points in $K$. Then one of the following
statements holds.

  1. (1)

If $\omega(p)$ contains only regular points then $\omega(p)$ is a periodic
orbit.

  2. (2)

If $\omega(p)$ contains both regular and singular points, then $\omega(p)$ is
formed by a set of orbits, every one of which tends to one singular point in
$\omega(p)$ when $t\to\pm\infty$.

  3. (3)

If $\omega(p)$ does not contain regular points, then $\omega(p)$ is a unique
singular point.

###  2.3. Poincaré compactification

In this section we describe the behavior of the orbits of the planar vector
field studied in the so called Poincaré disc. Roughly speaking, the Poincaré
disc is the unitary and closed disc centered at the origin of the coordinates.
The interior of such disc is identified with the plane $\mathbb{R}^{2}$, the
circle $\mathbb{S}^{1}$, the boundary of the disc, is identified with the
infinity of the plane. Consequently, in $\mathbb{R}^{2}$ we can go to infinity
in as many directions as points in $\mathbb{S}^{1}$. See Chapter 5 of [2] for
details about the Poincaré compactification.

Identifying the plane $\mathbb{R}^{2}$ with the plane of $\mathbb{R}^{3}$ of
the form $(y_{1},y_{2},y_{3})=(x_{1},x_{2},1)$, consider the sphere
$\mathbb{S}^{2}=\\{y\in\mathbb{R}^{3}:y_{1}^{2}+y_{2}^{2}+y_{3}^{2}=1\\}$,
called the Poincaré sphere. $\mathbb{S}^{2}$ is tangent to $\mathbb{R}^{2}$ in
the point $(0,0,1)$. The sphere $\mathbb{S}^{2}$ can be divided in three
pieces: $H_{+}$, $H_{-}$ and $S^{1}$, where
$H_{+}=\\{y\in\mathbb{S}^{2}:y_{3}>0\\}$ the upper hemisphere,
$H_{-}=\\{y\in\mathbb{S}^{2}:y_{3}<0\\}$ is the lower hemisphere and
$\mathbb{S}^{1}=\\{y\in\mathbb{S}^{2}:y_{3}=0\\}$ is the equator of the sphere
$\mathbb{S}^{2}$.

Consider the central projection of the vector field $\mathcal{X}$ defined in
$\mathbb{R}^{2}$ over $\mathbb{S}^{2}$ given by the central projections
$f^{+}:\mathbb{R}^{2}\rightarrow\mathbb{S}^{2}$ and
$f^{-}:\mathbb{R}^{2}\rightarrow\mathbb{S}^{2}$. More precisely, $f^{+}$
(respectively, $f^{-}$) is the intersection of the straight line that passes
through the point $y$ and the origin with the upper hemisphere (respectively,
lower) of the sphere $\mathbb{S}^{2}$. The central projections can be written
as

$f^{+}(x)=\left(\frac{x_{1}}{\Delta(x)},\frac{x_{2}}{\Delta(x)},\frac{1}{\Delta(x)}\right),\qquad
f^{-}(x)=\left(-\frac{x_{1}}{\Delta(x)},-\frac{x_{2}}{\Delta(x)},-\frac{1}{\Delta(x)}\right),$

where $\Delta(x)=\sqrt{x_{1}^{2}+x_{2}^{2}+1}$.

From the central projections we obtain vector fields induced in the upper and
lower hemisphere of the sphere. The vector field induced in $H_{+}$ is defined
by

$\overline{X}(y)=Df^{+}(x)X(x),$

where $y=f^{+}(x)$, and in $H_{-}$ is given by
$\overline{X}(y)=Df^{-}(x)X(x)$, where $y=f^{-}(x)$. Notice that
$\overline{X}$ is a vector field in $\mathbb{S}^{2}{\setminus}\mathbb{S}^{1}$
that is tangent to $\mathbb{S}^{2}$ in each point.

Multiply the induced vector field $\overline{X}$ by the factor
$\rho(x)=y_{3}^{d-1}$, with $d$ equal to degree of the polynomial vector
field$X$, the obtained vector field is a vector field defined in whole sphere
$\mathbb{S}^{2}$ and it is known as the Poincaré compactification of the
vector field $\mathcal{X}$ in $\mathbb{S}^{2}$. We denote it by
$p(\mathcal{X})$.

In order to describe the analytic expression of the vector field
$p(\mathcal{X})$ we consider the local charts
$U_{k}=\\{y\in\mathbb{S}^{2}:y_{k}>0\\}$,
$V_{k}=\\{y\in\mathbb{S}^{2}:y_{k}<0\\},$ for $k=1,2,3$ and the corresponding
local maps $\phi_{k}:U_{k}\rightarrow\mathbb{R}^{2}$ and
$\psi_{k}:V_{k}\rightarrow\mathbb{R}^{2}$ defined by
$\phi_{k}(y)=\psi_{k}(y)=(y_{m}/y_{k},y_{n}/y_{k}),$ for $m<n$ and $m,n\neq
k$. Denote $\phi_{k}(y)=\psi_{k}(y)=(u,v)$ for each $k$, this implies that
$(u,v)$ has diferent roles depending on each local chart. Figure 2 is a
geometric representation of the local coordinates of $(u,v)$ in each local
chart. We observe that the points of $\mathbb{S}^{1}$ in each local chart have
their coordinate $v=0$.

Figure 2. The local charts $(U_{k},\phi_{k})$ with $k=1,2,3$ for the Poincaré
sphere.

Consequently, if $\mathcal{X}(x)=(P(x_{1},x_{2}),Q(x_{1},x_{2}))$, then
$\overline{X}(y)=Df^{+}(x)X(x)$ with $y=f^{+}(x)$ and

$D\phi_{1}(y)\overline{X}(y)=D\phi_{1}(y)\circ Df^{-}(x)X(x)=D(\phi_{1}\circ
f^{+})(x)X(x).$

Then $(\phi\circ
f^{+})(x)=\left(\dfrac{x_{2}}{x_{1}},\dfrac{1}{x_{1}}\right)=(u,v),$ and

$\overline{X}|_{U_{1}}:=D\phi_{1}(y)\overline{X}(y)=\left(\begin{array}[]{cc}\displaystyle{x_{2}}/{x_{1}}&\displaystyle{1}/{x_{1}\vspace*{0.2cm}}\\\
\displaystyle-1/{x_{1}^{2}}&0\end{array}\right)\left(\begin{array}[]{c}P(x_{1},x_{2})\\\
Q(x_{1},x_{2})\end{array}\right),$
$=\displaystyle{\frac{1}{x_{1}^{2}}(-x_{2}P(x_{1},x_{2})+Q(x_{1},x_{2}),-P(x_{1},x_{2}))},$
$=\displaystyle{v^{2}\left(-\frac{u}{v}P\left(\frac{1}{v},\frac{u}{v}\right)+\frac{1}{v}Q\left(\frac{1}{v},\frac{u}{v}\right),-P\left(\frac{1}{v},\frac{u}{v}\right)\right)}.$

Moreover, if $m(z)=(1+u^{2}+v^{2})^{\frac{1-d}{2}}$, as
$\rho(y)=y_{3}^{d-1}=\dfrac{1}{\Delta(x)^{d-1}}=\dfrac{v^{d-1}}{\Delta(z)^{d-1}}=v^{d-1}m(z),$
we have

$\rho(\overline{X}|_{U_{1}})(z)=v^{d+1}m(z)\left(-\frac{u}{v}P\left(\frac{1}{v},\frac{u}{v}\right)+\frac{1}{v}Q\left(\frac{1}{v},\frac{u}{v}\right),-P\left(\frac{1}{v},\frac{u}{v}\right)\right).$

After a reparametrization of the time we eliminate the factor $m(z)$ from the
compactified vector field and the analytic expression of the vector field
$p(X)$ in the local chart $(U_{1},\phi_{1})$ is

$\displaystyle\dot{u}=v^{d}\left(-uP\left(\frac{1}{v},\frac{u}{v}\right)+Q\left(\frac{1}{v},\frac{u}{v}\right)\right),\qquad\dot{v}=-v^{d+1}P\left(\frac{1}{v},\frac{u}{v}\right).$

Analogously, in the local chart $(U_{2},\phi_{2})$ we have

$\displaystyle\dot{u}=v^{d}\left(P\left(\frac{u}{v},\frac{1}{v}\right)-uQ\left(\frac{u}{v},\frac{1}{v}\right)\right),\qquad\dot{v}=-v^{d+1}Q\left(\frac{u}{v},\frac{1}{v}\right),$

and, in the chart $(U_{3},\phi_{3})$

$\displaystyle\dot{u}=P(u,v),\qquad\dot{v}=Q(u,v).$

We remark that the expression of the vector field $p(X)$ in the remaning local
charts $(V_{k},\psi_{k})$ is the same as $(U_{k},\phi_{k})$ except by
multiplication by the factor $(-1)^{d-1}$, for $k=1,2,3$.

The Poincaré disc $\mathbb{D}$ is the projection of the closed hemisphere
$H^{+}$ under $(y_{1},y_{2},y_{3})\to(y_{1},y_{2})$. In the Poincaré disc the
finite singular points (respectively, infinite) of $X$ or $p(X)$ are the
singular points of $p(X)$ that are in $\mathbb{S}^{2}\setminus\mathbb{S}^{1}$
(respectively, $\mathbb{S}^{1}$). It is important to remark that, if
$y\in\mathbb{S}^{1}$ is an infinite singular point, then $-y$ is also an
infinite singular point and that the local phase portrait of $-y$ is the local
phase portrait of $y$ multiplied by $(-1)^{d-1}$, it follows that the
orientation of the orbits changes when the degree is even. Due to the fact
that infinite singular points appear in pair of points diametrally opposite,
it is enough to study the local phase portrait of only half of the infinite
points, and using the degree of the vector field, it is possible to determine
the other half.

Finally we introduce the concept of topologically equivalent vector fields.
Let ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$ be two polynomial vector
fields on $\mathbb{R}^{2}$, and let $p({\mathcal{X}}_{1})$ and
$p({\mathcal{X}}_{2})$ be their respective polynomial vector fields on the
Poincaré disc $\mathbb{D}$. We say that they are topologically equivalent if
there exists a homeomorphism on the Poincaré disc $\mathbb{D}$ which preserves
the infinity ${\mathbb{S}}^{1}$ and sends the orbits of $p({\mathcal{X}}_{1})$
to orbits of $p({\mathcal{X}}_{2})$, preserving the orientation of all the
orbits.

###  2.4. Liénard systems and limit cycles

One important kind of differential systems in the context of this paper are
the so called Liénard systems, that is a differential equation of the form

$\frac{d^{2}x}{dt^{2}}+f(x)\frac{dx}{dt}+g(x)=0,$

where $f,g\in C^{1}(\mathbb{R})$. Applying the change of coordinate
$y=\dot{x}+F(x)$, where $F(x)=\int_{0}^{x}f(s)ds$, the Liénard equation is
equivalent to the planar system

$\frac{dx}{dt}=-\phi(y)-F(x),\qquad\frac{dy}{dt}=g(x).$ (5)

We denote by $Lip$ the set of functions $f:I\subset\mathbb{R}\to\mathbb{R}$,
such $f$ is a Lipschitz continuous function, that is, for each $f\in Lip$
there is a real constant $M$ such as

$||f(x)-f(y)||\leq M||x-y||,$

for all $x,y\in I$, where $I$ is an interval of $\mathbb{R}$.

The next theorem was proved in [6] and it will be one of the fundamental tools
in the proof of existence and uniqueness of the limit cycles for the
generalized Rayleigh systems.

######  Theorem 5.

If the following conditions are satisfied for system (5)

  1. (1)

$g(x)\in$ Lip in any finite interval; $xg(x)>0,\text{ }x\neq 0;\text{
}G(-\infty)=G(+\infty)$, where $G(x)=\int_{0}^{x}g(s)ds$,

  2. (2)

$F^{\prime}(x)=f(x)\in C^{0}(-\infty,+\infty);\text{ }F(0)=0;\text{
}\displaystyle\frac{f(x)}{g(x)}$ is non-increasing when $x$ grows in
$(-\infty,0)$ and $(0,+\infty);\text{ }\displaystyle\frac{f(x)}{g(x)}$ is non-
constant when $0<|x|<<1,$

  3. (3)

$\phi(y)\in$ Lip in any finite interval, $y\phi(y)>0,\text{ }y\neq 0,\text{
}\phi(y)$ is non-decreasing, $\phi(y)$ has left and right-derivatives,
$\phi^{\prime}_{+}(0)$ and $\phi^{\prime}_{-}(0)$ in the point $y=0$,
$\phi^{\prime}_{+}(0).\phi^{\prime}_{-}(0)\neq 0$ when $f(0)=0$.

then it has at most one limit cycle. If such limit cycle exists then it is
stable.

##  3\. Proof of Theorem 1

In this section we investigate the local behavior of the Rayleigh system at
infinity, i.e. near $\mathbb{S}^{1}$, the boundary of the Poincaré disc.

######  Proposition 6.

The local phase portraits at the infinite singular points of system (3) in the
Poincaré disc is topologically equivalent to the one described in Figure 3.

Figure 3. Case $a<0$.

###### Proof.

The Poincaré compactification of system (3) in the local chart $U_{1}$ is

$\begin{array}[]{l}\dot{u}=-au+v^{2n}(au-1-u^{2}),\\\
\dot{v}=-av+v^{2n+1}(a-u).\end{array}$

Doing $v=0$ we conclude that the origin of the local chart $(U_{1},\phi_{1})$
is the unique singular point at this local chart and the Jacobian matrix in
$(0,0)$ is

$\left[\begin{array}[]{cc}-a&0\\\ 0&-a\end{array}\right].$

So the origin from the local chart $U_{1}$ is an atractor node if $a>0$ and it
is a repeller node if $a<0$.

In the local chart $(U_{2},\phi_{2})$ the compactified vector field (3) is
given by

$\begin{array}[]{l}\dot{u}=au^{2n+1}+v^{2n}(1-au+u^{2}),\\\
\dot{v}=uv^{2n+1}\end{array}$

Then the origin of the local chart $U_{2}$ is a degenerate singular point and,
to investigate its local phase portrait we apply the vertical directional blow
up, see [1] for more details about blow ups. Using the change of variables
$u=x$ and $v=zx$, and the reparametrization of time, that eliminate the common
factor $x^{2n-1}$ of the two components of $\dot{x}$ and $\dot{z}$, we get the
system

$\begin{array}[]{l}\dot{x}=ax^{2}+xz^{2n}(1-ax+x^{2}),\\\
\dot{z}=-z(ax+z^{2n}-az^{2n}x),\end{array}$

Where, again, the origin is the unique singular point of the system and it is
degenerate so we apply the blow up once again, this time we take the change of
coordinates $x=w^{2n}r$ and $z=w$, that is equivalent to sucessive blow ups.
After a reparametrization of the time that eliminate the common factor
$w^{2n}$ from both equations, the system is written as

$\begin{array}[]{l}\dot{r}=r((2n+1)(1+ar)-w^{2n}(a(2n+1))r+r^{2}w^{2n}),\\\
\dot{w}=w(-1+ar(-1+w^{2n}))\end{array}$ (6)

Notice that the system (6) has two critical points, $(0,0)$ and $(-1/a,0)$.
The Jacobian matrix at $(0,0)$ is

$\left[\begin{array}[]{cc}1+2n&0\\\ 0&-1\end{array}\right],$

and so $(0,0)$ is a saddle point. The Jacobian matrix at $(-1/a,0)$ is

$\left[\begin{array}[]{cc}-(1+2n)&0\\\ 0&0\end{array}\right],$

then the critical point $(-1/a,0)$ is a semi-hyperbolic point. After the
reparametrization $r\rightarrow s-1/a,t\rightarrow-t$, the conditions of
Proposition 3 are satisfied, so, checking that $m$ is odd and $a_{m}<0$, we
conclude that $(-1/a,0)$ is a saddle point. Applying the blowing down, as
represented in Figure 4, we conclude the local behaviour of the origin of the
local chart $U_{2}$.

Figure 4. Blow up process to the system (6) when $a<0$.

Bringing together the local information about the infinite singular points in
both local charts and using the continuity of the solutions we conclude that
the orbits of system (3) near of $\mathbb{S}^{1}$ are topologically equivalent
to the one described in Figure 3, except by the reversing of all orbits when
$a\neq 0$. See that for $a=0$ system (2) is a linear center. ∎

######  Remark 7.

We point out that in [4] the statement equivalent to this last proposition has
a misprint. The authors say that if $a>0$ the infinity singular point in the
local chart $U_{1}$ is repeller when in fact it is an atractor. The same for
$a<0$ to be an atractor when it is a repeller.

The next result describes the local phase portrait of system (3) at the finite
singular points.

######  Lemma 8.

System (3) has a unique finite singular point, the origin which is a stable
node if $a\geq 2$, an unstable node if $a\leq-2$, a unstable focus if
$-2<a<0$, a stable focus if $0<a<2$ and a center if $a=0$.

###### Proof.

We can easily verify that the unique critical point of system (3) is the
origin and the Jacobian matrix of system (3) at the origin is given by

$\left[\begin{array}[]{cc}-a&1\\\ -1&0\end{array}\right].$

So the lemma follows from the local study described in Section 2. ∎

Next theorem guarantee the existence of at least one limit cycle for system
(3).

######  Theorem 9.

The generalized Rayleigh system (3) has exactly one limit cycle if $a\neq 0$.

###### Proof.

Assume that $a<0$ then from Lemma 8 there exists a unique singular point of
system (3), the origin and it is a repellor. Moreover, from Proposition 6, we
conclude that each solution of system (3) is moving away from $\mathbb{S}^{1}$
in the Poincaré disc.

As the Poincaré disc is a compact set with a unique singular point, it follows
from the Poincaré–Bendixson Theorem (Theorem 3) that there exists at least one
limit cycle.

The study of the case $a>0$ is analogous. ∎

Now we have each ingredient to prove Theorem 1.

###### Proof of the Theorem 1..

Assuming $a>0$ and applying in (3) the change of coordinates
$(x,y,t)\rightarrow(x,y,-t)$ (that only changes the orientation of the
orbits), we obtain

$\dot{x}=-y+a(1-x^{2n})x,\qquad\dot{y}=x.$

Taking $g(x)=x$, it is immediate that $g\in Lip$ in any finite interval of the
real line and $xg(x)>0$, for $x\neq 0$. Moreover,

$\int_{0}^{-\infty}sds=\int_{0}^{+\infty}sds=+\infty.$

If $f(x)=F^{\prime}(x)=-a(1-x^{2n})+2n\,a\,x^{2n}=-a+a(2n+1)x^{2n}$, then
$f(x)$ is a real valued continuous function defined in the real line such that
$F(0)=0$. Therefore

$\dfrac{f(x)}{g(x)}=\dfrac{-a+a(2n+1)x^{2n}}{x}$

is a non-decreasing real function in the intervals $(-\infty,0)$ and
$(0,+\infty)$ because the derivative of $f(x)/g(x)$ for $x\neq 0$ is positive,
and also it is a non-constant function in any neighborhood of the origin.

When $\phi(y)=y$, $y\phi(y)=y^{2}>0$ for all values of $y\neq 0$, $\phi(y)\in
Lip$ in any finite interval of the real line and $\phi(y)$ is non-decreasing
function.

In short, each one of the hypothesis of Theorem 5 is satisfied and the
existence of a unique limit cycle to system (3) when $a>0$ is guaranted.

The same result is valid when $a<0$ because system (3) with $a>0$ is
topologically equivalent to system (3) with $a<0$, through the change of
coordinates $(x,y,t)\rightarrow(-x,y,-t)$. So Theorem 1 is proved. ∎

######  Remark 10.

The existence of at least one limit cycle for system (3) when the parameter
$a\neq 0$ is sufficiently small was proved in [3], but for proving this result
the authors use the averaging theory of first order that does not guarantee
the uniqueness of the limit cycle, it only guarantee the existence of at least
one of such periodic orbit. Using Theorem 5 from [6] we have the uniqueness of
the limit cycles and the desired result for any non zero
$a\in\mathbb{R}\setminus\\{0\\}$ and for any positive integer $n$.

The proof of Theorem 2 is a consequence from the following results:
Proposition 6, Lemma 8, Theorem 9 and 1.

## Acknowledgments

The first author is partially supported by Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. The second
author is partially supported by the Ministerio de Ciencia, Innovación y
Universidades, Agencia Estatal de Investigación grant PID2019-104658GB-I00,
the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617,
and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third
author is partially supported by Projeto Temático FAPESP number 2019/21181–0
and by Bolsa de Produtividade-CNPq number 304766/2019–4.

## Data availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

## References

  * [1] Alvares, M., Ferragut, A., and Jarque, X. A survey on the blow up technique.  Int. J. Bifurcation and Chaos, 21 (2011), 3103–3138.
  * [2] Dumortier, F., Llibre, J., and Artés, J. Qualitative Theory of Planar Differential Systems.  Springer, Berlin, Heidelberg, 2006.
  * [3] López, M., and Martínez, R. A note on the generalized Rayleigh equation: limit cycles and stability.  Journal of Mathematical Chemistry, 51 (2013), 1164–1169.
  * [4] Neto, A. L., de Melo, W., and Pugh, C. On Lienard’s equation.  Lecture Notes in Math., 597 (1977), 335–357.
  * [5] Rayleigh, L. On the stability, or instability, of certain fluid motions.  Proceedings of the London Mathematical Society, 11 (1880), 57–70.
  * [6] Zhifen, Z. Proof of the uniqueness theorem of limit cycles of generalized Lienard equations.  Appl. Anal., 23 (1986), 63–76.


# Learning to Explore Informative Trajectories and Samples for Embodied
Perception

Ya Jing, Tao Kong  ByteDance Research
{jingya<EMAIL_ADDRESS>
###### Abstract

We are witnessing significant progress on perception models, specifically
those trained on large-scale internet images. However, efficiently
generalizing these perception models to unseen embodied tasks is
insufficiently studied, which will help various relevant applications (e.g.,
home robots). Unlike static perception methods trained on pre-collected
images, the embodied agent can move around in the environment and obtain
images of objects from any viewpoints. Therefore, efficiently learning the
exploration policy and collection method to gather informative training
samples is the key to this task. To do this, we first build a 3D semantic
distribution map to train the exploration policy self-supervised by
introducing the semantic distribution disagreement and the semantic
distribution uncertainty rewards. Note that the map is generated from multi-
view observations and can weaken the impact of misidentification from an
unfamiliar viewpoint. Our agent is then encouraged to explore the objects with
different semantic distributions across viewpoints, or uncertain semantic
distributions. With the explored informative trajectories, we propose to
select hard samples on trajectories based on the semantic distribution
uncertainty to reduce unnecessary observations that can be correctly
identified. Experiments show that the perception model fine-tuned with our
method outperforms the baselines trained with other exploration policies.
Further, we demonstrate the robustness of our method in real-robot
experiments.

###### Index Terms:

Embodied Perception, Trajectory Exploration, Hard Sample Selection

##  I Introduction

Pre-training on large-scale datasets to build reusable models has drawn great
attention in recent years, e.g., the deep visual models [2] pre-trained on
ImageNet [3] can be reused for detection [4, 6], and pre-trained language
model like BERT [5] can be used for image-text retrieval [7]. To better adapt
to downstream tasks, many researchers focus on fine-tuning models on small-
scale task-related datasets [9, 8]. However, generalizing the perception model
pre-trained on large-scale internet images to embodied tasks is insufficiently
studied, which will help various relevant applications (e.g., home robots). In
order to use as few annotations as possible, efficiently collecting training
data in embodied scenes becomes the main challenge.

Different from visual learning based on static data (e.g., images), the
embodied agent can move around and interact with 3D environment. Therefore,
efficiently collecting training samples means learning an exploration policy
to encourage the agent to explore the areas where the pre-trained model
performs poorly. Since the ground-truth labels in scenes are unavailable, the
underlying spatial-temporal continuity in the 3D world can be used self-
supervised. To use the consistency in semantic predictions, the previous
method [12] proposes a semantic curiosity policy, which explores inconsistent
labeling of the same object by the perception model. When an exploration
trajectory is learned, all observations on this trajectory are collected for
labeling to fine-tune the pre-trained perception model. Despite the advance,
this method utilizes a fuzzy inconsistency estimation (i.e., projecting
multiple objects at different heights to the same location in a 2D map). In
addition, the uncertainty of the predicted semantic distribution that reflects
what the pre-trained perception model does not know in the new environment and
the hard sample selection on the trajectory are ignored in [12].

Figure 1: The illustration of semantic distribution disagreement, e.g., the
bed is recognized to different objects/distributions across three viewpoints
($v1,v2,v3$), and semantic distribution uncertainty, e.g., the probability of
couch in observation $o1$ being predicted as bed and couch is relatively
close. bg means background. Figure 2: The architecture of our proposed
informative trajectory and sample exploration method. It contains two steps:
the exploration policy aims to encourage the agent to explore the objects with
semantic distribution disagreement or uncertainty, then the training stage
aims at gathering hard samples on trajectories based on semantic distribution
uncertainty to fine-tune the pre-trained model.

To solve these problems, we propose learning to Explore Informative
Trajectories and Samples (EITS) for embodied perception, as shown in Fig. 2.
It consists of two steps: learning exploration policy and collecting hard
samples to fine-tune the perception model. During the exploration, our agent
moves around and collects multi-view observations fused by Exponential Moving
Average to generate a 3D semantic map, which weakens the impact of
misidentification from an unfamiliar viewpoint. The generated 3D map can be
regarded as the pseudo ground truth due to the fusion of predicted results
from different viewpoints. Unlike previous work [12] that adopts predicted
labels to build the semantic map, our work builds a predicted probabilistic
distribution map, i.e., a 3D semantic distribution map. It can be used to
constrain not only the predicted labels but also the predicted distributions,
as shown in Fig. 1 (left). Then one curiosity reward is measured by the
semantic distribution disagreement between the semantic prediction of the
current perspective and the generated 3D semantic distribution map. We also
measure the uncertainty of the predicted semantic distribution as shown in
Fig. 1 (right) as another curiosity reward. These two rewards are used
together to learn the exploration policy by maximizing the disagreement and
uncertainty of semantic predictions. Therefore, our agent can move to the
areas where the semantic predictions are different from pseudo ground truth or
the probabilities being predicted as two categories are relatively close.
After obtaining the exploration strategy, we gather indistinguishable hard
samples on each trajectory for the subsequent training. The selection method
is the same as the uncertainty of semantic distribution. The gathered data is
labeled and utilized to fine-tune the pre-trained perception model, making it
generalized well to new environments. We also show that the perception model
could be sustainably improved by our explore-finetune process back and forth.
The method is evaluated on a challenging Matterport dataset [17] and our real-
robot environment. Experimental results show that our EITS approach
outperforms the state-of-the-art methods.

The contributions of this paper can be summarized as threefold. (1) We propose
a novel informative trajectory exploration method for embodied perception by
measuring the semantic distribution disagreement across viewpoints and the
uncertainty of each observation. In addition, we are probably the first to
exploit the uncertainty over semantic predictions to handle this task. (2) The
hard samples on the trajectory selected by the semantic distribution
uncertainty further sieves the observations recognized well by the pre-trained
model, which can enhance the performance better when fine-tuning the pre-
trained model. (3) The proposed method achieves the best result on the
challenging Matterport dataset.

##  II Related Work

###  II-A Robot Perception Learning

Visual perception is a crucial function of a robot. Some works [16, 39]
directly utilize the perception model pre-trained on COCO [40] images to
perform object goal navigation. To improve the performance in embodied tasks,
some researchers [24, 25, 26] focus on learning a policy to directly improve
metrics of interest end-to-end at test time by obtaining information about the
environment. Unlike them, we aim to explore informative samples self-
supervised to better fine-tune the pre-trained perception model. The
exploration in reinforcement learning [41, 42, 43] also aims to maximize an
intrinsic reward function to encourage the agent to seek previously unseen or
poorly understood parts of the environment. Different from them, we compute
the reward function by multi-view consistency in semantics. The active
interactive imitation learning [44, 45, 46] are also related to our work in
disagreement and uncertainty measuring to decide whether to request a label
from the human. However, our agent does not require human intervention when
learning the exploration policy, and the exploration purpose is to improve the
perception model. Recently, Chaplot et al. [12] measure the semantic curiosity
to learn the exploration policy for embodied perception. But they ignore the
uncertainty over semantic predictions and hard sample selection on the learned
trajectory. Besides, some works [14, 13] attempt to learn both exploration and
perception utilizing pseudo labels in a completely self-supervised manner
without requiring any extra labels. In this paper, we propose effectively
generalizing the pre-trained perception model to embodied tasks, where
informative trajectories and samples are gathered by utilizing a 3D semantic
distribution map to measure the semantic distribution disagreement and the
semantic distribution uncertainty. Then the gathered data is labeled to fine-
tune the perception model.

###  II-B Semantic Mapping

3D mapping aiming to reconstruct a dense map of the environment has achieved
great advances in recent years. Fuentes-Pacheco et al. [32] do a very detailed
survey. Researchers also consider adding semantic information to the 3D map
[14]. Similar to them, we adopt the same setting and learn 3D semantic mapping
by differentiable projection operations. In this paper, we propose a 3D
semantic distribution map, which is used to learn the exploration policy.

###  II-C Embodied Task

Embodied agents can move around and interact with the surrounding environment.
Many environments are photo-realistic reconstructions of indoor [17, 33] and
outdoor [34, 35] scenes, where the ground-truth labels for objects are also
provided. Recently, many researchers have used these simulated environments in
visual navigation [16, 36], visual question answering [37] and visual
exploration [23]. Visual navigation usually involves point/object goal
navigation [16] and vision-and-language navigation [38] where the path to the
goal is described in natural language. Visual question answering [37] should
intelligently navigate to explore the environment, gather necessary visual
information, and then answer the question. Unlike them, our agent aims to
gather data for labeling to generalize the pre-trained perception model to
unseen environments efficiently.

##  III Approach

We aim to train an embodied agent with a perception model pre-trained on
internet images to explore informative trajectories and samples effectively.
Then the perception model fine-tuned on the gathered data can generalize well
to a new environment. As shown in Fig. 2, our proposed method consists of two
main parts. The exploration part aims to learn the active movement of an agent
to obtain informative trajectories via semantic distribution disagreement and
semantic distribution uncertainty self-supervised. Then we take advantage of
the semantic distribution uncertainty to collect hard samples on the learned
trajectory. After images are collected and semantically labeled, we fine-tune
the perception model on these images.

###  III-A 3D Semantic Distribution Mapping

Note that for each time step $t$, our agent’s observation space consists of an
RGB observation $I_{t}\in\mathbb{R}^{3\times W_{I}\times H_{I}}$, a depth
observation $D_{t}\in\mathbb{R}^{W_{I}\times H_{I}}$, and a 3-DOF pose sensor
$x_{t}\in\mathbb{R}^{3}$ which denotes the $x$-$y$ coordinates and the
orientation of the agent. The agent has three discrete actions: move forward,
turn left and turn right.

The easiest way to associate semantic predictions across frames on a
trajectory is to project the predictions on the top-down view to build a 2D
semantic map as [12]. However, due to the embodied agent moving in a 3D
environment, the height information is lost when projecting the predictions
onto a 2D map. These will result in projecting multiple objects at different
heights to the same location, e.g., if a potted plant is on the table, the
potted plant and table will be projected to the same location. Therefore, the
noise will be generated when calculating the disagreement across different
viewpoints. In this paper, we utilize the 3D semantic distribution map to
measure the semantic distribution disagreement. The semantic map $M$ is a 4D
tensor of size $K\times L_{M}\times W_{M}\times H_{M}$, where $L_{M}$,
$W_{M}$, $H_{M}$, denote the 3 spatial dimensions, and $K=C+2$, where $C$ is
the total number of semantic object categories. The first two channels in $K$
represent whether the corresponding voxel (x-y-z location) contains obstacles
and is the explored area, respectively. The other channels denote the
predicted semantic probability distribution among $C$ categories from the pre-
trained perception model. The map is initialized with all zeros at the
beginning of an episode, $M_{0}=[0]^{K\times L_{M}\times W_{M}\times H_{M}}$.
The agent always starts at the center of the map facing east at the beginning
of the episode, $x_{0}=(L_{M}/2,W_{M}/2,0.0)$ same as [16].

Fig. 2 shows the 3D semantic mapping procedure at a time step. The agent takes
action and then sees a new observation $I_{t}$. The pre-trained perception
model (e.g., Mask RCNN [4]) is adopted to predict the semantic categories of
the objects seen in $I_{t}$, where the semantic prediction is a probability
distribution among $C$ categories for each pixel. The depth observation
$D_{t}$ is used to compute the point cloud. Each point in the point cloud is
associated with the corresponding semantic prediction, which is then converted
into 3D space using differentiable geometric transformations based on the
agent pose to get the voxel representation. This voxel representation in the
same location is aggregated over time using Exponential Moving Average to get
the 3D semantic distribution map:

$\begin{split}M_{t}=\left\\{\begin{array}[]{ll}M_{t-1},&t=1\\\
\lambda*M_{t-1}+(1-\lambda)*m_{t},&t>1\\\ \end{array}\right.\end{split}$ (1)

where $m_{t}$ means the voxel representation at time step $t$ and $\lambda$
aims to control the relative importance of $M_{t-1}$ and $m_{t}$. The map can
integrate the predicted semantics of the same object from different viewpoints
to alleviate the misrecognition caused by the unfamiliar viewpoint. Therefore,
the map representation can be used as pseudo ground truth labels of objects in
the scene.

###  III-B Exploring Informative Trajectory

The goal of exploration policy $a_{t}=\pi(I_{t},\theta)$ is exploring objects
that are poorly identified by the current perception model based on the
observation $I_{t}$, where $a_{t}$ means the action and $\theta$ represents
the parameters of the policy model. Hence, we can collect valuable
observations in the explored areas to fine-tune the perception model. We
propose two novel distribution-based rewards to train the exploration policy
by maximizing the disagreement and uncertainty during moving.

The semantic distribution disagreement reward is defined as the Kullback-
Leibler divergence between the current prediction and the 3D semantic
distribution map, which encourages the agent to explore the objects with
different semantic distributions across viewpoints:

$r_{d}=KL(m_{t},M_{t-1}).$ (2)

Unlike semantic curiosity [12] which maximizes the label inconsistency based
on the 2D semantic map, our semantic distribution disagreement aims to explore
the objects with different distributions from the 3D semantic distribution
map.

In addition, we propose a semantic distribution uncertainty reward $r_{u}$ to
explore the objects whose predicted probabilities belonging to two categories
are relatively close, as Eq. 4 explains.

$\begin{split}r_{u}=\left\\{\begin{array}[]{ll}1,&u>\delta\\\ 0,&u<\delta\\\
\end{array}\right.\end{split}$ (3)

To train the policy, we first input the semantic map to a global exploration
policy to select a long-term goal (i.e., an x-y coordinate of the map). Then a
deterministic Fast Marching Method [18] is used for path planning, which uses
low-level navigation actions to achieve the goal. We sample the long-term goal
every 25 local steps, same as [16] to reduce the time horizon for exploration
in reinforcement learning. The Proximal Policy Optimization (PPO) is used to
train the policy.

###  III-C Efficient Sample Selection and Continue Training

After obtaining the trajectory, the easiest way is to label all observations
on the trajectory. Although the trained exploration policy can find more
objects with inconsistent and uncertain predictions, there are still many
observations that the pre-trained model can accurately identify. To
efficiently fine-tune the perception model, we propose a sample selection
method by measuring the uncertainty $u$ of the semantic distribution:

$u=Second_{max}(P_{i}),$ (4)

where $P_{i}\in{\mathbb{R}}^{C}$ is the predicted class probability of $i$th
object in a single image, the $Second_{max}$ means the second largest score in
$\\{p_{i}^{0},p_{i}^{1},...,p_{i}^{C-1}\\}$. If $u$ is larger than threshold
$\delta$, we select the corresponding image. Considering that the semantic
distribution disagreement relies heavily on multi-view observations in the
trajectory will reduce the efficiency of selection, and thus it is not
utilized to select hard samples.

We label the selected images and use them to fine-tune the perception model.

TABLE I: Comparison with the state-of-the-art methods for object detection (Bbox) and instance segmentation (Segm) using AP50 as the metric. n means the exploration policy is progressively trained for n times.  Task |  Method |  Chair |  Couch |  Potted Plant |  Bed |  Toilet |  Tv |  Average
---|---|---|---|---|---|---|---|---
|  Pre-trained |  21.05 |  25.23 |  22.58 |  24.24 |  22.62 |  29.67 |  24.23
|  Re-trained |  23.77 |  27.36 |  26.20 |  25.21 |  24.82 |  34.48 |  26.97
|  Random |  29.98 |  31.65 |  23.91 |  28.66 |  31.78 |  40.44 |  31.07
Bbox |  Active Neural SLAM [23] |  32.02 |  32.74 |  31.94 |  30.31 |  26.30 |  38.68 |  32.00
|  Semantic Curiosity [12] |  33.51 |  33.11 |  32.91 |  29.57 |  25.76 |  39.97 |  32.46
|  Ours (n=1) |  33.57 |  34.36 |  32.79 |  31.54 |  28.38 |  43.81 |  34.07
|  Ours (n=3) |  33.34 |  34.48 |  35.28 |  32.12 |  31.87 |  43.11 |  35.03
|  Pre-trained |  12.72 |  22.98 |  16.71 |  23.82 |  23.85 |  29.75 |  21.64
|  Re-trained |  14.99 |  24.68 |  18.36 |  24.32 |  25.15 |  34.23 |  23.62
|  Random |  18.22 |  27.25 |  8.82 |  28.19 |  29.08 |  39.39 |  25.16
Segm |  Active Neural SLAM [23] |  17.89 |  29.24 |  15.22 |  29.66 |  27.29 |  38.61 |  26.32
|  Semantic Curiosity [12] |  18.18 |  30.06 |  18.39 |  29.03 |  26.70 |  40.01 |  27.06
|  Ours (n=1) |  19.18 |  30.14 |  15.56 |  31.03 |  28.19 |  43.43 |  27.92
|  Ours (n=3) |  19.28 |  30.13 |  16.22 |  31.27 |  28.92 |  44.76 |  28.42

##  IV Experiments

###  IV-A Implementation details

We use the Matterport3D [17] dataset with Habitat simulator [19] in our main
experiments. The scenes in the Matterport3D dataset are 3D reconstructions of
real-world environments, split into a training set (54 scenes) and a test set
(10 scenes). We assume that the perfect agent pose and depth image can be
obtained in our setup.

The exploration policy consists of convolutional layers followed by fully
connected layers. The pre-trained Mask RCNN is frozen while training the
exploration policy. We use the PPO with a time horizon of 20 steps, 8 mini-
batches, and 4 epochs in each PPO update to train the policy. The reward,
entropy, and value loss coefficients are set to 0.02, 0.001, and 0.5,
respectively. We use Adam optimizer with a learning rate of $2.5\times
10^{-5}$. The maximum number of steps in each episode is 500. The $\lambda$
and $\delta$ are experimentally set to 0.3 and 0.1, respectively. To fairly
compare with previous methods, we set the number of training steps to 500k in
all experiments.

We pre-train a Mask-RCNN model with FPN [21] using ResNet-50 as the backbone
on the COCO [40] dataset labeled with 6 overlapping categories with the
Matterport3D, i.e., ‘chair’, ‘couch’, ‘potted plant’, ‘bed’, ‘toilet’ and
‘tv’. Then we fine-tune this model on the gathered samples with a fixed
learning rate of 0.001. All other hyper-parameters are set to default settings
in Detectron2 [22]. We randomly collect the samples in test scenes of
different episodes to evaluate the final perception model. The AP50 score is
adopted as the evaluation metric, which is the average precision with at least
$50\%$ IOU.

We further deploy our method to a real robot. Our robot is equipped with a
Kinect V2 camera, a 2D LiDAR, and an onboard computer (with an Intel i5- 7500T
CPU and an NVIDIA GeForce GTX 1060 GPU). Note that the LiDAR is only used with
wheel odometers to perform localization. We test our method in a built
60$m^{2}$ house with a dining room, a living room, and a bedroom.

TABLE II: Effects of setting different thresholds in hard sample selection on object detection task.  Method |  Training Image |  Chair |  Couch |  Potted Plant |  Bed |  Toilet |  Tv |  Average
---|---|---|---|---|---|---|---|---
$\delta$ = 0.1 |  20k |  33.57 |  34.36 |  32.79 |  31.54 |  28.38 |  43.81 |  34.07
$\delta$ = 0.2 |  13k |  33.38 |  34.61 |  31.34 |  31.84 |  26.24 |  41.64 |  33.18
$\delta$ = 0.3 |  9k |  32.79 |  35.03 |  31.10 |  31.81 |  22.83 |  41.11 |  32.44

###  IV-B Main Results

####  IV-B1 Simulation Environment

To demonstrate the effectiveness of our method, we compare our fine-tuned
object detection and instance segmentation results with the state-of-the-art
methods as shown in Tab. I. Note that these methods all use around 20k
training images. Pre-trained means the perception model was pre-trained on the
raw COCO dataset. Re-trained means we re-train the pre-trained model utilizing
COCO dataset labeled with 6 overlapping categories with the Matterport3D.
Random is a baseline exploration policy that samples actions randomly. It can
be seen that our model achieves the best performance and can further improve
the performance when progressively training the exploration strategy three
times based on the latest fine-tuned perception model.

TABLE III: Ablation studies on the object detection task. SDD and SDU means the semantic distribution disagreement reward and the semantic distribution uncertainty reward in trajectory exploration, respectively. HSS means the hard sample selection. SC means the Semantic Curiosity [12].  Method |  Chair |  Couch |  Potted Plant |  Bed |  Toilet |  Tv |  Average
---|---|---|---|---|---|---|---
Ours w/o SDD |  32.08 |  34.51 |  32.05 |  29.91 |  27.95 |  42.76 |  33.21
Ours w/o SDU |  33.49 |  34.39 |  32.95 |  31.19 |  27.31 |  42.18 |  33.59
Ours w/o HSS |  32.44 |  33.69 |  32.22 |  30.85 |  27.67 |  41.78 |  33.11
SC + HSS |  33.87 |  33.52 |  32.69 |  31.08 |  27.93 |  41.08 |  33.36
Ours |  33.57 |  34.36 |  32.79 |  31.54 |  28.38 |  43.81 |  34.07

Specifically, compared with the pre-trained model, our fine-tuned model gives
10.80$\%$ AP50 gains on the box detection metric. Compared with the previous
best competitor Semantic Curiosity [12] which rewards trajectories with
inconsistent labeling behavior and encourages the embodied agent to explore
such areas, our model significantly outperforms it by 2.57$\%$ absolute AP50
point on object box detection and 1.36$\%$ on instance segmentation. The
improved performances over the best competitor indicate that our proposed
informative trajectory exploration and hard sample selection method is very
effective for this task. In addition, we can see that our method is more
friendly to instances with simple shapes, e.g., Bed and Tv. These instance’s
shapes are easier to be reconstructed through 3D mapping. Objects with much
more complicated shapes, e.g., Potted Plant, are more likely to involve
mapping errors, which in turn decreases the performance of instance
segmentation.

####  IV-B2 Real Robot

We also deploy our learned exploration policy on a real robot to explore
informative trajectories and hard samples in an unseen environment. In
practice, we gather 170 hard samples for fine-tuning the pre-trained model and
an additional 50 randomly collected samples for validation, with an average of
4 objects in each image. Benefiting from the gathered informative images, the
fine-tuned perception model can improve the detection and segmentation
performances from 79.1% AP50 and 76.7% AP50 to 97.3% AP50 and 96.1% AP50,
respectively.

###  IV-C Ablation Analysis

Our method comprises two modules: informative trajectory exploration and hard
sample selection. To investigate these two components, we perform a set of
ablation studies with $n$ = 1 for simplicity, as shown in Tab. III.

We first investigate the importance of rewarding semantic distribution
disagreement across viewpoints and semantic distribution uncertainty to
explore the trajectory. It can be seen that the AP50 accuracy on object
detection drops 0.71$\%$ (SC+HSS vs. Ours) by replacing our exploration policy
as SC. The exploration module proves the effectiveness of learning informative
trajectories for subsequent sample selection. Then we investigate the
importance of semantic distribution uncertainty based hard sample selection by
removing it (Ours w/o HSS). The AP50 accuracy on object detection drops
0.96$\%$, demonstrating that selecting hard samples enhances the perception
results. In addition, by comparing the results between Ours and Ours w/o SDD,
Ours and Ours w/o SDU (semantic distribution disagreement and uncertainty in
informative trajectory exploration), we can find that utilizing SDD and SDU
can generate more effective trajectories.

TABLE IV: Effects of progressively training the exploration policy for $n$ times on the object detection task.  Method |  Chair |  Couch |  Potted Plant |  Bed |  Toilet |  Tv |  Average
---|---|---|---|---|---|---|---
n = 1 |  33.57 |  34.36 |  32.79 |  31.54 |  28.38 |  43.81 |  34.07
n = 2 |  32.18 |  33.32 |  36.06 |  31.38 |  30.81 |  44.53 |  34.71
n = 3 |  33.34 |  34.48 |  35.28 |  32.12 |  31.87 |  43.11 |  35.03
Figure 3: Qualitative examples of learned trajectories and sampled images from
the Matterport3D environment and the real robot. The first row shows the
explored informative trajectories trained by semantic distribution
disagreement and uncertainty rewards. The second row shows the gathered hard
images by semantic distribution uncertainty estimation. Figure 4: Qualitative
examples of instance segmentation by different models.

We compare the effectiveness of setting different thresholds $\delta$ in hard
sample selection as shown in Tab. II. In this experiment, we sample the images
from explored trajectories with 6 episodes and fixed steps in each training
scene, resulting in different numbers of sampled training images at different
thresholds. We can find that decent performances can be achieved by training
very few hard samples, which demonstrates the effectiveness of selecting hard
samples. Tab. IV shows the experimental results when progressively training
the exploration policy multiple times based on the latest fine-tuned
perception model. Note that they all use 20k training images.

To exploit measures of uncertainty in semantic distributions, we utilize the
entropy of categorical distribution (ECS) in place of the heuristic in Eq. 4
as shown in Tab. V. We experimentally set the threshold of entropy to 0.4. The
improved performance indicates that the uncertainty between all distributions
is more effective than between the two categories.

TABLE V: Effects of utilizing the entropy of categorical distribution (ECS) in place of the heuristic in Eq. 4 to measure semantic distribution uncertainty on the object detection task.  Method |  Chair |  Couch |  Potted Plant |  Bed |  Toilet |  Tv |  Average
---|---|---|---|---|---|---|---
Ours |  33.57 |  34.36 |  32.79 |  31.54 |  28.38 |  43.81 |  34.07
ECS |  31.40 |  32.70 |  34.23 |  30.90 |  30.75 |  46.06 |  34.34

###  IV-D Qualitative Results

To verify whether the proposed exploration policy and hard sample selection
method can obtain the observations with inconsistent or uncertain semantic
distributions, we visualize the explored trajectories and sampled images from
the Matterport3D dataset and real-world environment, as shown in Fig. 3. We
can see that our model is able to gather inconsistent and uncertain detections
via semantic distribution disagreement and uncertainty estimation. For
example, the couch is detected as different objects (chair/couch) or
distributions from different viewpoints on the first row. Besides, the couch
is detected as couch and chair with almost close scores on the second row. By
collecting these observations that are poorly identified by the pre-trained
perception model for labeling, the model can be fine-tuned better.

Fig. 4 shows the segmentation masks obtained by three different models, i.e.,
pre-trained, Semantic Curiosity [12], and our EITS, demonstrating our proposed
method’s benefits. As the figure shows, our generated segmentation masks have
more obvious object shapes and finer outlines in the first column. Besides,
our model, fine-tuned exclusively on hard samples, can detect the missed
objects by the pre-trained and Semantic Curiosity [12] models, as shown in the
third and fifth columns.

##  V Discussion and Limitations

We propose to generalize the perception model pre-trained on internet images
to the unseen 3D environments with as few annotations as possible. Therefore,
efficiently learning the exploration policy and selection method to gather
training samples is the key to this task. In this work, we propose a novel
informative trajectory exploration method via semantic distribution
disagreement and semantic distribution uncertainty. Then the uncertainty-based
hard sample selection method is proposed to further reduce unnecessary
observations that can be correctly identified. Extensive ablation studies
verify the effectiveness of each component of our method.

Although our method is more efficient than previous works, there are still
some limitations. Through exploring the informative trajectories and samples,
we can efficiently generalize the pre-trained model to the embodied task,
where labeling the segmentation mask is still costly. The weakly-supervised
methods (e.g., utilizing box annotations to train segmentation models) can be
utilized to fine-tune the perception model in the future. In addition, we
collect all samples before fine-tuning the perception model, which results in
our perception model not being updated. In the future, we can explore updating
the perception module when learning the exploration policy.

##  VI Acknowledgments

We would like to thank Minzhao Zhu, Yifeng Li, Yuxi Liu, Tao Wang and Yunfei
Liu for their help on the robot system, Hang Li for helpful feedback, and
other colleagues at ByteDance AI Lab for support throughout this project.

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# Strangeness production in STAR

###### Abstract

We present a summary of strangeness enhancement results comparing data from
Cu+Cu and Au+Au collisions at $\sqrt{S_{NN}}=200GeV$ measured by the STAR
experiment. Relative yields in central Cu+Cu data seem to be higher than the
equivalent sized peripheral Au+Au collision. In addition, strange particle
production from these two systems is compared in terms of a statistical model,
applying a Grand-Canonical ensemble and also applying a canonical correlation
volume for the strange particles. Thermal fit results from the Grand-Canonical
formalism shows little dependence on the system size but, when considering a
strange canonical ensemble, strangeness enhancement shows a strong dependency
on the correlation volume.

##  1 Introduction

Strangeness production is expected to be enhanced in heavy ion collision as
one of the possible signatures of the Quark-Gluon Plasma [1]. An increase of
the strange quark density should result in an increase of the strange hyperon
production thus, the measured integrated yield per participant should be
sensitive to the density of quarks in the system. Experimentally, strangeness
enhancement has been observed in A+A collisions with respect to p+p collisions
when comparing the normalized strange particle yields scaled by factors such
as the number of participants in the collision, or the number of wounded
nucleons [2, 3]. Thermal models with a canonical approach show that this
observed relative strangeness enhancement can also be explained by considering
a suppression due to limitations on the phase space available for strange
particles production in small systems [4, 5]. These models have been quite
successful in describing the level of enhancement and the observed hierarchy
with the number of strange valence quarks, but up to now, no model has been
able to describe simultaneously all features of the measured strangeness
enhancement. In this paper, we present the latest results from the analysis of
the STAR experiment with respect to the strangeness enhancement including data
from Au+Au and Cu+Cu collisions at RHIC maximum energy. In addition, we apply
a statistical thermal model fit to study the system size dependence of thermal
parameters.

##  2 Strangeness Enhancement

Fig. 1.: Strangeness enhancement plots, particle yields from Au+Au and Cu+Cu
data normalized by $<N_{part}>$ relative to yields from p+p as a function of
$<N_{part}>$. Left plot shows the relative enhancement of $\Lambda$, $\Xi$ and
$\Omega+\bar{\Omega}$, and the right plot shows the same data for
$\bar{\Lambda}$, $\bar{\Xi}$ and $\Omega+\bar{\Omega}$.

Figure 1 left side shows the system size dependence of the strangeness
enhancement ratio for $\Lambda$, $\Xi$ and $\Omega+\bar{\Omega}$ measured in
STAR for Au+Au and Cu+Cu at collision energy of 200 GeV per nucleon pairs.
Figure 1 right shows the same plot for $\bar{\Lambda}$, $\bar{\Xi}$ and
$\Omega+\bar{\Omega}$. Particle yields were measured for different event
centrality classes and normalized by the equivalent mean number of participant
nucleons $<N_{part}>$ and then divided by the equivalent ratio measured in p+p
collisions at the same energy. The $\Lambda$ and $\bar{\Lambda}$ yields were
corrected for feed-down from $\Xi$ decays. In both Au+Au and Cu+Cu data, a
large strangeness enhancement is observed even in the most peripheral
centrality bin. The data also shows a strong dependency with the system size
and do not seem to saturate as expected from a Grand-Canonical Thermal model.
Strangeness enhancement is higher for strange particles than for anti-
particles, which can be a result of the non zero net baryon density. The
striking result is that the strangeness enhancement observed in central Cu+Cu
data is higher than the peripheral Au+Au collision with equivalent
$<N_{part}>$. This is an indication that the production mechanism of these
strange particles does not scale with the pure geometrical parameterization of
the system size. Normalized proton yields were included in the plots of figure
1 as a reference to show that the relative discrepancy between the two systems
is unique to the strangeness sector. The measured proton yields in STAR are
inclusive [7], and in this plot these proton yields were subtracted by the
$\Lambda$ yield factorized by the decay branching ratio to correct for the
feed-down into the proton yield. The normalized proton yields do not show any
dependence with the system size and also no difference between Au+Au and Cu+Cu
data.

Fig. 2.: Energy dependence of the relative strangeness enhancement for
$\Lambda$, $\bar{\Lambda}$, $\Xi$, $\bar{\Xi}$ and $\Omega+\bar{\Omega}$,
observed at central collisions of Au+Au and Pb+Pb, compiling RHIC and SPS
results. SPS results shown here are from experiment NA57 [3].

Figure 2 shows the collision energy dependence of the strangeness enhancement
factor for the most central events for $\Lambda$, $\bar{\Lambda}$, $\Xi$,
$\bar{\Xi}$, $\Omega+\bar{\Omega}$. SPS data for the low energy points were
extracted from Pb+Pb data of experiment NA57 [3]. The observed strangeness
enhancement is decreasing with collision energy for the $\Lambda$ and $\Xi$,
as predicted by Grand-Canonical statistical model, but, an opposite trend is
observed for the anti-particles that yields higher enhancement in RHIC data
compared to SPS data. Also clear in this plot is the strange quark content
hierarchy observed in the enhancement as predicted by statistical thermal
models.

##  3 Statistical thermal model results

Data was analyzed using the THERMUS code [6] and considering particle ratios
that includes $p$ ($\bar{p}$), $\pi^{\pm}$, $K^{\pm}$, $\Lambda$
$(\bar{\Lambda})$, $\Xi^{\pm}$, $\Omega^{\pm}$, and $\phi$. Protons were
corrected for the feed-down of the $\Lambda$ decay that resulted in a
reduction of the inclusive proton yield of approximately $30\%$. Since STAR
does not yet have $\Sigma$ measurements, the feed-down contribution from this
particle was estimated using the ratio of $\Sigma/\Lambda=0.35$ [8]. Final
reduction of the inclusive proton yields was on the order of $35\%$ to $45\%$.
To estimate the error due to this feed-down correction, the relative $\Sigma$
yield to $\Lambda$ was varied over $50\%$. The overall effect on the final
thermal parameters was less than $10\%$.

Fig. 3.: Thermal model fit chemical freeze-out temperature $T_{ch}$, and
strangeness chemical potential $\mu_{S}$, as a function of $<N_{part}>$ for
Au+Au and Cu+Cu data. For comparison, the same fit was applied to p+p data and
included in this plot.

Figure 3 shows the chemical freeze-out temperature $T_{ch}$ and the
strangeness chemical potential $\mu_{S}$ as a function of the system size,
$<N_{part}>$. $T_{ch}$ is around 155 MeV and seems to show no dependence on
the system size and also no sensitivity to the colliding systems, yielding the
same results for Au+Au and Cu+Cu data. The baryon chemical potential $\mu_{B}$
and strangeness chemical potential $\mu_{S}$ also showed no difference between
the fits to Au+Au and Cu+Cu data. To study the validity of the statistical
thermal fit considering a Grand Canonical formulation, we have applied the
same fit to the p+p data. The chemical freeze-out temperature that results
from this fit is slightly lower, around 150 MeV.

Fig. 4.: Strangeness under-saturation parameter of the thermal model from fits
to particle ratios of Au+Au and Cu+Cu data for the different system sizes. The
result from a fit to the p+p data is also included.

Figure 4 shows the dependence on the system size of the strangeness under-
saturation factor $\gamma_{S}$ obtained from the statistical thermal fit with
a Grand-Canonical ensemble for the Au+Au and Cu+Cu data. Cu+Cu data shows the
same results and behavior of the peripheral Au+Au data. The $\gamma_{S}$
reaches unity only above $<N_{part}>$ greater than approximately 100. Most of
the Cu+Cu data is below that limit. The $\gamma_{S}$ factor can be interpreted
as a measure of the validity of the Grand-Canonical formalism to describe the
data in the strange sector. Under this assumption, from the results shown in
figure 4, it is clear that only the most central bins of Cu+Cu collision can
be well described with this approach. $\gamma_{S}$ from the fit to the p+p
data is also shown in figure 4, and shows a much lower value, around 0.6,
indicating that the strange particles ratios in p+p collisions cannot be well
described with this model.

Fig. 5.: Plot shows the strange particle to pion ratio as predicted by the
thermal model considering a Grand-Canonical ensemble with a strange canonical
system, as a function of the radius of the strangeness canonical size.

As an alternative to the Grand Canonical formulation (GC) where the
conservation of the quantum numbers are ensured on average, the THERMUS code
allows for a strangeness canonical (SC) ensemble where only the quantum
numbers of the strange particles are required to conserve exactly. In this
formalism, a correlation volume of strange particle production is defined as
the sub-volume where the strangeness chemical equilibrium is restricted. We
applied the fit to the Au+Au 200 GeV data using this approach and obtained a
strangeness correlation volume radius of approximately $5fm$, for the most
central event centrality class. This value is much higher than the value
presented for a similar analysis on SPS data [3] that was around $1fm$. To
understand the effect of this correlation volume on the final strange particle
yields, we studied the variation of the strange particle yields relative to
the pions for different correlation volumes. Figure 5 shows the results of the
model prediction in solid lines and the symbols represent the experimental
values of the most central Au+Au 200 GeV data. It is interesting to note that
the $\phi/\pi$ ratio does not depend on the correlation volume. The value of
the correlation volume obtained from the best fit to the data indicates that
the system is already in the region where the strange particle production is
saturated, and thus, consistent with a system where the strangeness is already
equilibrated.

##  4 Conclusions

Strange particle production is enhanced in heavy ion collisions compared to
p+p. This enhancement increases with the system size and does not seem to
saturate as expected by models considering Grand Canonical Statistical models.
When using $<N_{part}>$ as a scaling factor, Cu+Cu central collision show
higher strangeness enhancement than peripheral Au+Au collision. Application of
thermal model fit to Au+Au and Cu+Cu data seem to show almost no difference in
the thermal parameters. In addition, the thermal fit parameters considering a
Grand-Canonical approach seem to show no dependency on the system size except
for the $\gamma_{S}$ parameter. However, when considering a strangeness
Canonical formalism in the statistical thermal model, strangeness production
shows strong dependence on the correlation volume. Fits to the data indicate
that the correlation volume is large, around $5fm$, above the saturation
point.

## Acknowledgments

We wish to thank RHIC and STAR group and the U.S. DOE Office of Science for
the support in the participation of this conference.

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