# Dynamics of coupled modified Rössler oscillators: the role of
nonisochronicity parameter

C. Ramya  R. Gopal  R. Suresh<EMAIL_ADDRESS>V. K. Chandrasekar
<EMAIL_ADDRESS>Department of Physics, Centre for Nonlinear Science
and Engineering, School of Electrical and Electronics Engineering, SASTRA
Deemed University, Thanjavur 613 401, India

###### Abstract

The amplitude-dependent frequency of the oscillations, termed
_nonisochronicity_ , is one of the essential characteristics of nonlinear
oscillators. In this paper, the dynamics of the Rössler oscillator in the
presence of nonisochronicity is examined. In particular, we explore the
appearance of a new fixed point and the emergence of a coexisting limit-cycle
and quasiperiodic attractors. We also describe the sequence of bifurcations
leading to synchronized, desynchronized attractors and oscillation death
states in the coupled Rössler oscillators as a function of the strength of
nonisochronicity and coupling parameters. Further, we characterize the
multistability of the coexisting attractors by plotting the basins of
attraction. Our results open up the possibilities of understanding the
emergence of coexisting attractors, and into a qualitative change of the
collective states in coupled nonlinear oscillators in the presence of
nonisochronicity.

††preprint: AIP/123-QED

> Nonisochronicity is one of the essential and peculiar characteristic
> property of the nonlinear differential equations. We show here that for
> Rössler oscillator with nonisochronicity term, there exist many distinct
> dynamical states, due to the appearance of a new fixed point, which leads to
> rich and complex dynamics in the system. We observe that the coexistence of
> many attractors may introduce additional sensitivity and randomness in the
> chaotic system because an infinitesimal perturbation in initial values is
> sufficient for a trajectory to shift from one state to another. This can be
> of great interest for secure communication applications through encryption
> of information-based chaos, for example, and various control strategies in
> engineering contexts. We describe and characterize the regions of
> synchronization in coupled Rössler oscillators, basins of attraction
> corresponding to different attractors using master stability function
> formalism. We also observe that the system’s permutational symmetry is
> broken at a critical value of the coupling strength, which persuades multi-
> stable synchronized, desynchronized attractors, and oscillation death states
> in the coupled Rössler oscillators.

##  I Introduction

Systems that are described by the nonlinear differential equations are
omnipresent ranging from biological systems to physical systems and
economicsstrogatz2018 ; leung1989 ; puu2003 . The nonlinearities present in
these systems facilitate the emergence of rich dynamical states such as self-
sustained oscillations, turbulence, and chaos, which are impossible to obtain
in the systems described by the linear differential equations. In particular,
one of the important characteristics of nonlinear differential equations is
the amplitude-dependent frequency of nonlinear oscillations, known as
nonisochronicity, the term first used in ref.aronson1990 . Nonisochronicity is
central to inhomogeneities that can be seen in a variety of living situations
daido2006 ; tiberkevich2014  and due to this nonisochronous nature of the
nonlinear equations, different responses have been observed for different
input conditions in practical or experimental settings. In particular, when
considering the complex biological or other networks, the strong
nonisochronicity often results in rich synchronization patterns and the level
of isochronicity of oscillators is an important property of the phase
interaction function, where the strong dependence of the oscillatory period or
frequency on the amplitude results in high-level shear in the phase dynamics
and coupling can dramatically change the oscillatory periodsebek2016 . Also,
it has been reported that the higher nonisochronicity level can induce long,
irregular transient phase dynamics in the networks, where such long transient
dynamics have relevance in the functioning of biological
systemswickramasinghe2013 . The detection of isochrons in a system is also
prescribed recentlymauroy2012 ; mauroy2013 . Moreover, nonisochronicity plays
a significant role in the development of anomalous phase synchronization in
populations of nonidentical oscillatorsblasius2003 , which has also been
experimentally confirmeddana2006 ; wickramasinghe2011 . Further, in the
networks of nonlocally coupled Stuart-Landau oscillators, the nonisochronicity
leads to the emergence of various collective states such as amplitude chimera,
amplitude cluster, frequency chimera, frequency cluster states, imperfect and
mixed imperfect synchronized states, local and global chimera states, etc.,
which have been recently reported in the case of symmetry preserving and
symmetry breaking couplings premalatha2015 ; premalatha2016a ; premalatha2016b
; premalatha2018 ; senthilkumar2019 . Similarly, the emergence of
nonisochronicity induced chimera, and chimera-like states in nonlocally and
globally coupled complex Ginzburg-Landau modelkuramoto2002 ; sethia2014 ;
mishra2015 , van der Pol systemhens2015  are also demonstrated. Furthermore,
in ref.montbrio2003 , the authors reported that the nonisochronicity could
also be used to control the transition to synchronization in an ensemble of
nonidentical oscillators.

As mentioned earlier, the strong nonisochronicity parameter induces rich
collective dynamics in a coupled system of oscillators. However, the role of
nonisochronicity has been mainly studied in the case of Stuart-Landau
oscillator, coupled complex Ginzburg-Landau model and only a few studies are
reported in other dynamical systems. Nevertheless, there are many other
interesting nonlinear oscillatory systems and biological models that deserve
to be studied in detail in the presence of nonisochronicity. In compliance
with this, in a recent study chandrasekar2014 , the authors have reported the
advent of chimera states due to the presence of nonisochronicity term in a
diffusively coupled global network of Rössler oscillators. They also found
that the presence of multistabilitysommerer1993 ; feudel1998 ; do2008 ;
lai2017 ; wontchui2017 ; hens2012 ; jaros2015 ; patel2014  is the key
mechanism for the appearance of coexisting states. Moreover, the emergence of
chimera state in a star network of Rössler oscillators with nonisochronicity
term is demonstrated with various coupling topologiesmeena2016 .

Further, in Ref. chaurasia2017 , the authors have explored the possibility of
controlling chaos in an ensemble of modified Rössler oscillators, in which the
individual nodes are indirectly coupled through a single external oscillator.
Furthermore, the investigation of the emergence of chaotic phase
synchronization in three coupled modified Rössler oscillator systems with
parameter mismatches are reportednishikawa2009 . The authors showed the
switching between complete and partial phase synchronization as a function of
the coupling strength. The transition from in-phase to out-of-phase
synchronizations via the phase-flip transition in a network of relay-coupled
modified Rössler oscillators is demonstrated by Sharma et al.sharma2011 . In
all the studies mentioned above, the Rössler oscillator with nonisochronicity
term has been used to demonstrate the results. However, the basic knowledge of
the dynamical changes that occurred in the Rössler oscillator with
nonisochronicity parameter is still lagging. Additionally, the study of
nonisochronicity term induced bifurcations and collective dynamical states
exhibited in the coupled systems with different coupling topologies are
required exhaustive investigation.

Taking this as our motivation, in this paper, we study the dynamics of the
Rössler oscillator by adding the nonisochronicity term. Without which the
system exhibits limit-cycle oscillations for the chosen parameter values. When
we incorporate the nonisochronicity term into the oscillator equation, the
system manifests the coexistence of limit-cycle and quasiperiodic attractors.
This is due to the appearance of a new fixed point in response to the strength
of the nonisochronicity parameter. Next, the study is extended to a system of
two mutually coupled Rössler oscillators, and we investigate the influence of
the coupling parameter along with the nonisochronicity term. We show that when
one increases the strength of coupling, by fixing the nonisochronicity
parameter, the permutational symmetry
(($x_{1},y_{1},z_{1}$)$\longleftrightarrow$($x_{2},y_{2},z_{2}$)) of the
subsystems is broken at a critical value of the coupling strength, that
persuades multistable desynchronized attractors, which are emerged via
different bifurcations based on the strength of nonisochronicity and coupling
parameters.

When the coupling strength reaches a sufficiently larger value, the symmetry
is preserved again, which leads one of the attractors to exhibit synchronized
oscillations. Here, the synchronization is defined as the correlation of both
phase and amplitude of the oscillations. During the occurrence of
desynchronized oscillations, either amplitude or both phase and amplitude are
uncorrelated. In addition to the synchronized and desynchronized oscillations,
we discover the development of two distinct oscillation death (OD) states (in
which the subsystems exhibit inhomogeneous steady-state solutions) along with
the synchronized and desynchronized oscillations. The occurrence of steady
state solutions prasad2005 ; saxena2012 ; resmi2011 ; resmi2012 ; hens2013 ;
nandan2014 ; verma2018 ; verma2019  and multistability hens2012 ; jaros2015 ;
patel2014  has been reported in networks of original Rössler oscillators with
different coupling configurations. However, we emphasize that to the best of
our knowledge, the emergence of OD states and coexisting attractors are
unexplored yet in the coupled Rössler oscillators with nonisochronicity term.
Further, the multistability is characterized by estimating the bifurcation
diagrams, and the region of complete synchronization is identified using the
master stability function (MSF) formalism. Also, based on the emergence of
different attractors, the bifurcation plot is divided into several regions.
The development of coexisting attractors in those regions is corroborated by
plotting the basins of attraction.

The structure of the remaining paper is organized as follows: In Sec. II, we
will introduce the mathematical model of the Rössler oscillator with
nonisochronicity term, and discuss the stability of the fixed points of the
system with and without the nonisochronicity term. In Sec. III, we study the
dynamics of a Rössler oscillator with nonisochronicity term and show the
emergence of coexisting attractors that are emerged via various bifurcations.
Next, in Sec. IV, we will consider two mutually coupled oscillators for our
study and show the emergence of coexisting synchronized, desynchronized
attractors along with OD states using bifurcation diagrams. Further, the
bifurcation diagram is divided into different regions based on the combination
of coexisting attractors, and the multistability is confirmed by plotting the
basins of attraction in Sec. V. Moreover, the probability of occurrence of
each attractor in the basins is also discussed. Finally, the obtained results
are consolidated, and the future directions of research are discussed in Sec.
VI.

##  II Mathematical model and fixed point analysis

In order to demonstrate our results, we first consider a Rössler oscillator
with nonisochronicity term, which can be described by the mathematical
equation of the form

$\displaystyle\dot{x}$ $\displaystyle=$
$\displaystyle-\omega_{0}(1-\alpha(x^{2}+y^{2}))y-z,$ $\displaystyle\dot{y}$
$\displaystyle=$ $\displaystyle\omega_{0}(1-\alpha(x^{2}+y^{2}))x+ay,$ (1)
$\displaystyle\dot{z}$ $\displaystyle=$ $\displaystyle b+(x-c)z,$

where $x,y$, and $z$ are the state variables, $\omega_{0}$ is the natural
frequency of the system, $a,b$, and $c$ are the control parameters. $\alpha$
is a parameter to adjust the strength of nonisochronicity.

In order to find the fixed points, the three Rössler equations are set to zero
and the resulting equations of $x$ and $z$–components can be written in terms
of the $y$–variable as follows,

$\displaystyle x$ $\displaystyle=$
$\displaystyle\frac{acy^{2}}{ay^{2}+b},\quad z=\frac{ay^{2}+b}{c},$ (2)
$\displaystyle f(y)_{\alpha}$ $\displaystyle=$ $\displaystyle
cy\omega_{0}\left(1-\alpha\left(\frac{a^{2}c^{2}y^{4}}{\left(ay^{2}+b\right)^{2}}+y^{2}\right)\right)+ay^{2}+b=0.$

For $\alpha=0$, the function $f(y)_{\alpha}$ in Eq. (2) becomes

$f(y)_{0}=y(ay+c\omega_{0})+b,$ (3)

and the corresponding two roots (fixed points) are given by

$y(0)_{1,2}=\pm\frac{\sqrt{c^{2}\omega_{0}^{2}-4ab}}{2a}-c\omega_{0}.$ (4)

The respective two fixed points $y(0)_{1}$ and $y(0)_{2}$ are depicted as open
circles in Fig. 1 for the fixed values of the control parameters
$\omega_{0}=1,a=0.2,b=1.7$ and $c=5.7$. When $\alpha\neq 0$, the function
$f(y)_{\alpha}$ has three roots (fixed points) $y(\alpha)_{1,2,3}$. The fixed
points $y(\alpha)_{1,2}$ are the same fixed points that were present when
$\alpha=0$, however the positions are now dependent on the value of $\alpha$
(see filled triangles in Fig. 1). They are approximately given by the
following form

Figure 1: Fixed points of the $y$–component of a Rössler oscillator (II) in
the absence of nonisochronicity parameter ($\alpha=0$) and for $\alpha=0.01$.
The system parameters are fixed at $\omega_{0}=1$, $a=0.2$, $b=1.7$ and
$c=5.7$.

$\displaystyle
y(\alpha)_{(1,2)}\approx\bigg{(}a^{2}\beta^{5}\left(5a\beta-2c\omega_{0}\left(2\alpha
c^{2}+3\alpha\beta^{2}-2\right)\right)+$ $\displaystyle
b^{2}\beta^{2}(3a-2\alpha
c\omega_{0}\beta)+ab\beta^{3}\left(9a\beta+4c\omega_{0}\left(1-2\alpha\beta^{2}\right)\right)-b^{3}\bigg{)}$
$\displaystyle\times\bigg{(}6a^{3}\beta^{5}+a^{2}\beta^{3}\left(12b+c\omega_{0}\left(-5\alpha
c^{2}-7\alpha~\beta^{2}+5\right)\right)$
$\displaystyle+2ab\beta\bigg{(}3b+c\omega_{0}\beta\left(3-5\alpha\beta^{2}\right)\bigg{)}+b^{2}c\omega_{0}\left(1-3\alpha\beta^{2}\right)\bigg{)}^{-1}$
(5)

where $\beta=y(0)_{(1,2)}$. To find the fixed points of Eq. (II), the
nonisochronicity parameter $\alpha$ is considered as a small perturbation. The
other fixed point $y(\alpha)_{3}$ is newly emerged upon introducing $\alpha$.
This can be clearly seen from Fig. 1 that the new fixed point has emerged in
the positive $y$-direction due to the effect of $\alpha$. For the
aforementioned values of control parameters, the three fixed points are
unstable and the nature of their eigenvalues are

  1. 1.

FP$(\alpha)_{1}=(x(\alpha)_{1},y(\alpha)_{1},z(\alpha)_{1})$ – which has one
negative real eigenvalue (near zero) and one complex conjugate eigenvalue pair
with a positive real part.

  2. 2.

FP$(\alpha)_{2}=(x(\alpha)_{2},y(\alpha)_{2},z(\alpha)_{2})$ – which has one
complex conjugate eigenvalue pair with a negative real part and one positive
real eigenvalue.

  3. 3.

FP$(\alpha)_{3}=(x(\alpha)_{3},y(\alpha)_{3},z(\alpha)_{3})$ – which has one
negative real eigenvalue and one complex conjugate eigenvalue pair with a
positive real part.

Figure 2: The attractors that are emerged for different values of the
nonisochronicity parameter ($\alpha$) in the Rössler oscillator (II). Other
parameters are fixed as shown in Fig. 1. (a) Three dimensional projection of
the period-I limit-cycle attractor emerged for $\alpha=0$, (b) shows the
period-II attractor along with the coexisting period-I attractor and large-
amplitude quasiperiodic attractor for different sets of initial states, for
$\alpha=0.007$, (c) depicts the coexistence of period-I limit-cycle and
quasiperiodic attractors that are appeared in the system for $\alpha=0.01$.
(d) and (e) Depicts the power spectrum of the periodic and quasiperiodic
attractors (shown in Fig.2(c)), respectively.

When $\alpha=0$, the system shows a single limit-cycle attractor, for the
above chosen parameter values, which is rotating about the fixed point
$y(0)_{1}$. When introducing $\alpha$, due to the emergent of a new fixed
point $y(\alpha)_{3}$, the system shows additional limit-cycle and
quasiperiodic attractors. We explain this in detail in Sec. III. One can also
notice that in the case of $c^{2}\omega_{0}^{2}<4ab$, the system (II) has zero
(one) fixed point for $\alpha=0$ ($\neq 0$).

Figure 3: (a), (b) Bifurcation diagrams of a single Rössler oscillator model
(II) as a function of the nonisochronicity parameter $\alpha\in[0,0.013]$,
shows the emergence of different dynamical states. The evolution of three
fixed ponts are depicted as lines with open triangle, asterisk, and plus
symbol. Other system parameters are fixed as given in Fig. 1.

##  III Dynamics of an individual Rössler oscillator with nonisochronicity

First, we study the dynamics of Eq. (II) as a function of the nonisochronicity
parameter ($\alpha$). For our numerical investigation, we have integrated Eq.
(II) using the fourth-order Runge-Kutta method with a step size of 0.01. We
also confirm that the similar results can be observed when we use smaller
time-steps of 0.001. The emerging attractors are depicted in Fig. 2 for three
selective values of $\alpha$. First, in the absence of $\alpha$, as mentioned
earlier, the Rössler oscillator has two fixed points (FP(0)1 and FP(0)2), in
that FP(0)2 is saddle-focus whose one-dimensional unstable manifold repels the
trajectories along the stable manifold of the other fixed point FP(0)1, which
is also a saddle-focus. Therefore, a period-I limit-cycle attractor is
emerged, which rotates (counter-clockwise) around FP(0)1 fixed point. The
three-dimensional projection of the respective limit-cycle attractor in the
$(x,y,z)$ phase plane is displayed in Fig. 2(a) for $\alpha=0$. When we
incorporate and slowly increase the nonisochronicity parameter further, the
period-I limit-cycle is then bifurcated into a period-II attractor via period-
doubling bifurcation. At the same time, due to the inclusion of $\alpha$, one
more new fixed point FP($\alpha$)3 has emerged in the system, as shown in Fig.
1. Consequently, another period-I limit-cycle appears for different sets of
initial states, which rotate around the fixed point FP($\alpha$)3. In addition
to this, a large-amplitude quasiperiodic attractor is also appeared and
rotating around the fixed points FP($\alpha$)1, FP($\alpha$)2 and
FP($\alpha$)3. All three coexisting attractors that are emerged for different
sets of initial states are depicted in Fig. 2(b) for the value of
$\alpha=0.007$. The periodic (period-I attractor) and quasiperiodic nature of
the attractors are confirmed by estimating the power spectrum and are plotted
in Figs.2(d), and 2(e), respectively. The period-II attractor has become
unstable when $\alpha$ reaches a threshold value. Therefore, only the other
two attractors are stable and coexisted in the system, which is depicted in
Fig. 2(c) for $\alpha=0.01$.

Overall, the Rössler oscillator (II) exhibits coexisting limit-cycle and
quasiperiodic attractors when we incorporate the nonisochronicity parameter in
it. To acquire a clear picture of the dynamics of the system, the bifurcation
diagram of Eq (II) is depicted in Fig. 3(a) as a function of
$\alpha\in[0,0.013]$, which is obtained using the XPPAUT software
packageermentrout2002 . In this figure, the lines with different points
represent various stable attractors, and the broken lines indicate the
unstable periodic orbits. In particular, the lines with open circles in Fig.
3(a) indicate the maxima and minima of the period-I limit-cycle oscillations
equivalent to the attractor shown in Fig.2(a). The period-doubling
bifurcation, which occurred at the critical value of $\alpha=0.0016$, is
marked as PD in Fig. 3(a). The lines with open squares indicate the maxima and
minima of the respective period-II attractor, and this period-II orbit is then
become unstable when $\alpha>0.0082$. The newly emerged period-I limit-cycle
attractor for other sets of initial conditions is coexisted along with the
period-II orbit is also depicted in Fig. 3(a) as lines with solid circles. The
labels SN indicates the saddle-node bifurcation. The evolution of the three
fixed points (lines with open triangle, asterisk, and plus symbols) with
respect to $\alpha$ is also depicted in Fig. 3(a). Among them the newly
emerged fixed point is depicted with asterisk symbol.

The emergence of multistable attractors for different set of initial
conditions as a function of the nonisochronicity parameter can be confirmed
again in Fig. 3(b), in which the maxima of the system variable $x$ ($x_{max}$)
are plotted as a function of $\alpha$, shows the qualitative changes that
occurred in the system. The bifurcation of the perioic orbits and the
emergence of quasiperiodic states can be clealry visualized in Fig. 3(b) with
respect to $\alpha$. The inset in Fig. 3(b) elucidate the period-doubling
phenomenon.

Figure 4: Basins of attraction of the system (II) as a function of the system
variables $x$ and $y$ for (a) $\alpha=0.007$, and (b) $\alpha=0.01$ shows the
emergence of coexisting attractors for different initial states.

Further, to identify the initial states to which different attractors emerge
in the system, the basins of attraction of the Rössler oscillator is plotted
in Figs. 4(a) and 4(b) by fixing $z=-1$ for $\alpha$ = 0.007 and 0.01,
respectively (corresponding to the attractors depicted in Figs. 2(b) and
2(c)). In Fig. 4(a), the green (dark gray) colored region indicates the basin
of attraction of the period-II limit-cycle attractor, the area of orange
(light gray) color represents the basin of attraction of the period-I limit-
cycle attractor, and finally, the blue (black) color domain portrays the
initial states to which the large-amplitude quasiperiodic attractor emerge in
the system. We wish to emphasize here that, for our upcoming study, we choose
the initial states only from the limit-cycle region and omit the initial
states, which leads to the large-amplitude quasiperiodic oscillations.

Thus, an individual Rössler oscillator exhibits coexisting attractors with
respect to the strength of the nonisochronicity parameter. It is also of
interest to investigate the dynamics of the coupled system and how the
coupling parameter, along with the nonisochronicity term, influence the system
dynamics. Therefore, in Sec. IV, we will consider two mutually coupled system
of Rössler oscillators and study the dynamics as a function of the coupling
and nonisochronicity parameters.

Figure 5: Bifurcation diagram of the system (IV) illustrates the occurrence of
different bifurcations and multiple attractors as a function of the coupling
strength $\varepsilon\in[0,2.5]$ for $\alpha=0.01$. The vertical dashed lines
indicate the critical values of $\varepsilon$, in which different bifurcations
occur and separates the region from R1 to R8. Other parameters are fixed as
given in Fig. 1.

##  IV Dynamics of the coupled Rössler oscillator and the influence of
nonisochronicity and coupling parameters

The mathematical model of a system of two mutually coupled Rössler oscillators
can be given as

$\displaystyle\dot{x}_{1,2}$ $\displaystyle=$
$\displaystyle-\omega_{0}(1-\alpha(x^{2}+y^{2}))y_{1,2}-z_{1,2}+\varepsilon(x_{2,1}-x_{1,2}),$
$\displaystyle\dot{y}_{1,2}$ $\displaystyle=$
$\displaystyle\omega_{0}(1-\alpha(x^{2}+y^{2}))x_{1,2}+ay_{1,2},$ (6)
$\displaystyle\dot{z}_{1,2}$ $\displaystyle=$ $\displaystyle
b+(x_{1,2}-c)z_{1,2},$

where $\varepsilon~{}(\geq 0)$ represents the strength of diffusive coupling.
In Eq. (IV), if we vary the coupling strength, by fixing $\alpha$ as a
constant, the coupled system undergoes critical bifurcations. Consequently,
different desynchronized attractors have emerged due to the breaking of the
system’s permutational symmetry and coexisted for different sets of initial
states, including the birth of two different OD states. The one-parameter
bifurcation diagram is estimated as a function of $\varepsilon$ by fixing
$\alpha$ to display the development of different bifurcations and show the
transition to complete synchronization state of the coupled system.

Table 1: Table shows the emergence of different attractors in system(IV) as a function of coupling strength $\varepsilon\in[0,2.5]$ for nonisochronicity parameter $\alpha=0.01$. Attractor name | Oscillation type | Range of $\varepsilon\in[0,2.5]$
---|---|---
H1 | Homogeneous oscillations–1 | 0 $<\varepsilon\leq$ 0.001
QP1 | Quasiperiodic oscillations–1 | 0.001 $<\varepsilon\leq$ 0.147
QP2 | Quasiperiodic oscillations–2 | 0.001 $<\varepsilon\leq$ 0.22
H2 | Homogeneous oscillations–2 | 0.001 $<\varepsilon\leq$ 2.356
IH1 | Inhomogeneous oscillations–1 | 0.147 $<\varepsilon\leq$ 0.34
IH2 | Inhomogeneous oscillations–2 | 0.22 $<\varepsilon\leq$ 0.225
OD1 | Oscillation death state–1 | 0.225 $<\varepsilon\leq$ 0.521
IH3 | Inhomogeneous oscillations–3 | 0.245 $<\varepsilon\leq$ 0.341
OD2 | Oscillation death state–2 | 0.341 $<\varepsilon\leq$ 0.653
IH4 | Inhomogeneous oscillations–4 |  $\varepsilon>$ 0.521
S | Synchronized oscillations |  $\varepsilon>$ 2.356
Figure 6: Time evolution and the corresponding phase portraits of the system
(IV) obtained for different initial states for $\alpha=0.01$. [(a), (b)]
Homogeneous oscillations–2 (H2) for $\varepsilon=0.14$, and [(c), (d)] shows
the synchronized oscillations (S) of the subsystems for $\varepsilon=3.0$.
Figure 7: Time evolution and the respective phase portraits of the system (IV)
obtained for $\alpha=0.01$. [(a), (b)] Quasiperiodic oscillations–1 (QP1)
obtained for another set of initial states for $\varepsilon=0.14$, [(c), (d)]
inhomogeneous oscillations–1 (IH1) for $\varepsilon=0.25$, and [(e), (f)]
shows the OD state–2 (OD2) for $\varepsilon=0.4$.

The bifurcation diagram of the coupled system (IV) is depicted in Fig. 5 for
$\varepsilon\in[0,2.5]$, and $\alpha=0.01$. Again, the stable orbits are
identified using the lines with different points, and broken lines indicate
the unstable orbits. The vertical dashed lines represent the critical values
of $\varepsilon$, at which different bifurcations occur. Also, they separate
the bifurcation diagram into different regions based on the bifurcations that
occur. In the absence of coupling, both subsystems exhibit the coexistence of
period-I limit-cycle and large-amplitude quasiperiodic attractors, as depicted
in Fig. 2(c). In particular, the period-I limit-cycles of the subsystems
oscillate with equal amplitude, satisfying the permutational symmetry
($x_{1},y_{1},z_{1}$)$\longleftrightarrow$($x_{2},y_{2},z_{2}$), but the
phases are uncorrelated (exhibit constant phase shift), characterized as
homogeneous oscillations–1 and the attractors are named as H1. The solid
diamonds in Fig. 5 for $\varepsilon=0$ represent the maxima and minima of the
homogeneous oscillations. When we slowly increase the coupling strength beyond
$\varepsilon>0.001$, three different attractors have newly emerged in the
system and coexisted for different initial states. These three attractors are
further brought to the number of attractors as a function of the coupling
strength, via different bifurcation routes. To be more specific, the systems
exhibit desynchronized period-I limit-cycle attractor (H2) along with two
different quasiperiodic attractors, namely QP1 and QP2. The quasiperiodic
dynamics is confirmed again by estimating the power spectrum. In the
following, how these three attractors bifurcated and give birth to new
attractors via different bifurcations as a function of $\varepsilon$ is
explained.

Figure 8: Time evolution and the equivalent phase portraits of the system (IV)
obtained for different initial states for $\alpha=0.01$. [(a), (b)]
Quasiperiodic oscillations–2 (QP2) for $\varepsilon=0.14$, [(c), (d)]
inhomogeneous oscillations–2 (IH2) for $\varepsilon=0.221$, [(e), (f)] OD
state–1 (OD1) obtained for an another set of initial states for
$\varepsilon=0.25$, [(g), (h)] inhomogeneous oscillations–4 (IH4) for
$\varepsilon=0.8$, and [(i), (j)] shows the emergence of inhomogeneous
oscillations–3 (IH3) for $\varepsilon=0.25$.

  1. 1.

In the attractor H2, both the subsystems begin to evolve with different
amplitudes (due to the breaking of permutational symmetry), but still rotating
with the common center of rotation, which is called as homogeneous
oscillations–2. The line with open circles in Fig. 5 indicates the maxima and
minima of the attractor H2. The time series and related phase space plots of
the desynchronized attractor H2 are depicted in Figs. 6(c) and 6(d),
respectively, for $\varepsilon=0.14$. The attractor H2 is stable until
$\varepsilon=2.356$ and when we increase the coupling strength to
$\varepsilon>2.356$, the attractor H2 attains the complete synchronization
manifold, and the symmetry is again preserved in the subsystems. The emerged
synchronized attractor is labeled as S (lines with inverted triangles in Fig.
5), and the time series as well as the phase portrait plots of the attractor S
are depicted in Figs. 6(e) and 6(f), respectively, for $\varepsilon=3.0$.

  2. 2.

Secondly, the QP1 has emerged in the system, if we choose the initial states
from another basin of attraction. The two attractors are rotating around the
different centers of rotation. We named this attractor as QP1, and this
quasiperiodic attractor is stable in the range of the coupling strength
$\varepsilon\in(0.001,0.147]$. The time evolution and the phase space plots of
the QP1 attractor are plotted in Figs. 7(a) and 7(b) for $\varepsilon=0.14$.
When we increase the coupling strength further, the QP1 attractor lost its
stability through torus bifurcation (marked as TB1 in Fig. 5) and transformed
into stable desynchronized limit-cycle oscillations for $\varepsilon>0.147$.
The lines with crosses indicate the maxima and minima of the corresponding
limit-cycle attractor. Here, both the oscillators have a different center of
rotation, which is shown in Figs. 7(c) and 7(d) for $\varepsilon=0.25$. We
named these attractors as IH1 (inhomogeneous oscillations–1). The amplitude of
the attractor IH1 is slowly decreasing when one increases the coupling
strength, and for $\varepsilon=0.341$, the attractor IH1 is transformed into
OD state–2 (attractor OD2) that has emerged via reverse supercritical Hopf
bifurcation. The lines with open triangles in Fig. 5 represent attractor OD2,
and the critical coupling strength of the Hopf bifurcation point is labeled as
HB2 in Fig. 5. This steady-state is portrayed in Figs. 7(e) and 7(f) for
$\varepsilon=0.4$. Upon increasing the coupling strength further, the OD
state–2 and another unstable fixed point collides with each other and
disappeared via saddle-node bifurcation when the coupling strength reaches the
value of $\varepsilon=0.653$. The respective bifurcation point is marked as
SN2 in Fig. 5.

  3. 3.

Finally, QP2 attractor exhibits in the range of $\varepsilon\in(0.001,0.22]$.
The QP2 attractor also rotates about the different center of rotations. The
time series and phase space plots of the QP2 attractor are depicted in Figs.
8(a) and 8(b), respectively, for $\varepsilon=0.14$ indicating that the
amplitude of the QP2 attractor is relatively smaller than QP1. Next, if we
increase $\varepsilon$ further, the QP2 attractor is then bifurcated into
stable limit-cycle attractor through torus bifurcation (TB2) at
$\varepsilon=0.22$ (see Fig. 5). The newly emerged limit-cycles also have two
different centers of rotation and oscillate with different amplitudes
corroborated from the time evolution and phase portraits, as depicted in Figs.
8(c) and 8(d), respectively, for $\varepsilon=0.221$. We named this attractor
as IH2 (inhomogeneous oscillations–2). These limit-cycles are stable only for
a short range of $\varepsilon$ (filled triangles in Fig. 5) and then
bifurcated into an OD state–1 (OD1) for $\varepsilon=0.225$ via reverse
supercritical Hopf bifurcation, at which the two systems attain different
stable, steady-states. For instance, the corresponding states are portrayed in
Figs. 8(e) and 8(f) for $\varepsilon=0.25$. The critical Hopf bifurcation
point is marked as HB1 in Fig. 5 and the open squares depicted the attractor
OD1. After that, the OD1 attractor is lost stability and transformed into a
stable limit-cycle attractor (denoted by the solid squares in Fig. 5) for
$\varepsilon=0.521$ via supercritical Hopf bifurcation. The newly emerged
attractor also have two different origins of rotation, which are named as IH4
(inhomogeneous oscillations–4). The associated time series and phase portraits
of the IH4 attractor are depicted in Figs. 8(g) and 8(h), respectively, for
$\varepsilon=0.8$ confirming that the amplitudes are uncorrelated
(desynchronized). Finally, only two limit-cycle attractors IH4
(desynchronized) and S (synchronized) are stable and coexisted.

Figure 9: Basins of attraction of the system (IV) as a function of the state
variables $x_{1}$ and $x_{2}$ for different values of the coupling strength
$\varepsilon$ with $\alpha$=0.01,(a) plotted for $\varepsilon$=0.1, (b) for
$\varepsilon$ = 0.18, (c) for $\varepsilon$ = 0.221 and (d) is depicted for
$\varepsilon$ = 0.25. The inset shows the riddled nature of the basin in
certain ranges of initial states. Figure 10: Basins of attraction of the
system (IV) as a function of the state variables $x_{1}$ and $x_{2}$ for
different values of the coupling strength $\varepsilon$ with $\alpha$=0.01,(a)
plotted for $\varepsilon$=0.4, (b) for $\varepsilon$ = 0.6, (c) for
$\varepsilon$ = 0.8 and (d) is depicted for $\varepsilon$ = 3.0. Figure 11:
Probability of various attractors from H1 to S including the quasiperioc
attractor QP1 and QP2 as a function of the coupling strength and for
$\alpha=0.01$. Different colors and patterns indicate the probability of
different attractors.

Meanwhile, when the coupling strength reaches the value of
$\varepsilon=0.245$, an unstable periodic orbit is bifurcated into a stable
limit-cycle which is indicated as inhomogeneous oscillations–3 and the
corresponding attractors are named as IH3 (solid circles in Fig. 5), in which
one of the subsystems oscillate with period-II limit-cycle oscillations, and
the other subsystem exhibits period-I limit-cycle oscillations which are
evident from Figs. 8(i) and 8(j), plotted for $\varepsilon=0.25$. These
attractors are stable until the coupling strength reaches the value of
$\varepsilon=0.341$. After that, the attractor IH3 is merged with an unstable
orbit and disappeared via saddle-node bifurcation (denoted as SN1 in Fig. 5).

Figure 12: Emergence of multistable attractors for different initial states
are illustrated in a two-parameter diagram of the coupled Rössler oscillator
(IV) as a function of nonisochronicity parameter ($\alpha\in[0,0.013]$) and
coupling strength $\varepsilon\in[0,3]$. Distinct colors and patterns
represent the combination of different attractors. The combination of
attractors in regions R1 to R8 are already illustrated in Sec. V. The
combination of attractors in other regions from R9 to R17 are as follows: R9
indicates the coexistence of attractors H2 and QP1. Similarly, R10 =
H2+IH1+OD1+IH3, R11 = H2+IH3+OD2+IH4, R12 = H2+IH1+IH3, R13 = H2+IH1+OD1, R14
= IH1+S, R15 = IH1+OD1+S, R16 = OD1+OD2+S, R17 = OD2+IH4+S. The think black
continuous line indicates the complete synchronization region obtained using
MSF formalism (8). The parameter values are fixed as given in Fig. 1

Thus, the influence of coupling, along with nonisochronicity parameters, is
studied in terms of the bifurcation diagram. The emergence of different
bifurcations, and coexisting synchronized and desynchronized attractors are
discussed. In a nutshell, Table 1 shows the possible attractors that are
appeared in the system (IV) as a function of $\varepsilon\in[0,2.5]$ for
$\alpha=0.01$. Next, based on the emergence of different coexisting
attractors, the bifurcation diagram is divided into a number of regions, and
basins of attraction of each region are described in Sec. V.

##  V Basins of attraction in different regions of the bifurcation diagram

Based on the appearance of different bifurcations, Fig. 5 is divided into
eight regions from R1 to R8, in which different coexisting attractors have
emerged for different initial states as a function of the coupling strength.
In the following, we discuss the basins of attraction of those regions. It is
important to emphasize here that in order to plot the basins of attraction of
the coupled system, the initial states of the $x$-variables of the two
individual systems are uniformly distributed in the range of
$x_{1,2}\in[-7,7]$ and the initial states of the other state variables are
fixed at $y_{1,2}=0.5$ and $z_{1,2}=-1$.

Figure 13: The basins of attraction of the coupled Rössler oscillator (IV) for
(a) $\varepsilon=0.5$, and (b) for $\varepsilon=0.6$ with $\alpha=0.006$ shows
the coexisting attractors in the regions of R15 and R16, respectively.

In region R1, the system exhibits the coexistence of attractors H2, QP1, and
QP2. The basins of attraction of the respective attractors are depicted in
Fig. 9(a) for $\varepsilon$ = 0.1. In this figure, the blue (dark gray) color
region represents the basin of attraction of attractor H2. The light gray
color region indicates the basin of attraction of the QP1, and the initial
states to which the attractor QP2 has emerged are indicated by the region of
maroon (black) color. We can also note that the attractors H2 and QP1 are
having an almost equal number of occurrences. The attractor QP2 has occurred
for less number of initial states than the other two attractors. Next, in
region R2 of the bifurcation diagram, the coupled system exhibits the
coexistence of attractors H2, IH1, and QP2, in which IH1 has occurred
relatively for a large number of initial states than the other two attractors.
The corresponding basins of attraction are plotted in Fig. 9(b) for
$\varepsilon=$ 0.18, where at the pink (light gray) color portrays the basin
of attraction of IH1 attractor. Next, the attractors H2, IH1 and IH2 coexist
in the region R3 for different sets of initial states and the respective
basins of attraction is plotted in Fig. 9(c) for the value of coupling
strength $\varepsilon$ = 0.221. The dark gray color points illustrate the
initial states to which the IH2 has emerged in the system, and IH1 dominates
the other two attractors with a high number of occurrences. Further, the
system (IV) shows the coexistence of four different attractors including H2,
IH1, OD1, and IH3 in region R4. The basins of attraction of those attractors
are depicted in Fig. 9(d) for the value of coupling strength $\varepsilon$ =
0.25 corroborating that H2 has a high number of occurrences. The red (dark
gray) color indicates the basin of attraction of OD1, and the light green
(gray) color illustrates the basin of attraction of the attractor IH3.

Moreover, in region R5, the coupled system exhibits the coexistence of two OD
states (OD1 and OD2) with attractor H2. For instance, the corresponding basins
of attraction is plotted in Fig. 10(a) for $\varepsilon=0.4$ in which the dark
blue (black) color region describes the basin of attraction of the attractor
OD2. Next, the basins of attraction in the region R6 are plotted in Fig. 10(b)
for $\varepsilon=0.6$ in that the system shows the coexistence of the
attractors H2, OD2, and IH4. The aqua (light gray) color indicates the basin
of attraction of IH4. Furthermore, the basins of attraction of region R7 is
depicted in Fig. 10(c) for $\varepsilon$ = 0.8 at which only the attractors
A$1$ and IH4 are co-occurred. In all these Figs. 10(a),10(b) and 10(c) the
attractor H2 dominates the other attractors with a high number of occurrences.
Finally, in region R8, the newly emerged synchronized attractor S has
coexisted with IH4, and dominates the basin of attraction with a high number
of occurrences, which is plotted for $\varepsilon$ = 3.0 in Fig. 10(d) .

Therefore, from Figs. 9 and 10, one can observe that each attractor dominates
the other attractors with a high number of occurrences for large values of
initial states. Therefore it is also of interest to investigate the
probability of different attractors with respect to the coupling parameter
$\varepsilon$menck2013 ; schultz2017 ; brzeskit2019 . Figure 11 shows the
probability of various attractors for 500 uniformly distributed initial states
of $x_{1,2}\in$ [-7, 7] for $\alpha=0.01$ and as a function of $\varepsilon$.
From Fig. 11, one can easily uncover that the attractor H1 predominantly
occurs in the probability diagram with a high number of occurrences and shares
the basin of attraction with other coexisting attractors for
$\varepsilon<2.356$. After that, the S attractor dominates the basin to our
calculated range of $\varepsilon=3.0$ and shares the parameter space with IH4.

To clarify the coexistence of different attractors, we have plotted a two-
phase diagram in the parameter space of $\varepsilon\in[0,3]$ and
$\alpha\in[0,0.013]$ in Fig. 12, which depicts the multistability of
attractors for different values of $\varepsilon$ and $\alpha$. Each colored
and patterned regions indicate the coexistence of different attractors. One
can also notice that, apart from the regions R1 to R8 as discussed earlier,
there exist other combinations of coexisting attractors in regions from R9 to
R17 which are also indicated in Fig. 12. For an illustration, in region R15,
the attractors IH1, OD1 and S coexist and the respective basins of attraction
is plotted in Fig. 13(a) for $\varepsilon=0.5$ and $\alpha=0.006$. Similarly,
the basins of attraction of the region R16 is plotted in Fig. 13(b) for
$\varepsilon=0.6$ and $\alpha=0.006$, in which OD1 and OD2 are coexisting
along with the synchronized attractor S.

The region of the stable synchronized state for the coupled system (IV) is
identified using the well-known MSF formalism. The complete synchronization
manifold is defined by $x_{1}=x_{2}=x$, $y_{1}=y_{2}=y$ and $z_{1}=z_{2}=z$,
where $x,y,z$ are the solutions of the uncoupled system (II). Let,
$(\eta_{j},\xi_{j},\zeta_{j})$ are the deviations of $(x_{j},y_{j},z_{j})$,
$j=1,2$ from the synchronized solution $(x,y,z)$ and we have

$\displaystyle\dot{\eta}_{1,2}$ $\displaystyle=$
$\displaystyle-\omega_{0}(\xi_{1,2}-\alpha(2xy\eta_{1,2}+(x^{2}+3y^{2})\xi_{1,2}))-\zeta_{1,2}+$
$\displaystyle\epsilon(\eta_{2,1}-\eta_{1,2})$ $\displaystyle\dot{\xi}_{1,2}$
$\displaystyle=$
$\displaystyle\omega_{0}(\eta_{1,2}-\alpha(2yx\xi_{1,2}+(3x^{2}+y^{2})\eta_{1,2}))+a\xi_{1,2}$
$\displaystyle\dot{\zeta}_{1,2}$ $\displaystyle=$ $\displaystyle
x\zeta_{1,2}+z\eta_{1,2}-c\zeta_{1,2}.$ (7)

From Eq. (7) we get,

$\displaystyle\dot{\eta}$ $\displaystyle=$
$\displaystyle-\omega_{0}(\xi-\alpha(2xy\eta+(x^{2}+3y^{2})\xi))-\zeta-2\epsilon\eta$
$\displaystyle\dot{\xi}$ $\displaystyle=$
$\displaystyle\omega_{0}(\eta-\alpha(2yx\xi+(3x^{2}+y^{2})\eta))+a\xi$
$\displaystyle\dot{\zeta}$ $\displaystyle=$ $\displaystyle
x\zeta+z\eta-c\zeta,$ (8)

where
$(\eta,\xi,\zeta)=(\eta_{1}-\eta_{2},\xi_{1}-\xi_{2},\zeta_{1}-\zeta_{2})$ are
the transverse small perturbations on the synchronization manifolds. The
synchronized state is stable when all the transverse perturbations
$(\eta,\xi,\zeta)$ tend to zero when $t\rightarrow\infty$. This will happen
when all the transverse Lyapunov exponents are negative. Using the equation
(8) we have identified the region of stable complete synchronization state in
($\varepsilon,\alpha$) parameter space. The thick black continuous line in
Fig. 12 represents the region of stable complete synchronization.

##  VI Conclusions

In summary, we have systematically studied the influence of nonisochronicity
on the single and mutually coupled Rössler oscillator. First, we have examined
the fixed point analysis of a single Rössler oscillator with and without
nonisochronicity term. We showed that adding the nonisochronicity term can
increase the number of fixed points, which induces multistability of
coexisting limit-cycle and quasiperiodic attractors. Next, we have extended
our studies to a system of two mutually coupled Rössler oscillators and
investigated the influence of the coupling parameter along with the
nonisochronicity parameter. We showed that the permutational symmetry of the
coupled system is broken when one increases the coupling strength, persuades
multistability of desynchronized attractors, which are emerged via different
bifurcations. The broken symmetry is preserved again in the system when the
coupling strength reaches higher values, engenders the emergence of a
synchronized attractor. In addition to the synchronized and desynchronized
attractors, we have reported the emergence of two different OD states. The
results are corroborated using the bifurcation diagrams, and basins of
attraction. The region of complete synchronization is identified using the MSF
formalism.

The results presented in the paper are robust against system parameters, and
similar results can be observed even if the systems exhibit chaotic dynamics.
Also, we note that the results are obtained only in the two coupled systems.
However, it is also of interest to understand the interplay of nonisochronous
nature of the system and coupling topology in different network architectures
to induce rich collective dynamical behaviors and to identify the network
topologies that enhance or support the effects of nonisochronicity are still
open problems, and we are processing our research in this direction.

###### Acknowledgements.

C. Ramya acknowledges SASTRA for financial support in the form of Teaching
Assistantship. The work of R. S. is supported by the SERB-DST Fast Track
scheme for young scientists under Grant No. YSS/2015/001645. The work of V. K.
C. and R. G. are sponsored by the SERB-DST-CRG Grant No. CRG/2020/004353 and
CSIR EMR Grant No. 03(1444)/18/EMR-II. All the authors thank DST, New Delhi
for computational facilities under the DST-FIST programme
(SR/FST/PS-1/2020/135) to the Department of Physics.

DATA AVAILABILITY
The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# The ISCSLP 2022 Intelligent Cockpit Speech Recognition Challenge (ICSRC):
Dataset, Tracks, Baseline and Results

###### Abstract

This paper summarizes the outcomes from the ISCSLP 2022 Intelligent Cockpit
Speech Recognition Challenge (ICSRC). We first address the necessity of the
challenge and then introduce the associated dataset collected from a new-
energy vehicle (NEV) covering a variety of cockpit acoustic conditions and
linguistic contents. We then describe the track arrangement and the baseline
system. Specifically, we set up two tracks in terms of allowed model/system
size to investigate resource-constrained and -unconstrained setups, targeting
to vehicle embedded as well as cloud ASR systems respectively. Finally we
summarize the challenge results and provide the major observations from the
submitted systems.

Index Terms: Automatic speech recognition, intelligent cockpit, in-vehicle
speech recognition

##  1 Introduction

As cars become an indispensable part of human daily life, a secure and
comfortable driving environment is more and more desirable. The touch-based
interaction in the traditional cockpit is easy to distract the driver’s
attention, leading to inefficient operations and potential security risks.
Thus the concept of the intelligent cockpit is gradually on the rise [1, 2],
which aims to achieve a seamless driving and cabin experience for drivers and
passengers by integrating multimodal intelligent interactions, like speech,
gestures, body, etc., with different functions, like commands recognition,
entertainment, navigation, etc [3]. As a natural user interface (NUI), a
robust automatic speech recognition (ASR) system is crucial [4, 5]. In recent
years, ASR systems, particularly those end-to-end (E2E) neural approaches,
such as connectionist temporal classification (CTC) [6, 7], recurrent neural
network transducer (RNN-T) [8], attention-based encoder-decoder (AED) [9, 10,
11], have been gradually employed in many applications such as voice
assistant, video captioning and meeting transcription. Although speech
interface has been recently adopted in many cars, there are still some
significant challenges for ASR in the driving scenario.

First of all, the acoustic environment of the cockpit is complex. Since the
cockpit is a closed and irregular space, it has special room impulse response
(RIR), resulting in special reverberation conditions. In addition, there are
various kinds of noise during driving from both inside and outside, such as
wind, engine, wheel, background music and interfering speaker, etc. Although
recent advances on speech front-end processing and robust speech recognition
can be leveraged, the special acoustic conditions mentioned above should be
particularly considered. We notice that researchers have started to explore
different techniques to reduce noise in the vehicle for building a robust
speech recognition for intelligent cockpit [12, 13]. Separating different
speakers and realizing multi-talker ASR in a vehicle are also desired to truly
facilitate the needs of both driver and passenger(s).

Another challenge lies in the linguistic content to be recognized by a cockpit
ASR system. Specifically, the main content of intelligent cockpit speech
interaction is the user’s commands, which includes controlling the air
conditioner, playing songs, navigating, etc. These commands may involve a
large number of named entities such as contacts, singer names, and points of
interest (POI). Recognizing these phrases in the cockpit scene remains to be
improved because most named entities are rare in training data. For instance,
the commands for navigation involve lots of place names that are hard to
recognize while it’s important to recognize them correctly for better user
experience. This problem is essentially a domain mismatch problem in
linguistic aspects. To solve this problem, previous works have explored the
methods of fusing an external language model into the decoding process to
inject static contextual knowledge [14, 15, 16]. In contrast, injecting
contextual knowledge dynamically is also effective, such as attention-based
deep context [17, 18] and trie-based deep biasing [19, 20]. Besides the
language aspects, more challengingly, detecting to whom (voice assistant or
passengers) the speaker is talking is also essential in a full-duplex speech
interaction system. Moreover, the use of multi-modal integration, such as
audio-visual information, is definitely beneficial to smart cockpit systems
including ASR [21, 22].

Data sparsity is another obstacle to the related research in intelligent
cockpit scenario. This stands in stark contrast to the recent proliferating of
open-source speech data covering multiple domains and
languages111https://www.openslr.org. This is probably because the speech
collection process itself in cabin environments is difficult and covering
various acoustic conditions during recording is even more challenging [23].

To trigger speech recognition and related research in the intelligent cockpit
scenario, we launch the Intelligent Cockpit Speech Recognition Challenge
(ICSRC)222https://iscslp2022-icsrc.org in ISCSLP2022. In this challenge, a
20-hour Mandarin speech dataset recorded in a new-energy vehicle (NEV) is
available to participants. This dataset covers a variety of acoustic
conditions and linguistic contents. We particularly set up two tracks in terms
of allowed model/system size to investigate resource-constrained and
-unconstrained setups, targeting to vehicle embedded as well as cloud ASR
systems respectively. This paper introduces the details of the dataset,
arrangement on challenge tracks, baseline system and summary on the challenge
results. We hope the open data and the challenge will serve as a benchmark and
common test-bed for in-vehicle speech recognition.

##  2 Data

The released dataset of the challenge contains 20 hours of speech data in
total, which is divided into 10 hours for evaluation (Eval) in the model
training phase and another 11 hours as Test set for challenge scoring and
ranking. Both Eval and Test sets have 50 non-overlapped speakers with balanced
gender coverage. The Eval set is released to the participants at the beginning
of the challenge and can be used for finetuning model, while the Test set is
released at the final challenge scoring stage. For the Training set,
participants can only use the constrained data allowed by the challenge, which
are open source corpora selected from openslr.org. More details of the
released data are shown in Table 1.

Table 1: Details of the released Eval and Test data. Dataset | Uttrance | Duration (h) | Format
---|---|---|---
Eval | 8,780 | 10.01 |  16kHz, 16bit
Test | 9,509 | 11.06

The data is collected in the NEV with a Hi-Fi microphone placed on the display
screen of the car. During recording, a speaker sits on the front passenger
seat. The distance between the microphone and the speaker is around 0.5 m. All
speakers are native Chinese speaking Mandarin without strong accents. During
driving, the driver may drives on different road conditions (city streets and
highways) with various driving speed, open and close windows and play music,
which cover various acoustic conditions. The data distribution in terms of
signal-to-noise ratios (SNR) is shown in Figure 1. We can see that the SNR
covers a wide range and a large part of the data is collected with strong
noise. According to the linguistic content, the dataset can be categorized
into five categories listed as follows and the statistics are further detailed
in Table 2.

  * •

Air Conditioner: includes commands to control the air conditioner, like “调高温度
(turn up the temperature)”;

  * •

Phone Call: includes commands to make a phone call, like “呼叫张奥 (call Ao
Zhang)”;

  * •

Music: includes commands to control media player, like “播放周杰伦的歌 (play Jay
Zhou’s song)”;

  * •

Point of Interest (POI): includes commands to navigation, like “导航到博物馆 (route
to the museum)”;

  * •

Others: instead of commands, like reading news, conversations between
passengers.

Table 2: The percentage (%) of each category for the dataset. Category | Air Cond. | Phone Call | Music | POI | Others
---|---|---|---|---|---
Eval | 15 | 10 | 15 | 15 | 45
Test | 18 | 12 | 13 | 16 | 41
Figure 1:  SNR statistics of the released Eval and Test data.

##  3 Tracks, Evaluation and Baselines

In the challenge, we design two tracks for participants to select, which
evaluated in typical error metric of ASR. We also release a baseline system on
github for quick start.

###  3.1 Track setting

In real applications, the demand for intelligent cockpit speech recognition
system is twofold. With better Internet connection, one can run on the cloud
side with larger model and higher performance. Another is desired to run
locally on the vehicle with smaller footprint, lower latency as well as better
privacy. Therefore, we simply set up two tracks in the challenge in terms of
the limitation on the model size:

  * •

Track I (Limited model size): The number of model parameters cannot exceed 15M
and the finite state transducer (FST) file of the system should be less than
25M if participants build a system with FST.

  * •

Track II (Unlimited model size): The number of model parameters is not
limited.

###  3.2 Evaluation Metrics

In the challenge, the performance of an ASR system is measured by the typical
Character Error Rate (CER). The CER indicates the percentage of Mandarin
characters that are incorrectly predicted. Specifically for a given hypothesis
output, it measures the ratio of character errors, including insertions (Ins),
substitutions (Subs) and deletions (Del), to the total characters spoken in an
utterance under the dynamic programming algorithm. According to the challenge
rule, if the CERs of two systems on the Test set are coincidentally the same,
the one with lower model size and computation complexity will be judged as the
superior one.

###  3.3 Baseline

We release baseline
recipes333https://github.com/smallsmartao/ICSRC_2022_baseline based on the
WeNet [24] end-to-end speech recognition toolkit for quick start and
reproducible research. Specifically, our baseline ASR system follows the
standard configuration of the state-of-the-art Conformer structure – a
12-block encoder and a 6-block decoder, where the dimension of attention and
feed-forward layer is set to 256 and 2048 respectively. In all attention sub-
layers, the head number of the multi-head attention is set to 4. As for the
acoustic feartures, we use 80-dimensional mel-filterbanks (FBank) with a 25ms
frame length and a 10ms shift. The whole network is trained using the open-
source AISHELL-1 [25] corpus for 240 epochs and warm-up is used for the first
25,000 iterations. The training objective is a logarithmic linear combination
of the CTC and attention objectives. For inference, we adopt the attention-
rescoring decoding strategy as default. We also apply SpecAugment [26] for all
data during model training.

##  4 Results and Discussion

Twenty five teams finally submitted their results to the challenge, and the
CER for the top 4 teams of each track is summarized in Table 3. Observing the
detailed performance by subsets of different category, we can see that the
phone call and POI subsets, which involve contacts and location addresses, are
still challenging for a general speech recognizer with the current limited
training data. Specifically, there are clear performance gaps between the air
conditioner subset which is mostly composed of fixed control commands and the
other subsets which are much richer in linguistic content.

The major techniques used by the top ranking teams, including network
structure, data augmentation strategy, front-end processing, language modeling
and training strategy, are summarized in Table 4 and discussed in the the
following subsections accordingly.

We notice that the winner goes to team Toong (T044) in both tracks with the
lowest CER of 10.66% and 8.94% respectively, surpassing the official baseline
(53.57%/47.98%) with a large margin. The good performances are obtained by a
CTC model for Track I and a CTC/AED hybrid model for Track II respectively.
Particularly, the superior performance is attributed largely to a 6-gram
language model with iterative interpolation and pruning strategy. Moreover,
system fusion is also beneficial as reported by the winner team.

Table 3: CERs on test set for top ranking teams and baselines (BSL) Track |  | Team
---
Code
| Air
---
Cond.
| Phone
---
Call
Music | POI | Others | All
1 | T044 | 1.86 | 14.26 | 9.74 | 21.23 | 10.93 | 10.66
T002 | 3.45 | 21.06 | 10.86 | 27.18 | 11.36 | 12.67
T009 | 2.63 | 15.78 | 12.57 | 24.31 | 14.54 | 13.39
T013 | 1.42 | 15.75 | 13.49 | 25.66 | 14.89 | 13.57
BSL | 45.30 | 61.98 | 62.03 | 67.48 | 50.37 | 53.57
2 | T044 | 1.54 | 13.78 | 7.22 | 19.11 | 8.66 | 8.94
T009 | 1.35 | 12.18 | 8.78 | 17.34 | 11.03 | 9.86
T039 | 1.17 | 13.96 | 9.79 | 21.56 | 10.25 | 10.20
T002 | 3.08 | 19.62 | 8.89 | 24.54 | 7.88 | 10.21
BSL | 40.86 | 57.03 | 57.20 | 61.82 | 44.12 | 47.98

###  4.1 Network Structure

The network structures used by the top 4 teams of both tracks are summarized
in Table 4. We can see that most teams adopt the transformer-based E2E neural
model, based on an encoder-decoder structure with an attention mechanism.
Specifically, the Conformer architecture [27] is mostly employed as the
encoder. Moreover, team T039 [28] employs the recently proposed Squeezefomer
[29] as the encoder, which re-examines the design choices for both the macro
and micro-architecture of Conformer. Considering the model size limitation in
Track I, most teams choose to adopt fewer network layers or smaller hidden
size. Note that the winner team T044 even abandons the decoder and only keeps
the encoder with CTC as the acoustic model. In Track II, without the
limitation on model size, the winner team T044 employs an extremely large
model with 574M parameters, which exceeds the model size of other teams
obviously, to bring great modeling ability. Team T009 [30] adopts self-
conditioned folded encoder [31] and inter-CTC loss [32] to perform iterative
refinement implicitly by stacking multiple shared encoder layers with fewer
parameters, which leads to 6.59% relative CER reduction on the Test set
(12.89% to 12.04%). Team T039 adds a predictor and RNN-T loss to the
transformer-based model and the integrated model, which leads to 0.48%
absolute CER reduction compared with the original transformer-based model.
Besides, ROVER [33] is also adopted by this team to fuse the multiple outputs
from different ASR models, which brings 14.93% relative CER reduction (from
11.90% to 10.2%) on the Test set.

Table 4: Major techniques from top performing teams. Here En and De stand for
encoder and decoder respectively.

Track | Rank | Team | Data Aug. | Network | Loss | Model Size (M) | LM & TLG Size | CER (%)
---|---|---|---|---|---|---|---|---
1 | 1 | T044 |  | Spd,Ptch,Ns,
---
Rvb,SpcAug
En: 5 Cfmr Blk | CTC | 12.34 |  | 6-gram
---
25M
10.66
2 | T002 |  | Spd,Ptch,Ns,
---
Rvb,SpcAug
| En: 5 Cfmr Blk
---
De: 4 Tfmr Blk
CTC Att | 14.6 |  | 3-gram
---
24M
12.67
3 | T009 |  | Spd,Ptch,Ns,
---
Rvb,SpcAug,TTS
| En: 12 Cfmr Blk
---
De: 2 Tfmr Blk
| CTC Att
---
interCTC
14.82 |  | 4-gram
---
24M
13.39
4 | T013 |  | Spd,Ptch,Ns,
---
Rvb,SpcAug
| En: 6 Cfmr Blk
---
De: 2 Tfmr Blk
CTC Att | 14.87 | - | 13.57
2 | 1 | T044 |  | Spd,Ptch,Ns,
---
Rvb,SpcAug
| En: 12 Cfmr Blk
---
De: 12 Tfmr Blk
CTC Att | 574 |  | 6-gram
---
5.7G
8.94
2 | T009 |  | Spd,Ptch,Ns,
---
Rvb,SpcAug,TTS
| En: 15 Cfmr Blk
---
De: 3 Tfmr Blk
| CTC Att
---
interCTC
148 |  | 4-gram+Tfmr
---
410M
9.86
3 | T039 |  | Spd,Ns,Rvb,
---
SpcAug,TTS
| En: 12 Sfmr Blk
---
De: 6 Tfmr Blk
| CTC Att
---
RNN-T
149 | LSTM | 10.20
4 | T002 |  | Spd,Ptch,Ns,
---
Rvb,SpcAug
| En: 15 Cfmr Blk
---
De: 6 Tfmr Blk
CTC Att | 46 |  | 3-gram
---
360M
10.21

Note: Spd–Speed perturb.; Ptch–Pitch shift; Ns–Noise augment.; Rvb–Reverb
simu.; Cfmr–Conformer; Tfmer-Transformer; Sfmer–Suqeezeformer [29]

###  4.2 Data Augmentation and Speech Enhancement

Since the acoustic environment of the cockpit is complex, reflected from the
released data, data augmentation is adopted by most teams to achieve better
data coverage. Noise and reverberation augmentation is widely used. Moreover,
most teams adopt speed perturbation and spec-augment to further augment the
training samples, which are the mainstream tricks with stable performance
improvement. Pitch shift is also an effective data augmentation method
reported by several teams, which changes the pitch of a speech signal without
altering its duration. It is worth noticing that team T009 makes particular
efforts in data augmentation besides the above typical tricks. The first
special trick is random cutting and splicing a speech utterance. In detail,
two pieces of audio of the same speaker are randomly selected from the
training set, and then each piece is cut into two splices and the splices from
the two pieces are randomly combined as new samples. Secondly, team T009 uses
the provided data to train a TTS model particularly for producing extra speech
for model training. With the recent advances in speech generation [34], TTS
based data augmentation has gradually become a promising solution for ASR in
constrained data scenarios [35, 36].

Speech enhancement is conducted by team T039, where a DCCRN [37] model is
adopted to remove noise from the released cabin speech. Team T033 adopts
volume normalization particularly on audio signal before model training.

###  4.3 Language Modeling

Considering the linguistic characteristics of cabin speech, language model
plays a significant role for a in-vehicle ASR system. Accordingly, most teams
use language models based on the provided data. As show in Table 4, we
summarize the language modeling methods of the top 4 teams in the challenge.

As for language modeling, it is obvious that fusing an n-gram language model
with the E2E acoustic model works well, and it brings up to 2% relative
improvement on the Test set, reported from several submitted system
descriptions. Moreover, with the premise that a larger N-gram can obtain
richer contextual information, the winning team T044 employs a 6-gram LM for
both tracks, which leads to substantial performance gain according to its
system description. For generating a high quality N-gram LM, they adopt a
3-stage based iterative strategy. Firstly, a 6-gram LM is trained on the
transcripts of the Training set. To eliminate the domain mismatch between the
general Training set and the released Eval set, they select the unincluded
N-gram phrases from the Eval set based on the well-trained 6-gram LM.
Secondly, they randomly replace the original phrases in the Training
transcripts with the selected unincluded phrases according to their parts-of-
speech and simulate more in-domain texts. Finally, a new LM is trained based
on the simulated text and is composed with the well-trained LM. They repeat
aforementioned stages multiple times until there are no unincluded phrased in
the Eval set. In Track I, the FST file of a system should be less than 25M,
which means that the participants need to particularly prune the graph as much
as possible and control the amount of data used to build the language model.
Thus, data filtering on the training data is used by Team T042, which produces
a language model better matched to the intelligent scene.

Different from Track I, Track II does not has model size limitation. Thus
several teams adopt a neural network language model (NNLM) with large model
size. NNLM with different structures, including LSTM and Transformer, are
adopted by team T039 and T035 for shallow fusion. Moreover, team T009 fuses
scores from an n-gram model and a transformer language model in the decoding
process. As reported by several teams, the contribution from data filtering is
not salient for NNLM, in contrast to the greater contribution to the n-gram
language model.

In summary, for both tracks, language model is proved to be an effective
method to bias the recognition results towards the desired domain. However,
from Table 3, we can find that the commands involving named entities are still
challenging for a general speech recognizer trained with the current training
data. More effective domain transfer or biasing method should be considered in
the future.

###  4.4 Training Strategy

In order to better explore the potential of the E2E models, several teams
study some special training strategies. Particularly, team T002 employs
curriculum learning strategy [38] to the training data ordered by noisy
conditions in terms of SNR. It turns out that the ASR model can achieve better
performance when provided with a better organized training set, for example,
the one here consisting of training samples that exhibit an decreasing SNR
level. In other words, it is beneficial to learn a model from easy to hard
samples. Inspired by knowledge distillation [39], team T009 considers the
model in Track II as the teacher model and learns a student model for Track I
accordingly. Specifically, the Track II model (teacher model with more
parameters) provides soft labels of current utterance for the Track I model
(student model) training, where Kullback Leibler (KL) divergence loss is
adopted between the output posterior distributions of the two models to
transfer the generalization ability. This method narrows the gap between the
small and large models in the two tracks, leading to 13.87% relative CER
reduction (12.04% to 10.36%) for the system in Track I.

##  5 Conclusions

This paper summarizes the outcomes of the ISCSLP 2022 Intelligent Cockpit
Speech Recognition Challenge by introducing the background, tasks, evaluation
metric, baseline system as well as the results. We hope that the initiative of
the ICSRC challenge and the released data will promote speech recognition and
related research in intelligent cockpit scenario. Although recorded in real
cabin conditions, the data made available is limited with only 20 hours in
this challenge. We plan to record and release more data in the future,
hopefully with both multi-channel audio signals and visual information. As
mentioned in Section 1, there are many other challenging tasks in in-vehicle
speech processing, including speech separation, multi-talker speech
recognition and user intent detection, etc. We believe speech interface will
play an increasingly important role with the rapid development of NEV
industry, triggering more research in speech related area.

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