#  The Effects of Signal-to-Noise Ratio on Generative Adversarial Networks
Applied to Marine Bioacoustic Data
††thanks: This work was supported by the Engineering and Physical Sciences
Research Council, Centre for Doctoral Training in Cloud Computing for Big Data
(EP/L015358/1)

###### Abstract

In recent years generative adversarial networks (GANs) have been used to
supplement datasets within the field of marine bioacoustics. This is driven by
factors such as the cost to collect data, data sparsity and aid preprocessing.
One notable challenge with marine bioacoustic data is the low signal-to-noise
ratio (SNR) posing difficulty when applying deep learning techniques such as
GANs. This work investigates the effect SNR has on the audio-based GAN
performance and examines three different evaluation methodologies for GAN
performance, yielding interesting results on the effects of SNR on GANs,
specifically WaveGAN.

###### Index Terms:

generative adversarial network, signal-to-noise ratio, bioacoustics,
underwater acoustics, evaluation methods

##  1 Introduction

Generative adversarial networks (GANs) are used in a variety of settings
within the audio domain, including speech synthesis [1], music generation [2],
and bioacoustic synthesis [3]. In marine bioacoustics, GANs have previously
been employed for tasks such as dataset generation and augmentation [4, 5, 6].
Some bioacoustic data, like those from deployed passive acoustic monitoring
systems, typically have a low signal-to-noise ratio (SNR). This can cause
challenges when generating synthetic data with GANs.

Despite this, there is little work on the effect of low SNR on GAN-generated
synthetic bioacoustics. As such, this work examines how the SNR of marine
bioacoustic data affects GAN performance. Further, this work investigates
three different methodologies for evaluating GAN performance.

##  2 Related Work

In 2018, Donahue et al. introduced WaveGAN [7], an audio-based GAN based on
DCGAN [8] adapted to a one-dimensional network. To address the issue of
chequerboard artefacts in DCGAN-generated images, WaveGAN phase-shuffles the
discriminator at each layer.

Evaluating GAN-generated bioacoustics typically focuses on improving model
performance whilst ignoring the synthetic data’s quality [4, 9, 5]. Work
assessing synthetic data production often makes use of Dynamic Time Warping
[6], visually comparing frequencies to that of similar real-world data [3],
spectrogram inspection [3], cluster analysis [10, 11], training discriminator
networks to obtain error networks [11], and Siamese Neural Networks (SNNs) to
generate similarity scores between the real and generated data [10].

There is currently no standardised evaluation method for synthetic audio data.
Additionally, the effect of SNR on audio generation has not been explored in
the marine context.

##  3 Data

Fig. 1: Example of the SNR data. (Left) $\text{SNR}=10$dB. (Middle)
$\text{SNR}=-15$dB. (Right) Background noise.

Neural networks which operate on raw audio are often sample rate specific.
WaveGAN is designed for audio sampled at 16kHz [7], making it suitable for
marine acoustics at frequencies $\leq$ 8kHz. To evaluate the efficacy of
WaveGAN at lower sampling rates, data was generated at 32kHz††Data generated
using: https://github.com/chrisdonahue/wavegan.

Audio of a specific SNR is generated via $\bm{x}=\bm{s}+\beta\bm{n}$, where
$\bm{s}$ is the upsweep signal, $\bm{n}$ is the noise signal, $\mathrm{SNR}$
is the SNR, and $\beta=\frac{A_{s}}{A_{n}\sqrt{\mathrm{SNR}}}$ is the SNR
scaling parameter. $A_{j}$ is the root mean square amplitude of signal $j$.

To investigate the effect of SNR on GANs, an upsweep frequency shape which
mimics cetacean whistles is used as the signal, chosen as it is a simple shape
but one which could reasonably be present within marine bioacoustic data [12].
Whistles of varying height, length, and gradient were generated at 32kHz. One-
second clips of underwater noise, devoid of anthropogenic activity or
whistles, were randomly selected, downsampled to 32kHz, $\beta$-scaled, and
combined with the generated whistle. SNR data is systematically generated to
cover the range $[-15,10]$dB, at 5k clips per SNR. An additional 5k noise
clips are sampled for analysis. See Figure 1 for example data.

##  4 Methodology

Diverse synthetic SNR data is generated using a WaveGAN model with a 256-d
generator latent space ($\bm{z}\in\mathbb{R}^{256}$ where $z_{i}\sim U(-1,1)$
for $i=1,...,256$). A model is trained for approximately 10k epochs, chosen
empirically based on prior convergence experiments. Each trained network is
used to generate 500 synthetic whistles for evaluation purposes.

To assess synthetic data quality, three methods are explored: frequency
spectra comparison; spectrogram pixel intensity comparison; SNN analysis, akin
to [10].

###  4.1 Frequency Spectra Comparison

Direct signal comparisons, such as Dynamic Time Warping, are unsuitable for
comparing GAN-generated signals to training data as they aim to imitate the
training data distribution rather than replicate the data itself [13]. To
assess the synthetic data generated in this study, first only the frequency
domain is considered to provide an overview of each the whilst and noise data.

Average frequency spectra are calculated to obtain approximate noise and
whistle audio samples. Pearson’s correlation coefficient is computed for each
average spectrum compared to the synthetic sample’s frequency spectrum sample,
denoted $c_{f,w}$ for whistles and $c_{f,n}$ for noise. If $c_{f,w}\geq
c_{f,n}$ the synthetic sample is scored 1 else 0. This is repeated for each
synthetic sample, with the mean score calculated for each SNR level.

###  4.2 Spectrogram Pixel Intensity Comparison

Converting data from the time domain to the frequency-time domain allows for
the simultaneous analysis of both domains by examining the pixel intensity
distribution in the resulting spectrogram. The similarity between two
spectrograms can thus be calculated using pixel intensity distribution
correlation. For simplicity, spectrograms used in this work are first
converted to greyscale.

Average pixel intensity distributions (PIDs) of the whistles and noise are
calculated to obtain an estimate of their respective distributions. Pearson’s
correlations between the PID of the synthetic sample and the average PIDs of
the corresponding whistle and noise data are calculated, denoted $c_{p,w}$ and
$c_{p,n}$ respectively. If $c_{p,w}\geq c_{p,n}$ then the synthetic sample
score $=1$ else $0$. The overall SNR score is obtained by computing the mean
of all synthetic sample scores for that particular SNR, repeated for each SNR.

###  4.3 Triplet Loss Learning via SNNs

Two SNN evaluation methods are explored. The first trains a separate SNN for
each SNR, utilising the same noise data. The second trains a single SNN on the
entire SNR and noise data. Hyperparameters are tuned using Optuna [14]
encompassing the number of convolution blocks, dense layers, learning rate and
embedding size. Data is split using a 60:20:20 train-test-validation ratio,
preprocessed into greyscale spectrograms. SNNs are trained on training and
validation data, test data is used for subsequent analysis.

#### Individual SNNs

Fig. 2: Example embeddings from the $SNN_{\rm{SNRdB}}$ models visualised using
$t$-SNE for the upper and lower SNR bounds. Fig. 3: SNN embeddings of real
(left) and synthetic (right) data visualised into two dimensions using $t$-SNE
. The $t$-SNE visualisations, show the synthetic data is not clustering to the
correct SNRs well, with A lot of the points clustered around the noise. It
appears that the synthetic embeddings are clustering to a lower SNR decibel.

Here, one SNN per SNR ($SNN_{\rm{SNRdB}}$) is trained, each processing a
dataset containing 10k spectrograms representing one-second audio files,
equally divided between whistles and noise. SNNs are trained using semi-hard
triplet loss [15]. After hyperparameter tuning, each model is re-trained to
obtain a loss of approximately 0.1, obtained through prior empiric test set
evaluation, aiming to prevent overfitting and bias between models.

Embeddings for 500 synthetic whistles are generated by each $SNN_{\rm{SNRdB}}$
which, along with the whistle and noise embeddings, are used to predict labels
for the synthetic data. Visualisations of these embeddings, reduced to two
dimensions via $t$-SNE are shown Figure 2 for the upper and lower SNR bounds.
There is a clear separation between the whistle and noise $SNN_{10\rm{dB}}$
embeddings, this is also true for the with $SNN_{-15\rm{dB}}$ embeddings. The
$SNN_{10\rm{dB}}$ synthetic data predominantly cluster with the whistle data,
except for one outlier. This outlier exhibits noisy characteristics deviating
from typical whistle data. The $SNN_{-15\rm{dB}}$ synthetic embeddings show
clustering with both the noise and whistle data. Synthetic data labels are
determined using the nearest centroid (NC) algorithm via Euclidean distance
measurement, and accuracy scores are generated.

#### Single SNN

Here, a single SNN ($SNN_{\rm{alldBs}}$) is trained on all SNR and noise data,
containing 130k spectrograms comprising of 5k noise and 5k whistles per SNR.
The network outputs a 56-D embedding and consists of 7 convolutional blocks
and 2 fully connected layers, trained using the Adam [16] optimiser with a
learning rate of $2\times 10^{-4}$. The network achieves a validation loss of
0.37.

In Figure 3, $t$-SNE plots of the real and synthetic embeddings are displayed.
$SNN_{\rm{alldBs}}$ clusters data effectively with higher SNRs centralised and
lower SNRs spiralling outwards and distinctly clustered noise data, providing
confidence in lower SNR results. Interestingly, synthetic data appears to
cluster with lower SNRs relative to their actual SNR, e.g. 10dB synthetic data
are clustered around the 3dB or 4dB whistle data. Lower synthetic SNR data
often cluster around the noise data or between the $-15$dB whistle and noise
data. This is unlike results in Figure 2, where synthetic data typically
clusters well with the corresponding whistle data. This suggests treating SNRs
as continuous rather than categorical motivating classification and regression
approaches.

The challenge in treating the task as a regression problem lies in handling
background noise data. This data can be viewed as an infinitesimally small
SNR, effectively $SNR_{dB}\rightarrow-\infty$ or equivalently $SNR\rightarrow
0$, or as a categorical variable, ‘noise’. In the previous metrics, the latter
case is applied since only one decibel level was considered at a time.

In the categorical approach, NC is utilised to predict decibel and noise
labels for the synthetic data. Initially, centroids are computed for each SNR
and noise data. Subsequently, the synthetic embeddings are passed to the
algorithm and Euclidean distance is employed to determine the predicted
labels. Conversely, in the regression approach, $K$-nearest neighbours (KNN)
is adopted. Instead of an accuracy, root mean square error (RMSE) is employed
for evaluation. The optimal $K$ value is determined by employing the elbow
method on the RMSE, $R^{2}$, and the maximum distance between a point and its
neighbours from fitting KNN to test data embeddings for $K\in[1,1000]$.

##  5 Results

###  5.1 Frequency Spectra Comparison

Figure 4 presents the results of the frequency spectrum analysis. The
correlation score increases with the SNR. The increase in correlation score
has an approximately linear trend up to 0dB. Beyond this point, within the
range $(0,10]$dB, the correlation scores stabilise around 0.9, except the
outlier score observed at 5dB.

Fig. 4: The scaled frequency and pixel analysis correlation scores between
real data and synthetic data.

###  5.2 Spectrogram Pixel Intensity Comparison

Figure 4 presents the results of the spectrogram pixel intensity analysis. The
correlation score does not exhibit any well-defined trend but primarily
decreases within $[5,10]$dB. As pixel intensity is based upon the gradient of
greyscale, for high SNR there is a large number of white pixels compared to
the noise data due to the prominent whistle in the audio clip. However, the
synthetic samples do not exhibit the same prominent white pixels, which
influences the correlation score. This discrepancy may arise from differences
in SNR between the synthetic whistles and the whistle data.

###  5.3 Triplet Loss Learning

#### Individual SNNs

Figure 5 displays results for the individual SNNs (along with other results
discussed later). This reveals an approximately linear relationship between
the SNR and accuracy score, as in Section 5.1. Notably, this method heavily
relies on ensuring consistent performance across 26 distinct SNN models
without overfitting is challenging. The obtained losses in $SNN_{05dB}$ and
$SNN_{06dB}$, are very small (see Figure 5), deviating from the general trend
suggesting potential overfitting.

Fig. 5: The NC accuracy scores for each SNR obtained via individual SNNs,
single SNN with the classic accuracy and noise label accuracy.

#### Single SNN

Figure 5 presents the results obtained via categorical analysis. Assessing
network accuracy based on whether the correct SNR decibel is predicted is
unsuitable in this context as it does not appropriately capture the
relationship between WaveGAN performance and SNR. When focusing on a binary
outcome for each SNR, either noise or not (denoted ‘Single SNN Noise
Accuracy’), the results indicate WaveGAN models produce synthetic data
resembling whistles consistently from -5dB onwards. However, this approach
tends to penalise lower SNRs.

This ambiguity in evaluating data in a categorical manner prompted a shift to
considering SNR labels as continuous variables, assigning noise an SNR of 0.
Here, SNRs are converted from decibels to the original SNR (e.g. -10dB = 0.1)
and the $K$-Nearest Neighbours (KNN) algorithm is applied to the test
embeddings. SNRs are thus predicted using both the mean and weighted mean of a
point’s $K$ neighbours, scaled based on the distance to the point.

Through the elbow method, the optimal $K=400$ is determined via weighted mean.
Synthetic embeddings are passed to KNN to obtain an SNR prediction value, then
converted back to dB to ensure a fair comparison between SNRs. For example,
both -14dB to -15dB and 9dB to 10dB have a difference of 1dB, though when
considered a ratio the difference is 0.0082 and 2.06 respectively, thus
penalising higher SNRs. The minimum non-zero SNR predicted is
$-37.5\text{dB}$, hence SNRs predicted at 0 are assigned a value of
$-40\text{dB}$, ensuring only lower SNRs are penalised. The root mean square
decibel error (RMSDE) is then calculated using the difference between
predicted and actual SNR in dB.

Fig. 6: The RMSDE of the synthetic embeddings. The RMSDE reaches a minimum
value in the range $[-2,2]$dB.

Figure 6 shows the RMSDE for each SNR. Most SNRs within [-10,10]dB have an
RMSDE within [5,10], indicating the average distance between the synthetic
data’s actual and predicted SNR is [5,10]dB. Notably, lower SNRs perform less
satisfactorily. Unlike other approaches, this evaluation takes into account
the WaveGAN generator’s ability to produce audio with the correct SNR.
Evaluating test embeddings using this metric, all SNRs have an RMSDE within
[0,0.8], enhancing confidence in the results shown. These findings reinforce
the observation that synthetic data tends to have a lower SNR compared to the
target SNR for WaveGAN models, amplified for lower SNRs.

##  6 Conclusion

Comparing frequency spectra provides a simple method for evaluating WaveGAN’s
general performance at an individual SNR level. However, this is limited for
lower SNRs due to the similarities in frequency spectra and marginal
differences in correlations. Additionally, it does not capture whether a
correct sound is created. Comparing this to pixel intensity evaluation, this
method particularly struggles due to the large amount of dark and white
colouration within the low and high SNR spectrograms with an inability to
capture whistle-like characteristics. Unlike these methods the SNN methods
exhibit a clear distinction between the noise and whistle data for lower SNR,
giving higher result confidence.

Investigation using SNNs suggests poor evaluation may be the result of WaveGAN
producing audio at a lower SNR relative to that with which it is trained. This
study has shown SNNs can be applied to evaluate individual SNR performance and
synthetic data quality, as well as to gauge if the correct SNR is generated by
the WaveGAN by assessing all SNRs collectively, generating an easily
interpretable RMSDE metric which indicates WaveGAN’s proficiency for
generating synthetic data at the correct SNR.

Further, this work demonstrates the value of utilising SNNs for audio
generation evaluation. SNNs do not require large data volumes, making them
suitable for assessing generated data in scenarios with limited real-life
examples. Based on the results obtained, it is recommended to train one SNN
over all SNR classes if using multi-class data, including one class for noise
if possible. Visual inspection using $t$-SNE is recommended to assess
synthetic data spread.

For all evaluation methods, it is evident that SNR significantly impacts
WaveGAN performance, with lower SNRs resulting in poorer performance. Notably,
WaveGAN on average produces a lower SNR than the training data, as
demonstrated in Figure 3. As such, it is recommended to train WaveGAN models
on data with an $\text{SNR}\geq-5\text{dB}$.

Future work should explore the impact of SNR on other audio-based GANs, such
as WaveNet [17], DiffWave [18], or MelGAN [19] following a similar analysis.
Additionally, it is worth considering alternative similarity metrics, such as
Jaccard similarity and Cosine similarity. Rather than assigning a binary score
of 1 or 0 based on correlation values, the raw correlation values between the
whistle data and synthetic samples could be utilised. Finally whilst this work
located a clear lower performance limit of $-15\text{dB}$, future work should
explore the existence of an upper limit.

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HGNN | 0.748$\pm$0.003 | 0.636$\pm$0.003 | 0.750$\pm$0.003 | 0.750$\pm$0.003 | 0.641$\pm$0.006 | 0.630$\pm$0.004 | 0.603$\pm$0.002
FHGCN | 0.545$\pm$0.063 | 0.599$\pm$0.098 | 0.574$\pm$0.067 | 0.699$\pm$0.112 | 0.506$\pm$0.013 | 0.503$\pm$0.008 | 0.503$\pm$0.006
UGCNII | 0.866$\pm$0.003 | 0.773$\pm$0.005 | 0.867$\pm$0.002 | 0.858$\pm$0.002 | 0.782$\pm$0.005 | 0.717$\pm$0.003 | 0.651$\pm$0.003
Max | 0.928$\pm$0.003 | 0.773$\pm$0.005 | 0.929$\pm$0.003 | 0.941$\pm$0.003 | 0.782$\pm$0.005 | 0.717$\pm$0.003 | 0.651$\pm$0.003
Avg. | 0.808$\pm$0.142 | 0.664$\pm$0.064 | 0.817$\pm$0.134 | 0.833$\pm$0.118 | 0.583$\pm$0.076 | 0.554$\pm$0.064 | 0.592$\pm$0.041
Rank Avg. | 2.800$\pm$1.077 | 4.000$\pm$0.775 | 1.800$\pm$0.748 | 2.000$\pm$0.894 | 5.600$\pm$0.917 | 6.700$\pm$0.458 | 5.100$\pm$1.578
Measure | Model | DHG-13 | triad | DHG-26 | h-motif |  DHG |  n2v |  h2v | deep-h | 3h-motif
AUROC | LR | 0.552$\pm$0.008 | 0.612$\pm$0.010 | 0.692$\pm$0.005 | O.O.T.* | 0.696$\pm$0.012 | 0.592$\pm$0.003 | 0.563$\pm$0.003 | 0.627$\pm$0.003 | O.O.T.*
RF | 0.977$\pm$0.001 | 0.755$\pm$0.040 | 0.978$\pm$0.001 | 0.977$\pm$0.004 | 0.578$\pm$0.014 | 0.528$\pm$0.008 | 0.715$\pm$0.023
DT | 0.903$\pm$0.004 | 0.672$\pm$0.035 | 0.904$\pm$0.004 | 0.893$\pm$0.019 | 0.512$\pm$0.009 | 0.504$\pm$0.005 | 0.543$\pm$0.011
KNN | 0.957$\pm$0.002 | 0.705$\pm$0.032 | 0.952$\pm$0.002 | 0.906$\pm$0.021 | 0.627$\pm$0.012 | 0.558$\pm$0.011 | 0.681$\pm$0.006
MLP | 0.917$\pm$0.014 | 0.690$\pm$0.020 | 0.938$\pm$0.008 | 0.967$\pm$0.005 | 0.639$\pm$0.032 | 0.596$\pm$0.030 | 0.761$\pm$0.013
XGB | 0.974$\pm$0.002 | 0.797$\pm$0.046 | 0.977$\pm$0.002 | 0.981$\pm$0.004 | 0.596$\pm$0.012 | 0.553$\pm$0.009 | 0.776$\pm$0.007
LGBM | 0.965$\pm$0.003 | 0.806$\pm$0.040 | 0.969$\pm$0.003 | 0.983$\pm$0.002 | 0.597$\pm$0.012 | 0.561$\pm$0.009 | 0.782$\pm$0.007
HGNN | 0.832$\pm$0.003 | 0.663$\pm$0.003 | 0.834$\pm$0.003 | 0.828$\pm$0.002 | 0.681$\pm$0.006 | 0.666$\pm$0.005 | 0.605$\pm$0.002
FHGCN | 0.592$\pm$0.079 | 0.712$\pm$0.070 | 0.623$\pm$0.077 | 0.823$\pm$0.040 | 0.510$\pm$0.018 | 0.504$\pm$0.011 | 0.503$\pm$0.008
UGCNII | 0.940$\pm$0.002 | 0.857$\pm$0.006 | 0.941$\pm$0.002 | 0.935$\pm$0.002 | 0.863$\pm$0.005 | 0.795$\pm$0.004 | 0.713$\pm$0.004
Max | 0.977$\pm$0.001 | 0.857$\pm$0.006 | 0.978$\pm$0.001 | 0.983$\pm$0.002 | 0.863$\pm$0.005 | 0.795$\pm$0.004 | 0.782$\pm$0.007
Avg. | 0.861$\pm$0.150 | 0.727$\pm$0.072 | 0.881$\pm$0.120 | 0.899$\pm$0.088 | 0.619$\pm$0.095 | 0.583$\pm$0.084 | 0.671$\pm$0.093
Rank Avg. | 2.900$\pm$1.578 | 4.200$\pm$0.980 | 1.700$\pm$0.640 | 2.000$\pm$1.000 | 5.400$\pm$0.800 | 6.500$\pm$0.671 | 5.300$\pm$1.269
* O.O.T: out-of-time ($>1$ day). |

(j)

bitcoin-2015

Dimension | 13 | 26 | 91 | 431
---|---|---|---|---
Measure | Model | DHG-13 | triad | DHG-26 | h-motif |  DHG |  n2v |  h2v | deep-h | 3h-motif
ACC | LR | 0.545$\pm$0.002 | 0.549$\pm$0.022 | 0.551$\pm$0.002 | O.O.T.* | 0.556$\pm$0.003 | 0.527$\pm$0.022 | 0.530$\pm$0.015 | 0.569$\pm$0.028 | O.O.T.*
RF | 0.915$\pm$0.002 | 0.694$\pm$0.030 | 0.921$\pm$0.001 | 0.934$\pm$0.007 | 0.529$\pm$0.010 | 0.521$\pm$0.007 | 0.609$\pm$0.039
DT | 0.888$\pm$0.002 | 0.671$\pm$0.029 | 0.894$\pm$0.002 | 0.907$\pm$0.016 | 0.509$\pm$0.008 | 0.504$\pm$0.005 | 0.547$\pm$0.011
KNN | 0.888$\pm$0.001 | 0.668$\pm$0.028 | 0.890$\pm$0.001 | 0.866$\pm$0.020 | 0.567$\pm$0.010 | 0.536$\pm$0.009 | 0.626$\pm$0.003
MLP | 0.825$\pm$0.010 | 0.591$\pm$0.030 | 0.830$\pm$0.011 | 0.923$\pm$0.003 | 0.549$\pm$0.027 | 0.545$\pm$0.026 | 0.630$\pm$0.040
XGB | 0.897$\pm$0.002 | 0.722$\pm$0.038 | 0.913$\pm$0.002 | 0.938$\pm$0.005 | 0.544$\pm$0.012 | 0.522$\pm$0.013 | 0.621$\pm$0.030
LGBM | 0.875$\pm$0.002 | 0.733$\pm$0.041 | 0.897$\pm$0.002 | 0.937$\pm$0.003 | 0.551$\pm$0.016 | 0.529$\pm$0.016 | 0.610$\pm$0.032
HGNN | 0.750$\pm$0.002 | 0.633$\pm$0.003 | 0.751$\pm$0.002 | 0.749$\pm$0.003 | 0.627$\pm$0.005 | 0.629$\pm$0.003 | 0.597$\pm$0.004
FHGCN | 0.551$\pm$0.064 | 0.608$\pm$0.100 | 0.553$\pm$0.067 | 0.712$\pm$0.110 | 0.504$\pm$0.011 | 0.504$\pm$0.010 | 0.504$\pm$0.011
UGCNII | 0.862$\pm$0.004 | 0.762$\pm$0.005 | 0.862$\pm$0.003 | 0.854$\pm$0.003 | 0.771$\pm$0.010 | 0.720$\pm$0.004 | 0.646$\pm$0.004
Max | 0.915$\pm$0.002 | 0.762$\pm$0.005 | 0.921$\pm$0.001 | 0.938$\pm$0.005 | 0.771$\pm$0.010 | 0.720$\pm$0.004 | 0.646$\pm$0.004
Avg. | 0.800$\pm$0.133 | 0.663$\pm$0.064 | 0.806$\pm$0.135 | 0.838$\pm$0.121 | 0.568$\pm$0.075 | 0.554$\pm$0.065 | 0.596$\pm$0.041
Rank Avg. | 3.000$\pm$0.894 | 4.000$\pm$0.775 | 1.900$\pm$0.700 | 1.700$\pm$0.900 | 6.000$\pm$0.775 | 6.400$\pm$0.800 | 5.000$\pm$1.612
Measure | Model | DHG-13 | triad | DHG-26 | h-motif |  DHG |  n2v |  h2v | deep-h | 3h-motif
AUROC | LR | 0.622$\pm$0.006 | 0.612$\pm$0.009 | 0.630$\pm$0.011 | O.O.T.* | 0.654$\pm$0.012 | 0.564$\pm$0.003 | 0.567$\pm$0.003 | 0.635$\pm$0.002 | O.O.T.*
RF | 0.971$\pm$0.001 | 0.766$\pm$0.035 | 0.974$\pm$0.001 | 0.978$\pm$0.003 | 0.550$\pm$0.012 | 0.529$\pm$0.009 | 0.725$\pm$0.021
DT | 0.885$\pm$0.002 | 0.681$\pm$0.032 | 0.893$\pm$0.002 | 0.909$\pm$0.016 | 0.509$\pm$0.008 | 0.504$\pm$0.005 | 0.547$\pm$0.011
KNN | 0.948$\pm$0.001 | 0.716$\pm$0.029 | 0.948$\pm$0.001 | 0.927$\pm$0.014 | 0.592$\pm$0.014 | 0.553$\pm$0.011 | 0.678$\pm$0.006
MLP | 0.909$\pm$0.006 | 0.655$\pm$0.021 | 0.915$\pm$0.008 | 0.968$\pm$0.002 | 0.600$\pm$0.026 | 0.587$\pm$0.033 | 0.770$\pm$0.007
XGB | 0.964$\pm$0.001 | 0.801$\pm$0.041 | 0.971$\pm$0.001 | 0.982$\pm$0.003 | 0.562$\pm$0.012 | 0.550$\pm$0.011 | 0.783$\pm$0.005
LGBM | 0.951$\pm$0.001 | 0.814$\pm$0.044 | 0.963$\pm$0.001 | 0.982$\pm$0.002 | 0.563$\pm$0.011 | 0.558$\pm$0.010 | 0.788$\pm$0.006
HGNN | 0.833$\pm$0.002 | 0.659$\pm$0.003 | 0.834$\pm$0.002 | 0.827$\pm$0.002 | 0.659$\pm$0.006 | 0.663$\pm$0.004 | 0.596$\pm$0.003
FHGCN | 0.598$\pm$0.074 | 0.712$\pm$0.062 | 0.597$\pm$0.078 | 0.825$\pm$0.049 | 0.508$\pm$0.020 | 0.507$\pm$0.017 | 0.505$\pm$0.015
UGCNII | 0.937$\pm$0.003 | 0.847$\pm$0.006 | 0.937$\pm$0.002 | 0.931$\pm$0.002 | 0.856$\pm$0.010 | 0.801$\pm$0.004 | 0.704$\pm$0.005
Max | 0.971$\pm$0.001 | 0.847$\pm$0.006 | 0.974$\pm$0.001 | 0.982$\pm$0.003 | 0.856$\pm$0.010 | 0.801$\pm$0.004 | 0.788$\pm$0.006
Avg. | 0.862$\pm$0.132 | 0.726$\pm$0.074 | 0.866$\pm$0.133 | 0.898$\pm$0.099 | 0.596$\pm$0.096 | 0.582$\pm$0.085 | 0.673$\pm$0.095
Rank Avg. | 2.700$\pm$0.781 | 4.300$\pm$1.005 | 2.100$\pm$0.831 | 1.600$\pm$0.917 | 5.700$\pm$0.781 | 6.400$\pm$0.917 | 5.200$\pm$1.470
* O.O.T: out-of-time ($>1$ day). |

(k)

bitcoin-2016


# The Myth of Complete AI-Fairness

Virginia Dignum

(Umeå University, Sweden
Email<EMAIL_ADDRESS>
Just recently, IBM invited me to participate in a panel titled “Will AI ever
be completely fair?” My first reaction was that it surely would be a very
short panel, as the only possible answer is ‘no’. In this short paper, I wish
to further motivate my position in that debate: “I will never be completely
fair. Nothing ever is. The point is not complete fairness, but the need to
establish metrics and thresholds for fairness that ensure trust in AI
systems”.

The idea of fairness and justice has long and deep roots in Western
civilization, and is strongly linked to ethics. It is therefore not strange
that it is core to the current discussion about the ethics of development and
use of AI systems. Given that we often associate fairness with consistency and
accuracy, the idea that our decisions and decisions affecting us can become
fairer by replacing human judgement by automated, numerical, systems, is
therefore appealing. However, as Laurie Anderson recently said111As quoted by
Kate Crawford on Twitter
https://twitter.com/katecrawford/status/1377551240146522115; 1 April 2021. “If
you think technology will solve your problems, you don’t understand technology
— and you don’t understand your problems.” AI is not magic, and its results
are fundamentally constrained by the convictions and expectations of those
that build, manage, deploy and use it. Which makes it crucial that we
understand the mechanisms behind the systems and their decision capabilities.

The pursuit of fair AI is currently a lively one. One involving many
researchers, meetings and conferences (of which
FAccT222https://facctconference.org/ is the most known) and refers to the
notion that an algorithm is fair, if its results are independent of given
variables, especially those considered sensitive, such as the traits of
individuals which should not correlate with the outcome (i.e. gender,
ethnicity, sexual orientation, disability, etc.). However, nothing is ever
100% fair in 100% of the situations, and due to complex networked connection,
to ensure fairness for one (group) may lead to unfairness for others.
Moreover, what we consider fair often does depend on the traits of
individuals. An obvious example are social services. Most people believe in
the need for some form of social services, whether it is for children, for the
elderly, for the sick or the poor. And many of us will benefit from social
services at least once in our lives. Decision making in the attribution of
social benefits is dependent on individual characteristics such as age,
income, or chronic health problems. Algorithmic fairness approaches however
overemphasize concepts such as equality and do not adequately address caring
and concern for others.

Many years ago, I participated in a project at my children basic school that
was aimed at helping kids develop fairness standards, roughly modelled along
Kohlberg’s stages of moral development. It became clear quite quickly that
children aged 6-12 easily understand that fairness comes in many ‘flavours’:
if given cookies to divide between all kids of the class, the leading
principle was equality, i.e. giving each kid the same amount of cookies. But
they also understood and accepted the concept of equity: for instance in
deciding that a schoolmate with dyslexia should be given more time to perform
a school exam. Unfortunately, for the average algorithm, common sense and
world knowledge is many light years away from that of a six year old, and
switching between equity and equality depending on what is the best approach
to fairness in a given situation, is rarely a feature of algorithmic decision
making.

## So, how does fairness work in algorithms and what is being done to correct
for unfair results?

Doctors deciding on a patient’s treatment, or judges deciding on sentencing,
must be certain that probability estimates for different conditions are
correct for each specific subject, independent of age, race or gender.
Increasingly these decisions are mediated by algorithms. Algorithmic fairness
can be informally described as the probability of being classified in a
certain category should be similar to for all that exhibit those
characteristics, independently of other traits or properties. In order to
ensure algorithmic fairness, given often very unbalanced datasets, data
scientists use calibration (i.e. the comparison of the actual output and the
expected output). Moreover, if we are concerned about fairness between two
groups (e.g. male and female patients, or African-American defendants and
Caucasian defendants) then this calibration condition should hold
simultaneously for the set of people within each of these groups as well [4].
Calibration is a crucial condition for risk prediction tools in many settings.
If a risk prediction tool for evaluating defendants is not calibrated towards
race, for example, then a probability estimate could carry different meaning
for African-American than for Caucasian defendants, and hence the tool would
have the unintended and highly undesirable consequence of incentivizing judges
to take race into account when interpreting the tool’s predictions [7]. At the
same time, ideally the incidence of false positives (being incorrectly
classified as ‘X’) and false negatives (failing incorrectly to be classified
as ‘X’) should be the same independently of other traits or properties. That
is, fairness also means that, for instance, male and female candidates have
the same chance of being offered a, for them irrelevant, service or product,
or failing to receive for them relevant services or products.

Unfortunately, research shows that it is not possible to satisfy some of these
expected properties of fairness simultaneously: calibration between groups,
balance for false negatives, and balance for false positives. This means that
if we calibrate data, we need to be prepared to accept higher levels of false
positives and false negatives for some groups, and to deal with their human
and societal impact [6]. Taking the diagnostic example, a false positive means
that a patient is diagnosed with a disease they don’t have. With a false
negative that disease goes undiagnosed. The impact of either, both personal as
well as societal, can be huge. In the same way, being wrongly classified as
someone with a high risk to re-offend(false positive) has profound personal
consequences, given that innocent people are held without bail, while
incorrect classification as someone with a low risk to re-offend (false
negative) has deep societal consequences, where people that pose a real
criminal threat are let free333This example is at the core of the well-known
Propublica investigations of the COMPAS algorithms used by courts in the US to
determine recidivism risk: www.propublica.org/article/how-we-analyzed-the-
compasrecidivism-algorithm..

Given these technical difficulties in achieving perfectly fair, data-driven,
algorithms, it is high time to start a conversation about the societal and
individual impact of false positives and false negatives, and, more
importantly, about what should be the threshold for acceptation of algorithmic
decisions, that, by their nature, will never be completely ‘fair’.

## Fairness is not about bias but about prejudice

A commonly voiced explanation for algorithmic bias is the prevalence of human
bias in the data. For example, when a job application filtering tool is
trained on decisions made by humans, the machine learning algorithm may learn
to discriminate against women or individuals with a certain ethnic background.
Often this will happen even if ethnicity or gender are excluded from the data
since the algorithm will be able to exploit the information in the applicant’s
name, address or even the use of certain words. For example, Amazon’s
recruiting AI system filtered out applications by women, because they lacked
‘masculine’ wording, commonly used in applications by men.

There are many reasons for bias in datasets, from choice of subjects, to the
omission of certain characteristics or variables that properly capture the
phenomenon we want to predict, to changes over time, place or situation, to
the way training data is selected.

Much has been done already to categorize and address the many forms of machine
bias [8]. Also, many tools are available to support to unbias AI systems,
including IBM’s AI Fairness 360444https://github.com/Trusted-AI/AIF360 and
Google’s What If Tool555https://pair-code.github.io/what-if-
tool/index.html#about. Basically, these tools support the testing and
mitigation of bias through libraries of methods and test environments.
According to Google “[…] with the What If Tool you can test performance in
hypothetical situations, analyse the importance of different data features,
and visualize model behaviour across multiple models and subsets of input
data, and for different ML fairness metrics.” Note the focus on performance, a
constant in much of the work on AI.

However, not all bias is bad, in fact, there are even biases in the way we
approach bias. Bias in human data in not only impossible to fully eliminate,
it is often there for a reason. Bias is part of our lives partly because, we
do not have enough cognitive bandwidth to make every decision from ground zero
and therefore need to use generalizations, or biases, as a starting point.
Without bias, we would not been able to survive as a species, it helps us
selecting from a myriad of options in our environment. But not all biases are
bad. But that doesn’t mean we shouldn’t address them.

Bias is not the problem, prejudice and discrimination are. Whereas prejudice
represents a preconceived judgment or attitude, discrimination is a behaviour.
In society, discrimination is often enacted through institutional structures
and policies, and embedded in cultural beliefs and representations, and is
thus reflected in any data collected. The focus need be on using AI to support
interventions aimed at reducing prejudice and discrimination, e.g. through
education, facilitation of intergroup contact, targeting social norms
promoting positive relations between groups, or supporting people identify
their own bias and prejudices.

Facial analysis tools and recognition software have raised concerns about
racial bias in the technology. Work by Joy Buolamwini and Timnit Gebru has
shown how deep these biases go and how hard they are to eliminate [1]. In
fact, debiasing AI often leads to other biases. Sometimes this is known and
understood, such as the dataset they created as alternative to the datasets
commonly used for training facial recognition algorithms: using what they
called ’parliaments’, Buolamwini and Gebru created a dataset of faces balanced
in terms of race and gender, but notably unbalanced in terms of age, lighting
or pose. As long as this is understood, this dataset is probably perfectly
usable for training an algorithm to recognise faces of a certain age,
displayed under the same lighting conditions and with the same pose. It will
however not be usable if someone tries to train an algorithm to recognise
children’s faces. This illustrates that debiasing data is not without risks,
in particular because it focus on those characteristics that we are aware of,
which are ‘coloured’ by our own experience, time, place and culture.

Moreover, AI bias is more than biased data. It starts with who is collecting
the data, who is involved in selecting and/or designing the algorithms and who
is training the algorithms and labelling the data. Moreover, decisions about
which and whose data is collected and which data is being used to train the
algorithms and how the algorithm is designed and trained also influence the
fairness of the results. From labelling farms to ghost workers, the legion of
poorly paid, badly treated and ignored human labourers, working behind the
scenes of any working AI system, is huge and little is being done to
acknowledge them and to improve their working conditions. Books such as ‘Ghost
work’ [5] by Mary L. Gray and Siddharth Suri, or ‘Atlas of AI’ [2] by Kate
Crawford, are raising the issue but, as often is the case, the ‘folk is
sleeping’: it is easier to use the systems and profit from their results, than
to question how these results are being achieved and at what cost. The
question is thus: how fair is algorithm fairness for those that label, train
and calibrate the data it needs to produce fair results and, more importantly,
if we expect them to provide us with unbiased data, shouldn’t we be treating
them fairly?

## Beyond fairness

The fact that algorithms and humans, cannot ever be completely fair, does not
mean that we should’ve up and accept it. Improving fairness and overcoming
prejudice is partly a matter of understanding how the technology works. A
matter of education. Moreover, using technology properly, fair treatment of
those using and being affected by it, requires participation. Still many
stakeholder are not invited to the table, not joining the conversation.

Lack of fairness in AI systems is often also linked to a lack of explanatory
capabilities. If the results of the system cannot be easily understood or
explained, it is difficult to assess its fairness. Many of the current tools
that evaluate bias and fairness help identify where biases may occur, whether
in the data or the algorithms or even in their testing and evaluation. Even if
not all AI systems can be fully explainable, it is important to make sure that
their decisions are reproducible and the conditions for their use are clear
and open to auditing.

Current AI algorithms are built for accuracy and performance, or for
efficiency. Improving the speed of the algorithm, minimizing its computational
requirements and maximizing the accuracy of the results are the mantras that
lead current computer science and engineering education. However, these are
not the only optimization criteria. When humans and society are at stake,
other criteria need be considered. How do you balance safety and privacy?
Explainability and energy resources? Autonomy and accuracy? What do you do
when you cannot have both? Such moral overload dilemmas are at the core of
responsible development and use of AI [3].

Addressing them requires multidisciplinary development teams and involvement
of the humanities and social sciences in software engineering education. It
also requires a redefinition of incentives and metrics for what is a ‘good’
system. Doing the right thing, and doing it well means that we also need to
define what is good and for whom.

Finally, it is important to keep continuous efforts to improve algorithms and
data, define regulation and standardisation, and develop evaluation tools and
corrective frameworks. But the same time, we cannot ignore that no technology
is without risk, no action is without risk. It is high time to start the
conversation on which AI-risks we find acceptable for individuals and for
society as a whole, and how we distribute these risks, as well as the benefits
of AI.

## References

  * [1] Joy Buolamwini and Timnit Gebru.  Gender shades: Intersectional accuracy disparities in commercial gender classification.  In Conference on fairness, accountability and transparency, pages 77–91. PMLR, 2018.
  * [2] Kate Crawford.  The Atlas of AI.  Yale University Press, 2021.
  * [3] Virginia Dignum.  Responsible artificial intelligence: How to develop and use AI in a responsible way.  Springer Nature, 2019.
  * [4] Anthony W Flores, Kristin Bechtel, and Christopher T Lowenkamp.  False positives, false negatives, and false analyses: A rejoinder to machine bias: There’s software used across the country to predict future criminals. and it’s biased against blacks.  Fed. Probation, 80:38, 2016.
  * [5] Mary L Gray and Siddharth Suri.  Ghost work: How to stop Silicon Valley from building a new global underclass.  Eamon Dolan Books, 2019.
  * [6] Jon Kleinberg, Sendhil Mullainathan, and Manish Raghavan.  Inherent trade-offs in the fair determination of risk scores.  arXiv preprint arXiv:1609.05807, 2016.
  * [7] Geoff Pleiss, Manish Raghavan, Felix Wu, Jon Kleinberg, and Kilian Q Weinberger.  On fairness and calibration.  arXiv preprint arXiv:1709.02012, 2017.
  * [8] Tobias Schnabel, Adith Swaminathan, Ashudeep Singh, Navin Chandak, and Thorsten Joachims.  Recommendations as treatments: Debiasing learning and evaluation.  In international conference on machine learning, pages 1670–1679. PMLR, 2016.


Dynamics analysis of the fractional-order Lagrange system

Mihai IVAN

Abstract. The main purpose of this paper is to study the fractional-order
model with Caputo derivative associated to Lagrange system. For this
fractional-order system we investigate the existence and uniqueness of
solutions of initial value problem, asymptotic stability of its equilibrium
states, stabilization problem using appropriate controls and numerical
integration via the fractional Euler method.
111AMS classification: 26A33, 53D05, 65P20, 70H05.
Key words and phrases: fractional-order Lagrange system, stability, fractional
Euler method, numerical integration.

##  1 Introduction

The dynamic behavior of differential systems (in particular, Hamilton-Poisson
systems) have been studied due to their deep applications in many areas of
science and engineering including robotics, spatial dynamics, complete
synchronization and secure comunications. Three remarkable classes of systems
formed by differential equations on ${\bf R}^{3}$ are the general Euler top
system, the systems of Maxwell-Bloch type and the family of Lotka-Volterra
systems. In the recent decades, a series of dynamical systems belonging to
these three classes have been studied from the point of view of Poisson
geometry by many researchers, for instance [12, 30, 29, 28, 18, 13, 24, 19,
11, 23].

An interesting example of a differential system of the Euler top type is the
Lagrange system [30]. It is described by the following differential equations
on $~{}{\bf R}^{3}:$

$\dot{x}^{1}(t)=x^{2}(t)x^{3}(t),~{}~{}~{}\dot{x}^{2}(t)=x^{1}(t)x^{3}(t),~{}~{}~{}\dot{x}^{3}(t)=x^{1}(t)x^{2}(t),$
(1.1)

where $\dot{x}^{i}(t)=dx^{i}(t)/dt$ and $t$ is the time.

The system $(1.1)$ is used in theoretical physics for the study of $SU(2)-$
monopoles.

The fractional calculus deals with derivations and integration of arbitrary
order and has deep and natural connections with many fields of applied
mathematics, mathematical physics, engineering, biological systems, control
processing, chaos synchronization, [25, 27, 1, 3, 10, 26, 9, 5, 4, 6, 2]. The
fractional models associated to nonlinear dynamical systems have been
investigated in the papers [22, 14, 20].

The Lie groups, Lie algebroids and Leibniz algebroids have proven to be
powerful tools for geometric formulation of the Hamiltonian mechanics [7, 15,
16]. Also, they have been used in the investigation of many fractional
dynamical systems [10, 17, 22, 21].

This paper is structured as follows. The fractional-order Lagrange system is
defined in Section 2. The existence and uniqueness of solutions of initial
value problem for the fractional model $(2.2)~{}$ is discussed. For the
asymptotic stabilization problem of fractional model $~{}(2.2),$ we associate
the fractional Lagrange system with controls $~{}c_{1}$ and $c_{2},~{}$
denoted by $(3.3).~{}$ In Propositions $~{}(3.1)-(3.3)~{}$ are established
sufficient conditions on parameters $~{}c_{1}$ and $c_{2}~{}$ to control the
chaos in the fractional-order Lagrange system $~{}(3.3).~{}$ In Section 4, we
give the numerical integration of fractional-order Lagrange system $(3.3),$
via the fractional Euler’s method. Finally, a numerical simulation of
fractional model $~{}(3.3)~{}$ is shown to verify the theoretical results.

##  2  The fractional-order Lagrange system

We recall the Caputo definition of fractional derivatives, which is often used
in concrete applications. Let $f\in C^{\infty}(\textbf{R})$ and
$q\in\textbf{R},q>0.$ The $q-$order Caputo differential operator [9], is
described by $~{}D_{t}^{q}f(t)=I^{m-q}f^{(m)}(t),~{}q>0,~{}$ where
$~{}f^{(m)}(t)$ represents the $m-$order derivative of the function
$f,~{}m\in\textbf{N}^{\ast}$ is an integer such that $m-1\leq q\leq m$ and
$I^{q}$ is the $q-$order Riemann-Liouville integral operator [27, 9], which is
expressed by:

$I_{t}^{q}f(t)=\displaystyle\frac{1}{\Gamma(q)}\int_{0}^{t}{(t-s)^{q-1}}f(s,y(s))ds,~{}q>0,~{}~{}~{}s\in[0,T],$
(2.1)

where $~{}\Gamma$ is the Euler Gamma function. If $q=1$, then
$D_{t}^{q}f(t)=df/dt.$

The fractional-order Lagrange system associated to dynamics $(1.1)$ is defined
by the following set of fractional differential equations:

$\left\\{\begin{array}[]{lcl}D_{t}^{q}{x}^{1}(t)&=&x^{2}(t)x^{3}(t)\\\\[2.84544pt]
D_{t}^{q}{x}^{2}(t)&=&x^{1}(t)x^{3}(t),~{}~{}~{}q\in(0,1),\\\\[2.84544pt]
D_{t}^{q}{x}^{3}(t)&=&x^{1}(t)x^{2}(t).\end{array}\right.$ (2.2)

The initial value problem of the fractional-order Lagrange system $(2.1)$ can
be represented in the following matrix form:

$D_{t}^{\alpha}x(t)=x^{1}(t)Ax(t)+x^{3}(t)Bx(t),~{}~{}~{}~{}~{}x(0)=x_{0},$
(2.3)

where $0<q<1,~{}x(t)=(x^{1}(t),x^{2}(t),x^{3}(t))^{T},~{}t\in(0,\tau)$ and

$A=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&1&0\\\
\end{array}\right),~{}~{}~{}B=\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&0\\\ \end{array}\right).$

###### Proposition 2.1

The initial value problem of the fractional-order Lagrange system $(2.1)$ has
a unique solution.

Proof. Let $f(x(t))=x^{1}(t)Ax(t)+x^{3}(t)Bx(t).$ It is obviously continuous
and bounded on $D=\\{x\in{\bf
R}^{3}|~{}x^{i}\in[x_{0}^{i}-\delta,x_{0}^{i}+\delta]\\},i=\overline{1,3}$ for
any $\delta>0.$ We have
$~{}f(x(t))-f(y(t))=x^{1}(t)Ax(t)-y^{1}(t)Ay(t)+x^{3}(t)Bx(t)-y^{3}(t)By(t)=g(t)+h(t),$
where $~{}g(t)=x^{1}(t)Ax(t)-y^{1}(t)Ay(t)~{}$ and
$~{}h(t)=x^{3}(t)Bx(t)-y^{3}(t)By(t).$ Then
$(a)~{}~{}|f(x(t))-f(y(t))|\leq|g(t)|+|h(t)|.$

It is easy to see that
$~{}g(t)=(x^{1}(t)-y^{1}(t))Ax(t)+y^{1}(t)A(x(t)-y(t)).~{}$ Then
$|g(t)|\leq|(x^{1}(t)-y^{1}(t))Ax(t)|+|y^{1}(t)A(x(t)-y(t))|\leq$
$\leq\|A\|(|x(t)|\cdot|x^{1}(t)-y^{1}(t)|+|y^{1}(t)|\cdot|x(t)-y(t)|,$
where $\|\cdot\|$ and $|\cdot|$ denote matrix norm and vector norm
respectively.

Using the inequality $~{}|x^{1}(t)-y^{1}(t))|\leq|x(t)-y(t))|~{}$ one obtains
$(b)~{}~{}~{}|g(t)|\leq(\|A\|+|y^{1}(t)|)\cdot|x(t)-y(t)|.$
Similarly, we have
$(c)~{}~{}~{}|h(t)|\leq(\|B\|+|y^{3}(t)|)\cdot|x(t)-y(t)|.$
According to $(b)$ and $(c),$ the relation $(a)$ becomes
$(d)~{}~{}~{}|f(x(t))-f(y(t))|\leq(\|A\|+\|B\|+|y^{1}(t)|+|y^{3}(t)|)\cdot|x(t)-y(t)|.$
Replacing $\|A\|=1,~{}\|B\|=\sqrt{2}$ and using the inequalities
$~{}|y^{i}(t)|\leq|x_{0}|+\delta,~{}i=1,3~{}$ from the relation $(d),$ we
deduce that
$(e)~{}~{}~{}|f(x(t))-f(y(t))|\leq
L\cdot|x(t)-y(t)|,~{}~{}~{}~{}~{}\hbox{where}~{}L=1+\sqrt{2}+2(|x_{0}|+\delta)>0.$

The inequality $(e)$ shows that $f(x(t))$ satisfies a Lipschitz condition.
Based on the results of Theorems $1$ and $2$ in [8], we can conclude that the
initial value problem of the system $(2.2)$ has a unique solution. $\Box$

For the fractional Lagrange system $(2.1)$ we introduce the following
notations:

$f_{1}(x)=x^{2}x^{3},~{}~{}~{}f_{2}(x)=x^{1}x^{3},~{}~{}~{}f_{3}(x)=x^{1}x^{2}.$
(2.4)

###### Proposition 2.2

The equilibrium states of the fractional Lagrange system $(2.1)$ are given as
the union of the following three families:

$E_{1}:=\\{e_{1}^{m}=(m,0,0)\in{\bf R}^{3}|~{}m\in{\bf
R}\\},~{}~{}~{}E_{2}:=\\{e_{2}^{m}=(0,m,0)\in{\bf R}^{3}|~{}m\in{\bf R}\\},$

$E_{3}:=\\{e_{3}^{m}=(0,0,m)\in{\bf R}^{3}|~{}m\in{\bf R}\\}.$

Proof. The equilibrium states are solutions of the equations
$~{}f_{i}(x)=0,i=\overline{1,3}$ where $~{}f_{i},~{}i=\overline{1,3}$ are
given by (2.2). $\Box$

Let us we present the study of asymptotic stability of equilibrium states for
the fractional system $(2.1)$. Finally, we will discuss how to stabilize the
unstable equilibrium states of the system $(3.1)$ via fractional order
derivative. For this study we apply the Matignon’s test.

The Jacobian matrix associated to system $(2.1)$ is:

$J(x)=\left(\begin{array}[]{ccc}0&x^{3}&x^{2}\\\ x^{3}&0&x^{1}\\\
x^{2}&x^{1}&0\\\ \end{array}\right).\\\ $

###### Proposition 2.3

([25]) Let $x_{e}$ be an equilibrium state of system $(2.1)$ and $J(x_{e})$ be
the Jacobian matrix $J(x)$ evaluated at $x_{e}.$ Then $~{}x_{e}$ is locally
asymptotically stable, iff all eigenvalues of the matrix $J(x_{e})$ satisfy
the inequality:
$~{}~{}|arg(\lambda(J_{(}x_{e})))|>\displaystyle\frac{q\pi}{2}.$ $\Box$

###### Proposition 2.4

The equilibrium states $e_{i}^{m}\in E_{i},~{}i=\overline{1,3}~{}$ are
unstable $(\forall)q\in(0,1).$

Proof. The characteristic polynomial of the matrix
$~{}J(e_{1}^{m})=\left(\begin{array}[]{ccc}0&0&0\\\\[2.84544pt] 0&0&m\\\
0&m&0\\\ \end{array}\right)$ is
$~{}p_{J(e_{1}^{m})}(\lambda)=\det(J(e_{1}^{m})-\lambda
I)=-\lambda(\lambda^{2}-m^{2}).~{}$ For $m=0,$ the characteristic polynomials
of $J(e_{0})$ is $~{}p_{J(e_{0})}(\lambda)=\lambda^{3}.$

The characteristic polynomials of $J(e_{i}^{m}),~{}i=2,3$ are
$p_{J(e_{i}^{m})}(\lambda)=-\lambda(\lambda^{2}-m^{2}).$

The equations $~{}p_{J(e_{0})}(\lambda)=0$ and
$~{}p_{J(e_{i}^{m})}(\lambda)=0$ for $i=\overline{1,3}$ have the root
$\lambda_{1}=0.$ Since $~{}arg(\lambda_{1})=0<\frac{q\pi}{2}$ for all
$q\in(0,1),$ by Proposition 2.3 follows that the equilibrium states $e_{0}$
and $e_{i}^{m},i=\overline{1,3}$ are unstable for all $q\in(0,1).$ $\Box$

##  3  Stability analysis of the fractional-order Lagrange system with
controls $~{}(3.3)~{}$

In the case when $x_{e}$ is a unstable equilibrium state of the fractional-
order system $(2.1)$, we associate to $(2.1)$ a new fractional system, called
the fractional-order Lagrange system with controls and given by:

$\left\\{\begin{array}[]{lcl}D_{t}^{q}{x}^{1}(t)&=&x^{2}(t)x^{3}(t)+u_{1}(t)\\\\[2.84544pt]
D_{t}^{q}{x}^{2}(t)&=&x^{1}(t)x^{3}(t)+u_{2}(t),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}q\in(0,1),\\\\[2.84544pt]
D_{t}^{q}{x}^{3}(t)&=&x^{1}(t)x^{2}(t)+u_{3}(t),\end{array}\right.$ (3.1)

where $u_{i}(t),i=\overline{1,3}$ are control functions.

In this section we take the control functions $u_{i}(t),i=\overline{1,3},$
given by:

$u_{1}(t)=c_{1}x^{1}(t),~{}~{}~{}u_{2}(t)=c_{2}x^{2}(t),~{}~{}~{}u_{3}(t)=c_{2}x^{3}(t),~{}~{}~{}c_{1},c_{2}\in{\bf
R}.$ (3.2)

With the control functions $(3.2),$ the system $(3.1)$ becomes:

$\left\\{\begin{array}[]{lcl}D_{t}^{q}{x}^{1}(t)&=&x^{2}(t)x^{3}(t)+c_{1}x^{1}(t)\\\\[2.84544pt]
D_{t}^{q}{x}^{2}(t)&=&x^{1}(t)x^{3}(t)+c_{2}x^{2}(t),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}q\in(0,1),\\\\[2.84544pt]
D_{t}^{q}{x}^{3}(t)&=&x^{1}(t)x^{2}(t)+c_{2}x^{3}(t)\end{array}\right.$ (3.3)

where $c_{1},c_{2}\in{\bf R}^{\ast}$ are real constants.

The system $(3.3)$ is called the fractional-order Lagrange system with
controls $c_{1},c_{2}.$

If one selects the parameters $c_{1},c_{2}$ which then make the eigenvalues of
the Jacobian matrix of fractional model $(3.3)$ satisfy one of the conditions
from Proposition 2.3, then its trajectories asymptotically approaches the
unstable equilibrium state $x_{e}$ in the sense that
$\lim_{t\rightarrow\infty}\|x(t)-x_{e}\|=0$, where $\|\cdot\|$ is the
Euclidean norm.

The Jacobian matrix of the fractional-order Lagrange system $(3.3)$ with
controls $c_{1},c_{2}$ is:

$J(x,c_{1},c_{2})=\left(\begin{array}[]{ccccc}c_{1}&x^{3}&x^{2}\\\
x^{3}&c_{2}&x^{1}\\\ x^{2}&x^{1}&c_{2}\\\ \end{array}\right).$

###### Proposition 3.1

Let be the fractional-order Lagrange system $(3.3).$

$(i)~{}$ If $~{}c_{1}<0,c_{2}<0,$ then $e_{0}$ is asymptotically stable
$~{}(\forall)~{}q\in(0,1);$

$(ii)~{}$ If $~{}c_{1}>0,c_{2}>0$ or $~{}c_{1}c_{2}<0,$ then $e_{0}$ is
unstable $~{}(\forall)~{}q\in(0,1).$

Proof. The characteristic polynomial of $~{}J(e_{0},c_{1},c_{2})$ is
$~{}p_{0}(\lambda)=-(\lambda-c_{1})(\lambda-c_{2})^{2}.$ The roots of the
equation $p_{0}(\lambda)=0~{}$ are $\lambda_{1}=c_{1},~{}\lambda_{2,3}=c_{2}.$
$(i)~{}$ We suppose $~{}c_{1}<0~{}$ and $~{}c_{2}<0.$ In this case we have
$Re(\lambda_{i})<0$ for $i=\overline{1,3}.$ Since
$|arg(\lambda_{i})|=\pi>\displaystyle\frac{q\pi}{2},i=\overline{1,3}$ for all
$q\in(0,1)$, by Proposition 2.3, it implies that $e_{0}^{m}$ is locally
asymptotically stable.
$(ii)~{}$ We suppose $~{}c_{1}>0,c_{2}>0$ or $~{}c_{1}c_{2}<0.$ Since
$J(e_{0},c_{1},c_{2})$ has at least a positive eigenvalue, it follows that
$e_{0}$ is unstable. Hence, $(i)$ and $(ii)$ hold. $\Box$

###### Proposition 3.2

Let be the fractional-order Lagrange system $(3.3)~{}$ and $~{}q\in(0,1).$
$(i)~{}$ If $~{}c_{1}<0,c_{2}<0,$ then $~{}e_{1}^{m}$ is asymptotically stable
$~{}(\forall)m\in(c_{2},-c_{2})~{}$ and unstable
$~{}(\forall)~{}m\in(-\infty,c_{2})\cup(-c_{2},\infty).$
$(ii)~{}$ If $~{}c_{1}>0,c_{2}>0$ or $~{}c_{1}c_{2}<0,$ then $e_{1}^{m}$ is
unstable $~{}(\forall)m\in{\bf R}^{\ast}.$

Proof. The Jacobian matrix of $(3.3)$ at $e_{1}^{m}$ is
$~{}J(e_{1}^{m},c_{1},c_{2})=\left(\begin{array}[]{ccccc}c_{1}&0&0\\\\[5.69046pt]
0&c_{2}&m\\\ 0&m&c_{2}\\\ \end{array}\right)$ whose characteristic polynomial
is $~{}p_{1}(\lambda)=-(\lambda-c_{1})[(\lambda-c_{2})^{2}-m^{2}].$ The roots
of the equation $p_{1}(\lambda)=0~{}$ are
$\lambda_{1}=c_{1},~{}\lambda_{2,3}=c_{2}\pm m.$
$(i)~{}$ We suppose $~{}c_{1}>0,c_{2}>0.$ The eigenvalues
$~{}\lambda_{i},~{}i=\overline{1,3}$ are all negative if and only if
$~{}m\in(c_{2},-c_{2}).$ Then $e_{1}^{m}$ is asymptotically stable.
$(ii)~{}$ We suppose $~{}c_{1}>0,c_{2}>0$ or $~{}c_{1}c_{2}<0.$ In these
cases,the matrix $J(e_{0},c_{1},c_{2})$ has at least a positive eigenvalue. It
follows that $e_{1}^{m}$ is unstable. Hence the assertions $(i)$ and $(ii)$
hold. $\Box$

###### Proposition 3.3

Let be the fractional Lagrange system $(3.3)$ and $~{}q\in(0,1).$
$(i)$ If $~{}c_{1}<0,c_{2}<0,~{}$ then $~{}e_{2}^{m}~{}$ and $~{}e_{3}^{m}~{}$
are asymptotically stable for all
$~{}m\in(-\sqrt{c_{1}c_{2}},\sqrt{c_{1}c_{2}})~{}$ and unstable
$~{}(\forall)m\in(-\infty,-\sqrt{c_{1}c_{2}})\cup(\sqrt{c_{1}c_{2}},\infty).$
$(ii)~{}$ If $~{}c_{1}>0,~{}c_{2}<0~{}$ or $~{}c_{2}>0,~{}c_{1}\in{\bf R},$
then $~{}e_{2}^{m}~{}$ and $~{}e_{3}^{m}~{}$ are unstable
$~{}(\forall)m\in{\bf R}^{\ast}.~{}$

Proof. We consider $~{}k\in\\{2,3\\}.~{}$ The Jacobian matrices of $(3.3)$ at
$~{}e_{k}^{m}~{}$ are
$~{}J(e_{2}^{m},c_{1},c_{2})=\left(\begin{array}[]{ccccc}c_{1}&0&m\\\\[5.69046pt]
0&c_{2}&0\\\ m&0&c_{2}\\\ \end{array}\right)~{}$ and
$~{}J(e_{3}^{m},c_{1},c_{2})=\left(\begin{array}[]{ccccc}c_{1}&m&0\\\\[5.69046pt]
m&c_{2}&0\\\ 0&0&c_{2}\\\ \end{array}\right).$ Its characteristic polynomials
are $~{}p_{k}(\lambda)=-(\lambda-
c_{2})[\lambda^{2}-(c_{1}+c_{2})\lambda+c_{1}c_{2}-m^{2}].~{}$ The roots of
the equation $~{}p_{k}(\lambda)=0~{}$ are
$\lambda_{1}=c_{2},~{}\lambda_{2,3}=\frac{(c_{1}+c_{2})\pm\sqrt{\Delta}}{2}\in{\bf
R},$ where $~{}\Delta=(c_{1}-c_{2})^{2}+4m^{2}.$
$(i)~{}$ We suppose $~{}c_{1}<0,c_{2}<0.$ We have
$~{}\lambda_{2},\lambda_{3}<0~{}$ if and only if
$~{}\lambda_{2}+\lambda_{3}<0~{}$ and $~{}\lambda_{2}\lambda_{3}>0~{}$. Then
$~{}c_{1}+c_{2}<0~{}$ and $~{}(c_{1}+c_{2})^{2}-\Delta>0.~{}$ It follows that
$~{}\lambda_{i}<0,i=\overline{1,3}~{}$ for all
$~{}m\in(-\sqrt{c_{1}c_{2}},\sqrt{c_{1}c_{2}}).$ Therefore, $~{}e_{k}^{m}~{}$
is asymptotically stable.
$(ii)~{}$ We suppose $~{}c_{1}>0~{}$ and $~{}c_{2}<0~{}$ or $~{}c_{2}>0~{}$
and $c_{1}\in{\bf R}.$ In this cases,$~{}J(e_{k}^{m},c_{1},c_{2})~{}$ has at
least a positive eigenvalue and so $~{}e_{k}^{m}~{}$ is unstable
$~{}(\forall)m\in{\bf R}^{\ast}~{}.$ Therefore, the assertions $(i)$ and
$(ii)$ hold. $\Box$

###### Example 3.1

By choosing the control parameters $c_{1},c_{2}$ that satisfy one condition
from Proposition 3.3, then the trajectories of the fractional-order Lagrange
system $(3.3)$ are driven to the stable equilibrium point $e_{3}^{m}~{}(m\neq
0)$. For example, we select $c_{1}=-1.75,c_{2}=-2,~{}$ then the stability
condition $(i)$ of Proposition 3.3 is achieved. This implies that, the
trajectories of the system $(3.3)$ converge to the equilibrium point
$e_{3}=(0,0,m)$ when $m\in(-1.8708,1.8708)~{}$ and $~{}q\in(0,1).$ For
example, the fractional-order Lagrange system is asymptotically stable at
$e_{3}=(0,0,1.75)$ for $~{}q\in(0,1)$. $\Box$

Using Matlab, in Table 3.1 we give a set of values for $~{}c_{1},c_{2},~{}$
the equilibrium states and corresponding eigenvalues of fractional-order
Lagrange system $(3.3).$

$\begin{array}[]{|l|l|l|l|l|}\hline\cr c_{1},c_{2}\in{\bf
R}^{\ast}&Eigenvalues&m,~{}q&e_{i}^{m}&Stability\\\ \hline\cr
c_{1}=-0.2,~{}c_{2}=-0.8&\lambda_{1}=-0.2,~{}\lambda_{2,3}=-0.8&q\in(0,1)&e_{0}&stable\cr\hline\cr
c_{1}=-2,~{}c_{2}=-1.85&\lambda_{1}=-2,~{}\lambda_{2}=-0.85&m=1&e_{1}^{m}&stable\cr&\lambda_{3}=-2.85&q\in(0,1)&&m\in(-1.85,1.85)\cr\hline\cr
c_{1}=-7.2,~{}c_{2}=-0.2&\lambda_{1}=-0.2,~{}\lambda_{2}=0.4182&m=-2&e_{2}^{m}&unstable\cr&\lambda_{3}=-7.8182&q\in(0,1)&&m\in(-\infty,-1.2)\cup(1.2,\infty)\cr\hline\cr
c_{1}=-1.75,~{}c_{2}=-2&\lambda_{1}=-2,~{}\lambda_{2}=-0.9911&m=1.75&e_{3}^{m}&stable\cr&\lambda_{3}=-2.7588&q\in(0,1)&&m\in(-1.8708,1.8708)\cr\hline\cr\end{array}$

Table 3.1. The controls $~{}c_{1},c_{2},~{}$ equilibrium states
$~{}e_{i}^{m}~{}$ and corresponding eigenvalues.

##  4 Numerical integration of the fractional-order Lagrange system $(3.3)$

Consider the following general form of the initial value problem (IVP) with
Caputo derivative [26]:

$D_{t}^{q}y(t)=f(t,y(t)),~{}~{}~{}y(0)=y_{0},~{}~{}~{}t\in I=[0,T],~{}T>0$
(4.1)

where $~{}y:I\rightarrow{\bf R}^{n},~{}f:{\bf R}^{n}\rightarrow{\bf R}^{n}~{}$
is a continuous nonlinear function and $q\in(0,1),$ represents the order of
the derivative.

The right-hand side of the IVP $(4.1)$ in considered examples are Lipschitz
functions and the numerical method used in this works to integrate system
$(4.1)$ is the Fractional Euler’s method.

Since $f$ is assumed to be continuous function, every solution of the initial
value problem given by $(4.1)$ is also a solution of the following Volterra
fractional integral equation:

$y(t)=y(0)+~{}I_{t}^{q}f(t,y(t)),$ (4.2)

where $I_{t}^{q}$ is the $q-$order Riemann-Liouville integral operator
$(2.1).$

$I_{t}^{q}f(t)=\displaystyle\frac{1}{\Gamma(q)}\int_{0}^{t}{(t-s)^{q-1}}f(s,y(s))ds,~{}q>0,~{}~{}~{}s\in[0,T].$
(4.3)

Moreover, every solution of $(4.2)$ is a solution of the (IVP) $(4.1).$

To integrate the fractional equation $(4.1),$ means to find the solution of
$(4.2)$ over the interval $~{}[0,T].$ In this context, a set of points
$~{}(t_{j},y(t_{j}))$ are produced which are used as approximated values. In
order to achieve this approximation, the interval $[0,T]$ is partitioned into
$n$ subintervals $~{}[t_{j},t_{j+1}]$ each equal width
$~{}h=\frac{T}{n},~{}t_{j}=jh$ for $~{}j=0,1,...,n.~{}$ For the fractional-
order $q$ and $~{}j=0,1,2,...,$ it computes an approximation denoted as
$~{}y_{j+1}~{}$ for $~{}y(t_{j+1}),~{}j=0,1,....$

The general formula of the fractional Euler’s method for to compute the
elements $~{}y_{j},$ is

$y_{j+1}=y_{j}+\displaystyle\frac{h^{q}}{\Gamma(q+1)}f(t_{j},y(t_{j})),~{}~{}~{}~{}~{}t_{j+1}=t_{j}+h,~{}~{}~{}j=0,1,...,n.$
(4.4)

For more details, see [26, 6, 2].

For the numerical integration of the system $(3.3),$ we apply the fractional
Euler method (FEM). For this, consider the following fractional differential
equations

$\left\\{\begin{array}[]{lcl}D_{t}^{q}x^{i}(t)&=&F_{i}(x^{1}(t),x^{2}(t),x^{3}(t)),~{}~{}~{}i=\overline{1,3},~{}~{}t\in(t_{0},\tau),~{}q\in(0,1)\\\
x(t_{0})&=&(x^{1}(t_{0}),x^{2}(t_{0}),x^{3}(t_{0}))\end{array}\right.\\\ $
(4.5)

where

$\left\\{\begin{array}[]{lcl}F_{1}(x(t))&=&x^{2}(t)x^{3}(t)+c_{1}x^{1}(t),\\\
F_{2}(x(t))&=&x^{1}(t)x^{3}(t)+c_{2}x^{2}(t),~{}~{}~{}~{}~{}c_{1},c_{2}\in{\bf
R}^{*},\\\ F_{3}(x(t))&=&x^{1}(t)x^{2}(t)+c_{2}x^{3}(t).\\\
\end{array}\right.$ (4.6)

Since the functions $F_{i}(x(t)),i=\overline{1,3}$ are continuous, the initial
value problem $(4.5)$ is equivalent to system of Volterra integral equations,
which is given as follows:

$x^{i}(t)~{}=~{}x^{i}(0)+~{}I_{t}^{q}F_{i}(x^{1}(t),x^{2}(t),x^{3}(t)),~{}~{}~{}~{}~{}i=\overline{1,3}.$
(4.7)

The system $(4.7)$ is called the Volterra integral equations associated to
fractional-order Lagrange system $(4.5)$.

The problem for solving the system $(4.5)$ is reduced to one of solving a
sequence of systems of fractional equations in increasing dimension on
successive intervals $[j,(j+1)]$.

For the numerical integration of the system $(4.6)$ one can use the fractional
Euler method (the formula $(4.4)$ ), which is expressed as follows:

$x^{i}(j+1)=x^{i}(j)+\frac{h^{q}}{\Gamma(q+1)}F_{i}(x^{1}(j),x^{2}(j),x^{3}(j)),~{}~{}~{}i=\overline{1,3},$
(4.8)

where $j=0,1,2,...,N,h=\displaystyle\frac{T}{N},T>0,N>0.~{}$

More precisely, the numerical integration of the fractional system $(4.5)$ is
given by:

$\left\\{\begin{array}[]{ll}x^{1}(j+1)&=x^{1}(j)+h^{q}~{}\displaystyle\frac{1}{\Gamma(q+1)}(x^{2}(j)x^{3}(j)+c_{1}x^{1}(j))\\\\[8.5359pt]
x^{2}(j+1)&=x^{2}(j)+h^{q}~{}\displaystyle\frac{1}{\Gamma(q+1)}(x^{1}(j)x^{3}(j)+c_{2}x^{2}(j))\\\\[8.5359pt]
x^{3}(j+1)&=x^{3}(j)+h^{q}~{}\displaystyle\frac{1}{\Gamma(q+1)}(x^{1}(j)x^{2}(j)+c_{2}x^{3}(j))\\\\[8.5359pt]
x^{i}(0)&=x_{e}^{i}+\varepsilon,~{}~{}~{}i=\overline{1,3}.\\\
\end{array}\right.$ (4.9)

Using [26, 9], we have that the numerical algorithm given by $(4.9)$ is
convergent.

###### Example 4.1

Let us we present the numerical simulation of solutions of the fractional-
order Lagrange system $(3.3)$ which has considered in Example $~{}3.1.$
For this we apply the algorithm $(4.9)$ and software Maple. Then, in $(4.9)$
we take: $~{}c_{1}=-1.75,c_{2}=-2,~{}h=0.01,\varepsilon=0.01,N=500,t=502.$

The orbits $(n,x^{i}(n)),i=\overline{1,3}~{}$ for the solutions of fractional-
order Lagrange system for the equilibrium state $e_{3}=(0,0,1.75)~{}$ have the
representations given in figures Fig. 1(a), 2(a), 3(a) (for $~{}q=0.65~{})$
and Fig. 1(b), 2(b), 3(b) (for $~{}q=1).~{}$ $\Box$

|  |
---|---|---

Fig.1.(a) $~{}(n,x^{1}(n))$ for $q=0.65$ |  |  Fig.1.(b) $~{}(n,x^{1}(n))$ for $q=1$
---|---|---

|  |
---|---|---

Fig.2.(a) $~{}(n,x^{2}(n))$ for $q=0.65$ |  |  Fig.2.(b) $~{}(n,x^{2}(n))$ for $q=1$
---|---|---

|  |
---|---|---

Fig.3.(a) $~{}(n,x^{3}(n))$ for $q=0.65$ |  |  Fig.3.(b) $~{}(n,x^{3}(n))$ for $q=1$
---|---|---

The numerical simulations confirm the validity of the theoretical analysis.

###### Remark 4.1

Appyling $(4.9)$ and Maple for the numerical simulation of solutions of
fractional Lagrange system $~{}(3.3)~{}$ for each pair $~{}(c_{1},c_{2})~{}$
of values given in the Table (3.1), it will be found that the results obtained
are valid. $\Box$

Conclusions. This paper presents the fractional-order Lagrange system $(3.3)$
associated to system $(1.1).$ The fractional-order Lagrange system $(3.3)$ was
studied from fractional differential equations theory point of view:
asymptotic stability, determining of sufficient conditions on parameters
$c_{1},c_{2}$ to control the chaos in the proposed fractional system and
numerical integration of the fractional model $(4.2).~{}$ By choosing the
right parameters $c_{1}$ and $c_{2}$ in the fractional model $~{}(3.3),~{}$
this work offers a series of chaotic fractional differential systems. $\Box$

Acknowledgments. The author has very grateful to be reviewers for their
comments and suggestions.

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Author’s adress

Mihai Ivan
West University of Timişoara. Seminarul de Geometrie şi Topologie.
Teacher Training Department. Timişoara, Romania.
E-mail<EMAIL_ADDRESS>