# Beyond the Frame:
Single and multiple video summarization method with user-defined length

Vahid Ahmadi Kalkhorani1, Qingquan Zhang2, Guanqun Song3, Ting Zhu4
1,3,4Computer Science and Engineering, The Ohio State University, Columbus,
OH, USA
2Gies College of Business, University of Illinois at Urbana-Champaign,
Champaign, IL, USA
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Video smmarization is a crucial method to reduce the time of videos which
reduces the spent time to watch/review a long video. This apporach has became
more important as the amount of publisehed video is increasing everyday. A
single or multiple videos can be summarized into a relatively short video
using various of techniques from multimodal audio-visual techniques, to
natural language processing approaches. Audiovisual techniques may be used to
recognize significant visual events and pick the most important parts, while
NLP techniques can be used to evaluate the audio transcript and extract the
main sentences (timestamps) and corresponding video frames from the original
video.Another apporach is to use the best of both domain. Meaning that we can
use audio-visual cues as well as video transcript to extract and summarize the
video. In this paper, we combine a variety of NLP techniques (extractive and
contect-based summarizers) with video processing techniques to convert a long
video into a single relatively short video. We desing this toll in a way that
user can specify the relative length of the summarized video. We have also
explored ways of summarizing and concatenating multiple videos into a single
short video which will help having most important concepts from the same
subject in a single short video. Out approach shows that video summarizing is
a difficult but significant work, with substantial potential for further
research and development, and it is possible thanks to the development of NLP
models.

Index Terms: natural language processing, summarizing, video processing

##  1 Introduction

Online video production and sharing have increased dramatically in recent
years, and this growth is projected to continue. Users may find it difficult
to quickly and effectively locate the material they want due to the abundance
of video data accessible. A useful remedy for this issue is video summary,
which enables viewers to immediately scan through extensive amounts of video
footage to find the most important details and relevant areas. Video summary
is the process of extracting the most important parts from a video while
excluding extraneous or unimportant material. A typical video summary is made
up of key images or shots that encapsulate the main points of the video. The
objective is to produce a condensed video that still delivers the crucial
information but is simpler and quicker to explore. The research community has
recently given video summarizing a lot of attention, and several methods have
been suggested. These methods may be generally divided into two categories:
conventional [1, 2] and deep learning-based [3, 4, 5] approaches. Traditional
video summarizing techniques choose essential frames or images by manually
extracting attributes like color, texture, and motion. The most crucial parts
of the film are then determined by training a machine learning model using
these attributes. To choose important frames or pictures, content-based,
feature-based, and graph-based techniques are frequently employed. [6]. The
visual content of the movie is examined using content-based approaches to
pinpoint crucial shots or frames. Techniques like histogram analysis, edge
detection, and motion analysis can be used for this purpose. To identify
important frames or shots, feature-based algorithms combine elements including
color, texture, and motion. The video is modeled using graph-based techniques,
where each node corresponds to a frame or shot and the edges show the
connections between them. While traditional methods have been successful in
some applications, they have limitations when dealing with complex scenes or
videos with large amounts of content. Deep learning-based methods have emerged
as a promising approach to video summarization, leveraging the power of neural
networks to learn complex features directly from the data. Deep learning-based
video summarization methods typically use convolutional neural networks (CNNs)
or recurrent neural networks (RNNs) to extract relevant features from the
video frames or shots. These features are then used to select the most
important frames or shots. Attention mechanisms can also be used to highlight
the most salient parts of the video. However, deep learning-based video
summarization methods face several challenges. First, they require large
amounts of labeled data for training, which can be difficult to obtain for
some applications. Second, they struggle to capture temporal dependencies,
which are crucial for understanding the context of the video. Finally, the
selection of the most important frames or shots can be subjective, and the
quality of the summary can vary depending on the user’s preferences.

###  1.1 Related works

Video summarization is a process of creating a short and concise summary of a
long video. It is a challenging task due to the large amount of data that
needs to be processed. In recent years, there has been a lot of research in
this area, and various methods have been proposed to tackle this problem. In
this section, we will discuss some of the most popular video summarization
methods.

One of the most popular video summarization methods is based on deep
reinforcement learning [7]. In this method, a deep neural network is trained
to select the most important frames from a video. The network is trained using
a reward function that encourages it to select frames that are most
representative of the video. The method has been shown to be effective in
summarizing videos of different genres.

Another popular method is based on curve simplification [1]. In this method,
the video is first divided into a set of shots. Then, keyframes are extracted
from each shot. The keyframes are then represented as curves, and a curve
simplification algorithm is used to select the most representative curves. The
method has been shown to be effective in summarizing videos of different
genres.

A low complexity video summarization method has been proposed for different
types of enterprise videos such as commercial clips, talk videos, or e-meeting
screen and slide share recordings [8]. The generated summary comprises salient
shots and/or slides that are representative of the video. The method has been
shown to be effective in summarizing enterprise videos.

Another method is based on camera motion [9]. In this method, the camera
motion is analyzed to identify the most important frames. The method has been
shown to be effective in summarizing videos of different genres.

A video summarization method based on camera motion and a graph-based approach
has also been proposed [10]. In this method, the video is first divided into a
set of shots, and then keyframes are extracted from each shot. The keyframes
are then represented as nodes in a graph, and a graph-based algorithm is used
to select the most representative nodes. The method has been shown to be
effective in summarizing videos of different genres.

Data CollectionText PreprocessingAbstractive SummarizationFrame SelectionPost-
processingVideo ConcatenationOutput Video SummaryVideo Data Figure 1: Summary
of video summarization methodology

##  2 Video summarization based on text suumarizers

###  2.1 Problem Formulation

Let $\textbf{V}=\\{\textbf{v}_{1},\textbf{v}_{2},...,\textbf{v}_{N}\\}$ be a
long target video clip that we want to summarize, where each frame
$\textbf{v}_{i}$ is a RGB image matrix. The goal of video summarization is to
select a subset of frames $S\subset\textbf{V}$ that best represents the video
while minimizing the summary length. Formally, we can define the video
summarization problem as

minimize: $\sum_{i=1}^{N}l_{v_{i}}I_{i}-P_{v_{i}}I_{i}$

subject to: $\sum_{i\in N}l_{v_{i}}=L$

$l_{i}\in[0,L],\forall i\in\\{1,2,...,N\\}$ where $I_{i}$ is a binary variable
indicating whether clip i is selected for the summary, and $l_{v_{i}}$ is the
length of the video clip $i$ and $P_{v_{i}}$ is the importance of each short
clip. The objective function minimizes the weighted summary length, which is
the sum of the weights of the selected frames. The constraint ensures that
exactly one frame is selected for the summary, and the binary variables
enforce the selection of frames.

This formulation can be extended to incorporate additional constraints or
objectives, such as the maximum importance of each video clip, or the maximum
length of each segment besides the total length of the video. For example, on
might decide to not include video segments with importance less than a certain
number which will help remove some portions of the video which are not
necessary such as problem-solving, announcements by the lecturer, etc.

###  2.2 Video transcript summarization

Text summarization is a field of natural language processing that aims at
generating a shorter version of a given text while retaining its most
important information. As can be seen clearly, the problem definition of text
summarization is very similar to video summarization and they can help each
other when the other modality (text/video) is available. In recent years,
there has been a growing interest in developing text summarizer models that
can automatically generate summaries for various types of texts, including
news articles, scientific papers, and legal documents. In this section, we
will briefly introduce different methods in this field and their potential in
solving our problem i.e. video summarization.

####  2.2.1 Extractive methods

One of the earliest approaches to text summarization is the extraction-based
method, which involves selecting the most important sentences or phrases from
the original text and combining them to form a summary. In their survey on
automatic text summarization, Hovy and Lin [11] provide an overview of various
extraction-based methods, including sentence ranking, graph-based methods, and
clustering-based methods. These methods typically involve analyzing the text
to identify important sentences based on features such as sentence length,
word frequency, and semantic similarity. The selected sentences are then
combined to create a summary. Some popular extractive summarization methods
include:

  * •

Frequency-based methods, such as TF-IDF and TextRank, that rank sentences
based on their frequency or importance in the text.

  * •

Graph-based methods, such as LexRank and SumBasic, which represent the text as
a graph and use graph algorithms to identify important sentences.

  * •

Machine learning-based methods, such as SVM and Random Forest, which use
supervised learning algorithms to classify sentences as important or
unimportant.

####  2.2.2 Abstractive methods

Another popular approach to text summarization is the abstraction-based
method, which involves generating a summary by paraphrasing the original text
using natural language generation techniques. This method is more challenging
than extraction-based methods, as it requires the model to understand the
meaning of the text and generate new sentences that convey the same
information. In their comprehensive study on automatic text summarization,
Narayan et al. [12] provide an overview of various abstraction-based methods,
including rule-based methods, machine learning-based methods, and deep
learning-based methods. These methods typically involve generating summaries
using natural language generation techniques, such as language models and
neural networks. Some popular abstractive summarization methods include:

  * •

Sequence-to-sequence models, such as Encoder-Decoder and Transformer, which
generate summaries using a neural network that learns to encode the input text
into an embedding vector and decode the summary using this embedding vector
and a decoder layer.

  * •

Reinforcement learning-based methods, which use a reward function to train a
model to generate summaries that are both informative and readable, the reward
of these models generally depends on the output text length and the importance
of each sentence that appeared in the summary. This required function plays an
important role in the performance and training process of these models and
needs special attention and design.

  * •

Hybrid methods, which combine extractive and abstractive methods to generate
summaries that are both concise and informative. These models are usually more
efficient compared to abstractive methods. As they analyze a shorter text
compared to the original text and yet are able to generate abstractive text
which is usually more smooth compared to abstracitve results.

Although the abstractive methods generate a better representation of the long
input text, the drawback for our problem (video summarization) is the
inefficient process of finding corresponding timestamps of each sentence in
the original text. Since the sentences in the original text do not appear
directly in the summarized text, we need to compare the sentences in the
output summarized text to every sentence in the original text and find the
best match. On the other hand, in extractive methods, we can stop the process
of finding timestamps as soon as we find a match. However, in the abstractive
methods, we need to continue this process up to the end, as the best match
might be at the end and next sentences. Another disadvantage of abstractive
methods in our case is that these methods do not preserve the order of
sentences and might change the order arbitrarily making the process of finding
timestamps even harder.

####  2.2.3 Hybrid methods

Recently, there has been a growing interest in developing hybrid text
summarizer models that combine both extraction-based and abstraction-based
methods. These models aim to leverage the strengths of both approaches and
generate more accurate and informative summaries. In their paper on text
summarizer using NLP, Singh and Kaur [8] describe a hybrid model that uses a
combination of sentence ranking and sentence compression techniques to
generate summaries.

###  2.3 Proposed approach

Algorithm 1 Single Video Summarizer

1:Prompt the user to enter the desired summary ratio (e.g., 10%, 20%, etc.)
and store it in a variable, ratio

2:Use a speech-to-text model or Youtube API to convert the video’s audio into
text.

3:Apply NLP tool to restore any missing punctuation to the text.

4:Use an abstractive summarization model to summarize the text to the desired
ratio, preserving only the most important sentences or phrases.

5:for all timestamp in timestamps do

6: Cut the video using the current timestamp.

7: Add the resulting video clip to a list of clips.

8:end for

9:Concatenate the resulting video clips.

10:Output the summarized video.

Figure 2 illustrates the overall proposed approach for video summarization.
Overally, we can divide this process into 1) preprocessing, which inclues
getting the data, preprocessing video and video transcript 2) processing
involving summarizing video transcript, finding the corresponding timestamps
in the original video, cutting the video into shorter video clips 3)
postprocessing including postprocessing the summarized text for cheatsheet and
abstractive summarization and concatenating the shorter video clip to obtain
the final summarized video.

####  2.3.1 Data collection

In this step, we need to have access to a source of video clips (mostly
lecture videos) and their transcript. Here open-source courses published on
YouTube are great sources of data. More specifically, we use MIT open-course
videos. YoutTube has another advantage that in the early stages of the
project, we could use its API to extract the video’s transcript directly
without using audio-to-text models. To this end, we send a query to youtube
with the id of the video-of-interset and requests for transcripts. YouTube API
returns the video transcript including the spoken sentences by the lecturer
and the start and end time of each sentence in a JSON format. We also get the
download link which we use to download the whole video. We also write a script
that downloads the YouTube video given the download link and starting and end
timestamps. This helps us reduce the time and space required to download the
lecture video.

####  2.3.2 Text prepossessing

One disadvantage of YouTube transcription is that it does not contain any
punctuation like ”.”, ”,”, ”?”, or ”!”, and this is of great importance as
text summarizers rely heavily on text punctuation to extract sentences and
their importance, this step is almost the first step of all text summarizers.
To tackle this challenge, we have used a specific model trained for this
purpose [13] namely rpunct (restore punctuation). This model is a BERT model
which is trained specifically to restore punctuations of a sentence. This
helps us to have a clean version of the video transcript which is ready to be
processed by text summarizer. The text summarizer then splits the text from
this punctuation and processes each text segment. One other advantage of
restoring punctuations is that we can preprocess the text and reduce the
number of sentences before passing that through the summarizer. For instance,
we know the questions lecturer asks from the audience has less importance
compared to the answer he/she gives to the same questions. Thus, we can remove
all sentences ending with a question mark.

####  2.3.3 Text summarizer

The next step in the process is to summarize the prepossessed text. To this
end, we employ a pre-trained BERT model with is fine-tuned for text-
summarization task [14]. This model takes the prepossessed video transcript
with a ratio $\in[0,100]\%$ and ranks sentences in the text based on their
importance and selects the top ratio% sentences. This model has the capability
of being run on GPU which makes the inference time much lower. On a CPU
machine (Intel Core-i7 with 8GB RAM) it takes around 1:30 to summarize a
1-hour lecture.

####  2.3.4 Video Clip Extraction and pre-processing

Given the summarized text of one video transcript, we need to find the stating
and end time that, each sentence is mentioned in the video. To do so, when we
are summarizing single video, we use compared every sentence in the summarized
text, with sentences in the original transcript and given the fact that we
have access to the starting and ending time of every single sentence in the
video, we can get the timestamps of sentences in the summarized text. To make
this process more efficient, we save the ending point of last found sentence
and start searching from this point for the next sentence, to make sure that
we are not searching the previous sentences multiple times.

In the case of multiple video summarization, since we are using an abstractive
method, the exact sentence may not appear in the original text, so we need to
have a comparison tool to find what portions of the video need to be cut. To
solve this challenge, we compare the embedding vector of each sentence in the
summarized text with the original text, and peak the highest similarity score
sentence.

Then, we use ffmpeg [15] to extract video clips from the original video. These
are usually short video clips corresponding every sentence in the summarized
text.

This process is show in Algorithm 1.

##  3 Diffusion video summary

Video ConcatenationVideo to Text ConversionPrepossessingText
SummarizationDivide and ConquerContext-Based Summary Video Clip Extraction and
Preprocessing Video Clip ConcatenationCheat-Sheet Creation Figure 2:
Methodology for Summarizing and Combining Two Videos

We extended the single video summarization problem to a more general and
potentially more useful problem of summarizing multiple videos (lectures).
Given a set of video lectures with 1-2 hours length each, we want to generate
a short video clip that summarizes all these videos together into a short
video clip. To final video clip will contain different segments from different
lectures from the same or multiple instructors.

###  3.1 Divide and Conquer

When summarizing multiple video we can take two approaches:

Approach 1. We concatenate the transcript of all videos together and then
summarize the resulting long transcript text. In this approach, after
converting audio to text, we concatenate all texts together. Then, we pass the
concatenated text to the rpunc model to restore punctuation. Then, we pass the
preprocessed text to the extractive summarizer model. This model extracts the
most important sentences of the joined video. In the next step we loop over
all videos’ texts and find the video that each sentence in the summarized text
belongs to. Then, we find the corresponding starting and end timestamps and
cut the video from these points and store them on disk. After cutting all
clips from all videos we concatenate them together. The pseudo-code for this
approach is shown in Algorithm 2. As shown in 11-20 of this pseudo-code we
have nested for loop whihc is not very efficient. Moreover, lines 9 and 10 in
this algorithm pass a large text to NLP models which are not computationally
efficient.

1:Prompt the user to enter the desired summary ratio (e.g., 10%, 20%, etc.)
and store it in a variable, ratio

2:for all video in videos do

3: Use a speech-to-text model or Youtube API to convert the video’s audio into
text and store it

4:end for

5:Concatenate all texts

6:Apply NLP tool to restore any missing punctuation to the text.

7:Use an extarctive summarization model to summarize the text to the desired
ratio, preserving only the most important sentences or phrases, and store it
in summarizedList

8:Concatenate all summarized texts

9:Apply NLP tool to restore any missing punctuation to the text.

10:Use an abstractive summarization model to summarize the text with the user
entered ratio

11:for all sentence in summarized text do

12: for all video in videos do

13: for all timestamp in timestamps do

14: if sentence in video then

15: Cut the video using the current timestamp.

16: store video clip

17: end if

18: end for

19: end for

20:end for

21:Concatenate the resulting video clips.

22:Output the summarized video.

Algorithm 2 Multi Video Summarizer - Approach 1

Approach 2. We summarize each video transcript individually first. Then
concatenate the summarized texts and summarized this shorter text.

We call the second approach divide-and-conquer inspired by the famous
algorithm with the same name. As can be seen, the second approach is more
efficient and elegant approach compared to approach 1. First, since every
video is summarized individually, in the last summarization step, we are
dealing with much smaller text, which decreases computational cost. Second, in
the first step, we are removing irrelevant part of each video which does not
contribute to the final meaning. Finally, by applying an abstractive
summarizer to the final extractive summary of the videos we can get rid of
repeated concepts in the video and have a smoother video which is necessary in
this case. To make the last point more clear, we need to point out that when
extractive methods report important sentences in order and sentences in the
final result will always have the same order of appearance as the original
text. This is not what we desire as it hurts the smoothness of the final
result. Suppose the case that we have the same lecture by the same lecturer in
two different semesters. If we do not use the abstractive method, in the first
half of the summarized video we will see the lecture goes from start to end,
and from start to end in the second part, which is obviously not desired. In
this approach, we calculated the embedding similarities between each sentence
in the text and picked the one with the maximum similarity score. We used a
pre-trained transformer-based model to generate sentence embeddings and
calculated the cosine similarity between each pair of embeddings. We then
selected the sentence with the highest similarity score as the summary
sentence. We repeated this process until we reached the desired summary
length.

##  4 Multi-modal output

As another extension to the project we are generating text-based summaries
beside the video summary which is the primary focus of this project. To do so,
we are employing a hybrid approach as discussed in section 2.2.3.

###  4.1 Context-based summary and cheat-sheet

To generate the text summary of the given video we employ an abstractive BERT
model where we convert the whole video transcript into a single paragraph. To
make this process more efficient we run the abstractive summarizer on the
extractive summary of the video(s) that we generated for the video
summarization (hybrid approach). This summarized paragraph can be used for
searching videos among thousands of videos without the need to provide a title
or keywords for each video by the publisher of the video. One example of the
process is shown in Fig. 3.

Figure 3: Text summary of a lecture on convolutional neural network published
in MIT opencourse channel
https://www.youtube.com/watch?v=uapdILWYTzE The video discusses the use of
neural networks to extract meaningful features from data, such as human faces,
by learning a hierarchy of features that can be used to detect the presence of
a face in a new image. Different filters or convolutional features are used to
detect different types of features in the image. These features are then
passed through fully connected layers to perform classification tasks, such as
image classification. The power of this architecture extends to various tasks,
such as in medicine where deep learning models can analyze medical image
scans. The behavior of the neural network can be visualized using semantic
segmentation maps to obtain a more detailed understanding of the image
classification process.

We have also used an abstractive method to extract key-points mentioned in the
video in a few bullet-points in form of a cheatsheet. One example of the
process is shown in Fig. 4.

Figure 4: Cheatsheet of a lecture on convolutional neural network published in
MIT opencourse channel
https://www.youtube.com/watch?v=uapdILWYTzE • A neural network-based approach
is used to extract meaningful features from data of human faces. • Filters are
designed to pick up different types of features in the image. • The
computation of class scores is performed using convolution, non-linearity, and
pooling. • The architecture is powerful and extends to many tasks beyond image
classification. • Deep learning models can have a significant impact in
medicine and healthcare. • Neural networks can be visualized through semantic
segmentation maps. Algorithm 3 Multi Video Summarizer - Approach 2

1:Prompt the user to enter the desired summary ratio (e.g., 10%, 20%, etc.)
and store it in a variable, ratio

2:for all video in videos do

3: Use a speech-to-text model or Youtube API to convert the video’s audio into
text.

4: Apply NLP tool to restore any missing punctuation to the text.

5: Use an extarctive summarization model to summarize the text to the desired
ratio, preserving only the most important sentences or phrases, and store it
in summarizedList

6:end for

7:Concatenate all summarized texts

8:Apply NLP tool to restore any missing punctuation to the text.

9:Use an abstractive summarization model to summarize the text with $1/N$
ratio, where $N$ is the number of videos

10:Find starting and end time of every sentence by comparing its embedding
vector with the summarized text in previous step

11:for all video in videos do

12: for all timestamp in timestamps do

13: Cut the video using the current timestamp.

14: Add the resulting video clip to a list of clips.

15: end for

16:end for

17:Concatenate the resulting video clips.

18:Output the summarized video.

##  5 Evaluation and Challenges

The evaluation of the summarized video was one of the most challenging parts.
To tackle this challenge we relied on the performance of the NLP models and
more specifically text summarizers. As our proposed approach highly depends on
the performance of the extractive and abstractive text summarization models,
the quality of output video has a one-to-one relation to the performance of
these models. Thus, the quality of the output video in terms of smoothness and
natural language is the same as these NLP models. In other words, our proposed
procedure has the chance to be improved with the advancements in NLP models.
Another challenge, we faced during the project was the computational and space
complexity of NLP models, especially in multi-video summarization tasks. Since
our goal was to summarize tens of videos into a short video, we had to
summarize tens of hours of videos (lectures) which is of course very time and
computation-consuming. To tackle this problem, we proposed the divide-and-
conquer approach where we summarize each video separately and then summarize
these summarise again. This has multiple advantages: First, we can save the
summary of the video at the upload time and save them for future usage. So,
there is no need to process a video multiple times, for multiple use cases.
Second, we can summarize the video in parallel in the first step instead of
processing the while text sequentially.

##  6 Future Work and Extended Considerations

The field of video summarization presents several exciting directions for
future research. One such direction involves the integration of video
summarization techniques with emerging technologies in IoT and heterogeneous
computing systems [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. For
instance, the development of algorithms that can efficiently process and
summarize video data from a multitude of IoT devices, such as security cameras
and environmental sensors, is a promising area of research. This approach
could leverage concurrent communication methods and heterogeneous data sources
to create more comprehensive and context-aware video summaries.

The application of machine learning, particularly deep learning models, in
video summarization has shown promising results [29, 30, 31, 32, 33, 34, 35].
Future work could explore the development of models that are optimized for
low-power devices [36, 37, 38], making video summarization more accessible in
mobile and embedded systems. This could include edge devices that perform on-
the-fly video summarization, aiding in real-time decision-making in various
scenarios like traffic management or emergency response.

Furthermore, the integration of video summarization methods with security and
privacy-preserving techniques [39, 40, 41, 42, 43, 44] presents a novel
research area. As video data often contains sensitive information, developing
summarization methods that respect user privacy and data security is critical.
Techniques such as physical-layer message authentication [45] and obfuscation-
robust character extraction [46] could be adapted to ensure the integrity and
confidentiality of summarized video data.

Another intriguing area of research involves the use of video summarization in
smart health and wellness applications [47, 48, 49, 50, 51, 52, 53]. For
instance, summarizing video data from fitness sessions or medical procedures
can provide quick and insightful overviews for both patients and healthcare
professionals. The challenge here is to develop summarization algorithms that
are sensitive to the nuances and critical moments in health-related videos.

Lastly, the exploration of video summarization in sustainable energy systems
and smart cities [54, 55, 56, 57] could lead to innovative applications.
Summarized video data could be utilized to monitor and manage energy
consumption, traffic flows, and public safety in a more efficient manner.

In conclusion, the future of video summarization is not confined to improving
the algorithms’ accuracy and efficiency but also extends to its application in
diverse and interdisciplinary domains. The integration of video summarization
with IoT, machine learning, security, and smart systems opens new pathways for
research and practical applications.

##  7 Conclusion

In conclusion, in this paper, we presented a robust of effective method to
summarize single and multiple videos. Our methodology provides a comprehensive
framework that integrates several different techniques and processes in NLP
and video processing domain, including different abstractive and context-based
summaries and text preprocessors. Our proposed method can solve two distinct
problems of summarizing a single video and multiple videos at the same time.
To select the most efficient way, we have designed two different processes for
each problem. In the single video summarization method, we used an extractive
method, while we are using a hybrid method (both extractive and abstractive)
in the multi-video summarization task. We have also generated context-based
summaries which can be used for cheat-sheet generation for offline review of
videos and effectively search in the context of videos without the need of
adding information about the content of each video manually by the user. The
presented approach has a lot of potential and usage in different use cases,
such as lecture video summarization, news broadcasts, and social-media
applications.

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# Prevalent Behavior of Smooth Strongly Monotone Discrete-Time Dynamical
Systems††thanks: Supported by NSF of China No.11825106, 11771414 and 11971232,
Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and
Technology of China.

Yi Wang  School of Mathematical Sciences
University of Science and Technology of China
Hefei, Anhui, 230026, P. R. China  Jinxiang Yao  School of Mathematical
Sciences
University of Science and Technology of China
Hefei, Anhui, 230026, P. R. China  Yufeng Zhang Corresponding author:
<EMAIL_ADDRESS>(Y. Zhang). School of Mathematical Sciences
University of Science and Technology of China
Hefei, Anhui, 230026, P. R. China

###### Abstract

For $C^{1}$-smooth strongly monotone discrete-time dynamical systems, it is
shown that “convergence to linearly stable cycles” is a prevalent asymptotic
behavior in the measure-theoretic sense. The results are then applied to
classes of time-periodic parabolic equations and give new results on
prevalence of convergence to periodic solutions. In particular, for equations
with Neumann boundary conditions on convex domains, we show the prevalence of
the set of initial conditions corresponding to the solutions that converge to
spatially-homogeneous periodic solutions. While, for equations on radially
symmetric domains, we obtain the prevalence of the set of initial values
corresponding to solutions that are asymptotic to radially symmetric periodic
solutions.

##  1 Introduction

Prevalence is a frequently used notion, first introduced by Christensen [2]
and Hunt, Sauer and Yorke [17], that describes properties of interest occurs
for “almost surely” in an infinite-dimensional space from a probabilistic or
measure-theoretic perspective. It is a natural generalization to separable
Banach spaces of the notion of Lebesgue measure zero for Euclidean spaces (see
definition in Section 2). In particular, on $\mathbb{R}^{n}$, it is equivalent
to the notion of “Lebesgue almost everywhere”. Over decades since its
development, prevalence have undergone extensive investigations. We refer to
[17, 20, 18, 5] (and references therein) for more details.

In dynamical systems, prevalence is parallel to another more classical notion
called genericity, which formulates in the topological sense the properties of
interest occurs residually (i.e., on a countable intersection of open dense
subsets in a Baire space). Many examples in dynamical systems are known to be
both generic in the topological sense and prevalent in measure-theoretic
sense; that is, these two notions of typicality could coincide with each other
in many circumstances (see, e.g. [18, 20]). However, on the other hand, there
are also many cases for which prevalence differs from topological genericity.
As a matter of fact, in many cases properties that are generic in the
topological sense are not prevalent in the measure-theoretic sense, and vise-
versa (see, e.g. [18, 20]).

Monotone dynamical systems are abundant and important sources of topological
genericity and prevalence. The theory of monotone dynamical systems grew out
of the series of ground-breaking work of M. W. Hirsch [12, 13, 14, 15] and H.
Matano [21, 22]; largely focusing on ordinary differential equations and
parabolic partial differential equations. Since then, the theory and
applications have been extended to discrete-time dynamical systems, non-
autonomous systems, as well as random/stochastic systems. One may refer to [3,
8, 16, 19, 27, 30, 31, 32] for the earlier and recent developments.

The core to the huge success of developing the theory and applications of a
strongly monotone semiflow is the so called Hirsch’s generic convergence
theorem [15], concluding that the generic precompact orbit approaches the set
of equilibria (also referred as generic quasi-convergence). Later on, for
$C^{1}$-smooth strongly monotone semiflows, the improved generic convergence
was obtained by Poláčik [25, 26] and Smith and Thieme [33]. Very recently,
generic Poincaré-Bendixson Theorem was established by Feng, Wang and Wu [7, 8]
for smooth flows strongly monotone with respect to cone of rank-$2$.

For strongly monotone discrete-time systems (mappings), however, there is no
result analogous to Hirsch’s generic convergence theorem is available if they
are merely continuous (or even Lipschitz). Whether any characterization of
typical dynamics is possible in this case or not is still an open problem,
unless certain smoothness assumption is imposed. Poláčik and Tereščák [28]
first proved that the generic convergence to cycles occurs provided that the
mapping $F$ is of class $C^{1,\alpha}$ (i.e., $F$ is a $C^{1}$-map with a
locally $\alpha$-Hölder derivative $DF$, $\alpha\in(0,1]$). Here, a cycle
means a periodic orbit of $F$. For the lower regularity of $F$, Tereščák [36]
and Wang and Yao [38] utilized different approaches to obtain the generic
convergence to cycles for $C^{1}$-smooth strongly monotone discrete-time
systems.

A drawback of topological genericity is that open dense subsets can be
arbitrarily small in terms of measure. Hirsch [13] further showed that his
generic convergence theorem also holds in a measure-theoretical sense of
Gaussian measure. However, since it is based on the normal distributions in
statistics, Gaussian measure is not translation-invariant.

Hence, a natural interesting question arises whether the prevalent dynamics
analogue hold for strongly monotone systems. To the best of our knowledge,
Enciso, Hirsch and Smith [6] first tackled the problem and investigated the
prevalent behavior of a strongly monotone semiflow $\Phi_{t}$. Among others,
they [6, Theorem 1] proved that the set of points that converge to a linearly
stable equilibrium is prevalent. Here, an equilibrium is linearly stable if
the spectral radius of the Frechet derivative $D\Phi_{t}$ at the equilibrium
is no more than $1$ for all $t>0$.

In the present paper, we shall focus on prevalent behavior of the strongly
monotone discrete-time dynamical systems. To be more specific, we will first
show that, for $C^{1}$-smooth strongly monotone discrete-time system, the set
of points converge to a linearly stable cycle is prevalent (see Theorem A).
Here, a cycle is linearly stable if the spectral radius of the derivative
$DF^{p}$ along the cycle (of period $p$) is no more than 1 (see Section 2).

It deserves to point out that there are a few major differences of research
approaches between discrete and continuous time strongly monotone systems. For
instance, Limit Set Dichotomy (see, e.g. [6, Theorem 4, p.121] or [16, 31]),
which is one of the fundamental building blocks for continuous-time systems,
are no longer valid for discrete-time systems (see, e.g. [27, 4, 35]). Thus,
the situation is quite different with discrete-time systems as one has no a
priori information on the structure of limit sets of typical trajectories.
Consequently, novel techniques are needed to understand the prevalent
dynamics. Our approach is based on a critical insight for the inherent
structure of discrete-time strongly monotone systems, called _Dynamics
Alternatives_ (see Lemma 3.6), which was first discovered by Poláčik and
Tereščák [28] for $C^{1,\alpha}$-mappings and was recently improved by the two
of present authors [38] for $C^{1}$-mappings. Together with a useful tool of
upper (respectively, lower) $\omega$-unstable sets introduced by Takáč [35],
we utilize the $C^{1}$-Dynamics Alternatives to accomplish our approach.

Moreover, motivated by our approach, we show the prevalence of asymptotic
symmetry in discrete-time strongly monotone systems on which a compact
connected group $G$ acts. This gives the prevalent analogues to the works by
Mierczyński and Poláčik [24] and Takáč [34] on the topological genericity of
asymptotic symmetry in continuous and discrete strongly monotone dynamical
systems admitting group actions (see also [37]). For these results, an
interesting feature of is that no smoothness assumption was imposed on the
systems. Combining with Theorem A, we further obtain that, if in addition the
discrete-time system is $C^{1}$-smooth, then the set of points that are
asymptotic to $G$-symmetric cycles is prevalent (see Theorem B).

Our main results will be applied to obtain the dynamics of prevalent initial
values for nonlinear time $\tau$-periodic parabolic equations with proper
boundary value problems. By applying Theorem A, we first obtain the prevalence
of the set of initial points corresponding to the solutions that are
asymptotic to a linearly stable subharmonic periodic solution (i.e., solution
whose minimal period is $k\tau$ ($k\in\mathbb{N}$), a nontrivial multiple of
the period of the equations). In particular, for the periodic parabolic
equations with Neumann boundary conditions on convex domains, we proved the
prevalence of the set of initial conditions corresponding to the solutions
that converge to spatially homogeneous $\tau$-periodic solutions. We further
apply Theorem B to obtain asymptotic symmetry of prevalent solutions for time-
period parabolic equations on a radially symmetric domain. In particular, due
to the close interplay between spatial geometry and the temporal evolution of
solutions, we obtained the prevalence of the set of initial values
corresponding to solutions that are asymptotic to radially symmetric
$\tau$-periodic solutions.

This paper is organized as follow. In Section 2, we agree on some notations,
give relevant definitions and state our main results. We give the proofs of
our main theorems in Section 3. Section 4 is devoted to the study the
prevalent dynamical behavior of the solutions for the nonlinear time-periodic
parabolic equations.

##  2 Notations and Main Theorems

In this section, we will fix some notations, make some preliminaries and
formulate our main theorems at the end of this section.

Let $(\mathbb{B},\|\cdot\|)$ be a Banach space with norm $\|\cdot\|$. A closed
convex subset $K\subset\mathbb{B}$ is called a _closed convex cone_ if (i) if
$x\in K$ and $\alpha>0$, then $\alpha x\in K$; (ii) if $x,y\in K$, then
$x+y\in K$; (iii) $K\cap(-K)=\\{0\\}$. For $x,y\in\mathbb{B}$, we write $x\leq
y$ iff $y-x\in K$; $x<y$ iff $x\leq y$ and $x\neq y$; $x\ll y$ iff $y-x\in$
$\rm{Int}$$K$. $\mathbb{B}$ is called an _ordered Banach space_ if it is
endowed with an order relation, induced by a cone $K$. In particular,
$\mathbb{B}$ is called a _strongly ordered Banach space_ if the interior
$\rm{Int}$$K$ of $K$ is nonempty. For $a,b\in\mathbb{B}$ with $a\leq b$, we
denote the _closed order interval_ by $[a,b]=\\{x\in\mathbb{B}:a\leq x\leq
b\\}$ and the _open order interval_ by $[[a,b]]=\\{x\in\mathbb{B}:a\ll x\ll
b\\}$. In particular, we write $[a,\infty]]=\\{x\in\mathbb{B}:x\geq a\\}$ and
$[[\infty,b]=\\{x\in\mathbb{B}:x\leq b\\}$. Let $X\subset\mathbb{B}$, we
denote by $\widehat{X}$ the set $X$ with the order-topology generated from
open order interval $[[a,b]]$. A subset $X\subset\mathbb{B}$ is called _order-
convex_ in $\mathbb{B}$ if $[a,b]\subset X$ whenever $a,b\in X$ and $a<b$.

Let $X\subset\mathbb{B}$ and $F:X\rightarrow X$ is a continuous map. The map
$F$ is called _monotone_ if $x\leq y$ implies $Fx\leq Fy$, and is called
_strongly monotone_ if $Fx\ll Fy$ whenever $x<y$.

A _semi-orbit_ of $x\in X$ is $\mathcal{O}^{+}(x):=\\{F^{n}x\\}_{n\geq 0}$,
and the _$\omega$ -limit set_ of $x$ is $\omega(x):=\bigcap_{k\geq
0}\overline{\mathcal{O}^{+}(F^{k}x)}$. We say that the map $F$ is _$\omega$
-compact_ in a subset $B\subset X$ if $\mathcal{O}^{+}(x)$ has compact closure
in $X$ for each $x\in B$ and $\cup_{x\in B}\omega(x)$ has compact closure in
$X$. A point $x\in X$ is called _$p$ -periodic_ for some integer $p\geq 1$, if
$F^{p}(x)=x$ and $F^{l}x\neq x$ for $l=1$, $2$, $\cdots$, $p-1$. A set $B$ is
called a _cycle_ if $B=\mathcal{O}^{+}(x)$ for some periodic points $x$. A
cycle is _linearly stable_ if the spectral radius of the derivative $DF^{p}$
along the cycle (of period $p$) is no more than $1$. A set $B\subset X$ is
_unordered_ if it does not contain points $x,y$ such that $x<y$. Given any
$x\in X$, we define the _upper_ and _lower $\omega$-limit sets_ of $x$ in $X$
by

$\omega_{+}(x):=\bigcap_{\begin{subarray}{c}u\in X\\\ u\gg
x\end{subarray}}\overline{(\bigcup_{\begin{subarray}{c}y\in X\\\ u\geq
y>x\end{subarray}}\omega(y))}\quad
and\quad\omega_{-}(x):=\bigcap_{\begin{subarray}{c}u\in X\\\ u\ll
x\end{subarray}}\overline{(\bigcup_{\begin{subarray}{c}y\in X\\\ u\leq
y<x\end{subarray}}\omega(y))},$

respectively. Clearly, $\omega_{+}(x)$ (resp. $\omega_{-}$) is non-empty if
$F$ is $\omega$-compact in every closed order interval $[a,b]$ in $X$.
Hereafter, we write

$\mathcal{U}_{+}=\\{x\in X:\omega_{+}(x)\neq\omega(x)\\}\quad
and\quad\mathcal{U}_{-}=\\{x\in X:\omega_{-}(x)\neq\omega(x)\\}$

as the _upper $\omega$-unstable set_ and _lower $\omega$-unstable set_ in $X$,
respectively.

Throughout the paper, we will assume the following standing assumptions:

(H1)

$\mathbb{B}$ is a separable strongly ordered Banach space with a closed-convex
cone $K$; and $F:\mathbb{B}\rightarrow\mathbb{B}$ is a strongly monotone map.

(H2)

Let $X\subset\mathbb{B}$ be an $F$-invariant order-convex open subset; and $F$
is $\omega$-compact in every closed order interval $[a,b]$ in $X$.

(H3)

$F:\mathbb{B}\rightarrow\mathbb{B}$ is a $C^{1}$-map, such that for any
$x\in\mathbb{B}$ the derivative $DF(x)$ is a compact strongly positive
operator, i.e., $DF(x)v\gg 0$ whenever $v>0$.

In the following, we give the definition of _prevalence_ (see, e.g. [18, 17,
6]). A Borel set $W\subset\mathbb{B}$ is _shy_ if there exists a nonzero
compactly supported Borel measure $\mu$ on $\mathbb{B}$ such that $\mu(x+W)=0$
for every $x\in\mathbb{B}$. More generally, a set is called _shy_ if it is
contained in a shy Borel set. A set is _prevalent_ if its complement is shy.
Given $A\subset\mathbb{B}$, we say that a set $W$ is _prevalent in_ $A$ if
$A\setminus W$ is shy.

In general situation, shy sets have the following properties: (i) Every subset
of a shy set is shy; (ii) Every translation of a shy set is shy; (iii) No
nonempty open set is shy. (iv) Every countable union of shy sets is shy ([17,
18]). In finite-dimensional spaces, a set $M$ is shy if and only if it has
Lebesgue measure zero (see, e.g. [18, Proposition 2.5] or [17, Fact 6]).

Denote by $\mathfrak{C}_{P}$ the set of states $x\in X$ such that $\omega(x)$
is a linearly stable cycle, i.e.,

$\mathfrak{C}_{P}:=\\{x\in X:\omega(x)\ is\ a\ linearly\ stable\ cycle\\}.$

Our first main theorem is as follows:

$\textbf{Theorem A}.$ _Assume that _(H1)_ , _(H2)_ and _(H3)_ hold. Then
$\mathfrak{C}_{P}$ is prevalent in $X$._

In order to state our next main theorem, we need to introduce an additional
assumption. Let $G$ be a compact metrizable topological group. A mapping
$\Gamma:G\times X\rightarrow X$ is called a _group action_ of $G$ on $X$ if it
is jointly continuous and $g\mapsto\Gamma(g)\equiv\gamma(g,\cdot)$ is a group
homeomorphism of $G$ into Hom($X$), the group of all homeomorphisms of $X$
onto itself. We say that $\gamma$ is _increasing_ if, for each $g\in G$, the
mapping $\Gamma(g):X\rightarrow X$ is increasing, i.e., $x\leq y$ in $X$
implies $\Gamma(g)x\leq\Gamma(g)y$. We say that $\gamma$ _commutes_ with a map
$F$ if, for each $g\in G$, the mapping $\Gamma(g)$ commutes with $F$. For
briefly, We write $\gamma(g,x)\equiv g\cdot x$. A point $x\in X$ is called _G-
symmetric_ if $g\cdot x=x$ for all $g\in G$. A subset $S\subset X$ is called
_G-symmetric_ if all points in $S$ are $G$-symmetric.

(H4)

$G$ is a compact connected metrizable topological group acting on $X$ in such
a way that its action is increasing and commutes with $F$.

$\textbf{Theorem B}.$ _Assume that _(H1)_ , _(H2)_ and _(H4)_ hold. Then the
subset_

$S=\\{x\in X:\omega(x)\ is\ G\text{-}symmetric\\}$

_is prevalent in $X$. Moreover, if in addition _(H3)_ holds, then the set of
initial points corresponding to semiorbits that are asymptotic to
$G$-symmetric linearly stable cycles, is prevalent in $X$._

######  Remark 2.1.

Under assumptions (H1)-(H3), it has been proved in Tereščák [36] and Wang and
Yao [38] that $\mathfrak{C}_{P}$ contains an open and dense subset of $X$
(hence, $\mathfrak{C}_{P}$ is generic in $X$). Our Theorem A shows that
$\mathfrak{C}_{P}$ is also prevalent in $X$. This entails the fact that
_“convergence to linearly stable cycles”_ in $C^{1}$-smooth discrete-time
strongly monotone systems are both prevalent in measure-theoretic sense and
generic in the topological sense.

######  Remark 2.2.

In [23], Mierczyński smartly constructed an example of a strongly monotone
discrete-time system, in which the subset $\mathcal{M}\cup\mathcal{N}$ is open
and dense (hence, generic), but not prevalent in $X$ (see [23, p.1492]). Here
$\mathcal{M}$ is the set of points whose orbits are eventually monotone, and
$\mathcal{N}$ is the set of those convergent $\xi$ for which there is a simply
ordered arc $L\ni\xi$, $L\subset X\backslash\mathcal{M}$ (see more detail of
$\mathcal{M}$ and $\mathcal{N}$ in [23, p.1477]). We here point out that
$\mathcal{M}\cup\mathcal{N}$ is actually contained in $\mathfrak{C}_{P}$,
which means that $\mathcal{M}\cup\mathcal{N}$ is not large enough to guarantee
its prevalence in $X$.

######  Remark 2.3.

Theorem B provides the prevalent analogues to the works by Mierczyński and
Poláčik [24] and Takáč [34] on the topological genericity of symmetry in
continuous and discrete strongly monotone dynamical systems admitting group
actions.

##  3 Proof of the main Theorems

In this section, we will prove our main Theorems. For this purpose, we need
several useful propositions, which describe the structures of the upper (resp.
low) $\omega$-unstable sets $\mathcal{U}_{+}$ (resp. $\mathcal{U}_{-}$), as
well as a useful sufficient condition to guarantee a set to be shy in $X$.

First, we focus on the structure of $\mathcal{U}_{+}$ and $\mathcal{U}_{-}$.
Some additional notations are introduced. A pair $(A,B)$ of subsets $A,B$ of
$X$ is called an _order decomposition_ of $X$ if it satisfies: (i)
$A\neq\emptyset$ and $B\neq\emptyset$; (ii) $A$ and $B$ are closed; (iii) $A$
is lower closed (i.e., $[[-\infty,a]\subset A$ whenever $a\in A$) and $B$ is
upper closed (i.e., $[b,\infty]]\subset B$ whenever $b\in B$); (iv) $A\cup
B=X$; (v) Int$(A\cap B)=\emptyset$.

An order decomposition $(A,B)$ of $X$ is called _invariant_ if $F(A)\subset A$
and $F(B)\subset B$. The set $H=A\cap B$ (possibly empty) is called the
_boundary_ of the order decomposition $(A,B)$ of $X$. A _$d$ -hypersurface_ is
a nonempty subset $H$ of $X$ such that $H=A\cap B$ for some order
decomposition $(A,B)$ of $X$. The boundary $H$ of the order decomposition
$(A,B)$ of $X$ satisfies $H=\partial A=\partial B$, where “$\partial$” is the
boundary symbol in $X$, and $H$ is invariant whenever $(A,B)$ is invariant.
Clearly, a $d$-hypersurface $H$ never contains two strongly ordered points
($x\ll y$). In particular, every nonempty, unordered, invariant set $G\subset
X$ is contained in some invariant $d$-hypersurface $H\subset X$ (see, e.g.
[35, Proposition 1.2]).

A system $\Gamma$ of invariant order decompositions of $X$ is called an
_invariant order resolution_ of $X$ if $\Gamma$ satisfies (i) If
$(A_{1}B_{1})$ and $(A_{2},B_{2})\in\Gamma$, then either $A_{1}\subset A_{2}$
or $A_{2}\subset A_{1}$; (ii) If $\mathcal{O}^{+}(x)$ is unordered, then
$O^{+}(x)\subset H=A\cap B$ for some $(A,B)\in\Gamma$. The existence of the
invariant order resolution can be guaranteed by [35, Theorem 4.2].

Now we give the following structures of the $\mathcal{U}_{+}$ and
$\mathcal{U}_{-}$, which are critical for the proof of the prevalence.

######  Lemma 3.1.

Assume that _(H1)_ , _(H2)_ hold. Let $\Gamma$ be the invariant order
resolution of $X$. Then $\mathcal{U}_{+}$ is the union of at most countably
many sets $M_{i}$; each $M_{i}$ are relatively open in $H_{i}$, where
$H_{i}=A\cap B$ for some $(A,B)\in\Gamma$. A corresponding result holds for
$x\in\mathcal{U}_{-}$.

###### Proof.

See Takáč [35, Corollary 5.6]. ∎

######  Remark 3.2.

If, in addition, $X$ is order-open ($X$ is open in the order-topology of
$\widehat{\mathbb{B}}$), then each $M_{i}$ is a Lipschitz manifold of
codimension one in $\widehat{\mathbb{B}}$ (see the detail in Takáč [35,
Corollary 5.6]).

######  Proposition 3.3.

Assume that _(H1)_ , _(H2)_ hold. Then

_(i)._ If $x\in\mathcal{U}_{+}$, then there exists some $a_{x}\gg x$ such that
$\omega(y)=\omega_{+}(x)$ for any $y\in[[x,a_{x}]]$; and hence,
$[[x,a_{x}]]\cap\mathcal{U}_{+}=\emptyset$.

_(ii)._ Let $x,y\in\mathcal{U}_{+}$ and $x\ll y$. Then the two corresponding
open sets obtained in _(i)_ are mutually disjoint, i.e.,
$[[x,a_{x}]]\cap[[y,a_{y}]]=\emptyset$.

A corresponding result holds for $x,y\in\mathcal{U}_{-}$.

###### Proof.

(i). See Takáč [35, Proposition 3.3]. (ii). Let $a_{x}$, $a_{y}$ be obtained
from (i) for $x$ and $y$, respectively. Suppose that
$b\in[[x,a_{x}]]\cap[[y,a_{y}]]\neq\emptyset$. Then $x\ll y\ll b\ll a_{x}$;
hence, $y\in[[x,a_{x}]]$, which implies that
$y\in[[x,a_{x}]]\cap\mathcal{U}_{+}\neq\emptyset$, contradicting (i) of
Proposition 3.3. ∎

######  Proposition 3.4.

Assume that _(H1)_ , _(H2)_ hold. Then both $\mathcal{U}_{+}$ and
$\mathcal{U}_{-}$ are Borel sets.

###### Proof.

We only prove that $\mathcal{U}_{+}$ is a Borel set, since the proof of
$\mathcal{U}_{-}$ is analogous. Let $\Gamma$ be the invariant order resolution
of $X$. Then, by Lemma 3.1, $\mathcal{U}_{+}$ is the union of at most
countably many subsets $M_{i}$. Each $M_{i}$ is relatively open with respect
to $H_{i}$, with $H_{i}=A_{i}\cap B_{i}$ for certain $(A_{i},B_{i})\in\Gamma$.
In other words,

$\mathcal{U}_{+}=\bigcup_{i\in\Lambda}M_{i}=\bigcup_{i\in\Lambda}(H_{i}\cap
V_{i})=\bigcup_{i\in\Lambda}((A_{i}\cap B_{i})\cap V_{i}),$

where $\Lambda$ is a countable index set; and $V_{i}$ ($i\in\Lambda$) are open
subsets in $X$. Noticing that $A_{i}$ and $B_{i}$ are closed due to the
definition of order decomposition, we obtain that $\mathcal{U}_{+}$ is Borel.
∎

Next, we introduce a sufficient condition which guarantees a Borel subset of
$X$ to be shy.

######  Lemma 3.5.

Let $W\subset X$ be a Borel subset and assume that there exists a vector $v\gg
0$ such that $L\cap W$ is countable for every straight line $L$ parallel to
$v$. Then $W$ must be shy in $X$.

###### Proof.

See Enciso, Hirsch and Smith [6, Lemma 1]. ∎

Before we give the proof, we mention the following critical insight for the
inherent structure of the system.

######  Lemma 3.6 (Dynamics alternatives).

Assume that _(H1)_ -_(H3)_ hold. For any $x\in X$, one of the following
situations must occurs:
_(Alta)._ $x\in\mathfrak{C}_{P}$;
_(Altb)._ there exists some $\delta>0$ such that for any $y\in X$ satisfying
$y<x$ or $y>x$, one has

$\limsup_{n\rightarrow\infty}\|F^{n}(y)-F^{n}(x)\|\geq\delta.$

###### Proof.

See Wang and Yao [38, Theorem 2.1] (or Poláčik and Tereščák [28, Theorem 4.1]
for $C^{1,\alpha}$-version). ∎

Now we are ready to prove the our main Theorems.

$\textbf{Proof of Theorem A}.$ Let $M=X\backslash\mathfrak{C}_{P}$ be the
complement of $\mathfrak{C}_{P}$ in $X$. We first show that
$M\subset\mathcal{U}\triangleq\mathcal{U}_{+}\cap\mathcal{U}_{-}$. For this
purpose, we utilize Dynamics Alternatives in Lemma 3.6, which state that, for
each $x\in X$, either (Alta) $x\in\mathfrak{C}_{P}$; or (Altb) there exists
some $\delta>0$ such that for any $y\in X$ satisfying $y>x$ or $y<x$, one has
$\limsup_{n\rightarrow\infty}\|F^{n}y-F^{n}x\|\geq\delta$.

By virtue of Lemma 3.6, any point $x\in M$ can only satisfy (Altb). Then,
together with the definition of upper and lower $\omega$-limit sets, we obtain
that $\omega(x)\neq\omega_{+}(x)$ and $\omega(x)\neq\omega_{-}(x)$, which
entails that $x\in\mathcal{U}_{+}\cap\mathcal{U}_{-}\triangleq\mathcal{U}$.

Next, we will prove that both $\mathcal{U}_{+}$ and $\mathcal{U}_{-}$ are shy.
Again, we just show that $\mathcal{U}_{+}$ is shy. The case for
$\mathcal{U}_{-}$ is similar. By virtue of Lemma 3.5, it suffices to show that
$\mathcal{U}_{+}\cap L_{v}$ is countable, where $L_{v}$ is any straight line
parallel to any fixed vector $v\gg 0$. To this end, fix any positive vector
$v\gg 0$. Recalling that $\mathcal{U}_{+}$ is Borel (by Proposition 3.4), we
can define a set-function $R$ from $\mathcal{U}_{+}\cap L_{v}$ to
$\mathcal{B}(L_{v})$ as

$R:\mathcal{U}_{+}\cap L_{v}\rightarrow\mathcal{B}(L_{v});\ x\mapsto
f(x)\triangleq L_{v}\cap[[x,a_{x}]],$

where $\mathcal{B}(L_{v})$ are the set of all Borel subsets of $L_{v}$, and
$[[x,a_{x}]]$ is from Proposition 3.3(i). By Proposition 3.3(ii), it is clear
that $R$ is injective; and moreover, for each $x\in\mathcal{U}_{+}\cap L_{v}$,
$[[x,a_{x}]]\cap L_{v}$ is mutually disjoint open in $L_{v}$. Consequently,
the image of $R$ is an at most countable subset in $\mathcal{B}(L_{v})$. Since
$R$ is injective, it follows that $\mathcal{U}_{+}\cap L_{v}$ is at most
countable. Thus, by Lemma 3.5, we have proved that $\mathcal{U}_{+}$ is shy.

Finally, noticing that
$M\subset\mathcal{U}\triangleq\mathcal{U}_{+}\cap\mathcal{U}_{-}$, we obtain
that $M$ is shy, which implies that $\mathfrak{C}_{P}$ is prevalent in $X$. We
have completed the proof. $\square$

$\textbf{Proof of Theorem B}.$ Let $M=X\backslash S$. Then one has
$M\subset\mathcal{U}$ (see Takáč[34, Theorem 0.3]). By the same arguments in
the proof of Theorem A, we get that $\mathcal{U}_{+}$ and $\mathcal{U}_{-}$
are shy sets; and hence, $M$ is shy in $X$, which entails that $S$ is
prevalent in $X$.

If in addition (H3) holds, then $\mathfrak{C}_{P}$ is prevalent in $X$ by
Theorem A. Recall that $S$ is prevalent in $X$. Then, we have
$\mathfrak{C}_{P}\cap S$ is prevalent in $X$. $\square$

##  4 Application to parabolic equations

In this section, we will apply our main theorems to investigate the prevalent
dynamics of parabolic equations with proper boundary value problems.

Example 1 (Prevalent dynamics of parabolic equations on bounded domains)
Consider the following initial-boundary value problem for parabolic equations

$\displaystyle\frac{\partial u}{\partial t}-\triangle u$ $\displaystyle=$
$\displaystyle f(t,x,u,\nabla u),\quad\quad\quad\ x\in\Omega,\ t>0,$
$\displaystyle\mathcal{B}u$ $\displaystyle=$ $\displaystyle
0,\quad\quad\quad\quad\quad\quad\quad\quad\ x\in\partial\Omega,\ t>0,$ (4.1)
$\displaystyle u(0,x)$ $\displaystyle=$ $\displaystyle u_{0}(x).$

Here $\Omega\subset\mathbb{R}^{m}$ is a bounded domain with boundary
$\partial\Omega$ of class $C^{2+\theta}$ for some $\theta\in(0,1)$. Of course,
$\Delta$ is the Laplacian. The nonlinearity $f:(t,x,u,\xi)\mapsto
f(t,x,u,\xi)$ are assumed to satisfy:

(N1) $f$ is continuous and locally Hölder continuous in $(t,x)$ uniformly for
$(u,\xi)$ in bounded subset of $\mathbb{R}\times\mathbb{R}^{m}$, and
$\frac{\partial f}{\partial u}$ and $\frac{\partial
f}{\partial\xi_{i}}(i=1,\ldots,m)$ exist and are continuous with respect to
$(u,\xi)$.

(N2) $f$ is periodic in $t$ of given period $\tau>0$.

We consider a time-independent regular linear boundary operator $\mathcal{B}$
on $\partial\Omega$ of _Dirichlet_ ($\mathcal{B}u=u$) or _Neumann_
($\mathcal{B}u=\frac{\partial u}{\partial v}$) type. Here $v$ is the unit
outward normal vector field on $\partial\Omega$.

We also assume that

(N3) There exists an $\epsilon>0$ and a continuous function
$c:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that

$|f(t,x,u,\xi)|\leq c(p)(1+|\xi|^{2-\epsilon})\quad(p\geq 0,\
(t,x,u,\xi)\in[0,\tau]\times\bar{\Omega}\times[-p,p]\times\mathbb{R}^{m}).$

(N4) There is some $\kappa>0$ such that
$uf(t,x,u,0)<0\quad((t,x)\in[0,\tau]\times\bar{\Omega},|u|\geq\kappa).$

Let $A$ be the realization of $\triangle$ under boundary condition, that is,
$A$ is the operator $u\mapsto\triangle u$ with domain

$D(A)=X^{1}\triangleq\\{u\in W^{2,p}(\Omega):\mathcal{B}u=0\\}.$

Let $X^{\alpha}$, $0\leq\alpha<1$, be the fractional power spaces associated
with $A$. We choose $\frac{1}{2}+\frac{m}{2p}<\alpha<1$ such that
$X^{\alpha}\subset C^{1+\gamma}(\bar{\Omega},\mathbb{R})$ with continuous
inclusion for $\gamma\in[0,2\alpha-\frac{m}{p}-1)$. Moreover,
$V:=(X^{\alpha},\|\cdot\|_{\alpha})$ is a strongly ordered Banach space with
the closed-convex cone $X^{\alpha}_{+}$ consisting of all nonnegative
functions in $X^{\alpha}$. For every $u_{0}\in V$, (4) admits a (locally)
unique regular solution $u(t,x,u_{0})$ in $V$. Set $X\triangleq\\{v\in
X^{\alpha}:-\kappa<v(x)<\kappa\ for\ all\ x\in\Omega\\}$ for $\kappa$ in (N4).
By virtue of (N4) and the standard a priori estimates ([9]), the solution $u$
is bounded in $V$, and hence becomes a globally defined classical solution on
$X$.

Let $F$ be the $\tau$-periodic Poincaré map of (4) as $F:V\rightarrow
V,u(t)\mapsto u(t+\tau)$. Due to Strong Maximum Principle, $F$ generates a
discrete-time strongly monotone system on $V$. One further knows that $F$ is
of $C^{1}$ (see, e.g. [9]); and moreover, $F$ is $\omega$-compact in every
closed order interval in $X$ since $F$ is a compact map with $F(X)\subseteq X$
(see, e.g. [27, 11]).

Thus, the discrete-time system generated by $F$ satisfies the standing
assumptions (H1)-(H3) in Section 2. Based on our Theorem A, we have the
following theorem.

######  Theorem 4.1.

Assume that _(N1)-(N4)_ hold for system _( 4)_. Then the set of initial
condition $u_{0}\in X$ corresponding to solution $u(t,x)$ converges to a
linearly stable $k\tau$-periodic solution _(_ for some integer $k>0$_)_ , is
prevalent in $X$.

Example 2 (Prevalent dynamics of parabolic equations on convex domains)
Consider the periodic-parabolic Neumann problem on a smooth convex bounded
domain $\Omega\subset\mathbb{R}^{m}$:

$\displaystyle\frac{\partial u}{\partial t}-\triangle u$ $\displaystyle=$
$\displaystyle f(t,u,\nabla u),\quad\quad\ \rm{in}\ \Omega\times\mathbb{R},$
(4.2) $\displaystyle\frac{\partial u}{\partial n}$ $\displaystyle=$
$\displaystyle 0,\quad\quad\quad\quad\quad\quad\ \rm{on}\
\partial\Omega\times\mathbb{R},$

where $f:(t,u,\xi)\mapsto f(t,u,\xi)$ is independent of $x$ and satisfies
assumptions (N1)-(N4).

Again, set $V:=(X^{\alpha},\|\cdot\|_{\alpha})$ and $X\triangleq\\{u\in
X^{\alpha}:-\kappa<u(x)<\kappa\ for\ all\ x\in\Omega\\}$. Let $F$ be the
$\tau$-periodic Poincaré map as $F:V\rightarrow V,u(t)\mapsto u(t+\tau)$. So
$F$ also generates a $C^{1}$-smooth discrete-time dynamical system on $V$
satisfying (H1)-(H3). We call a solution $u(t,x)$ is _spatially homogeneous_
if $u(t,x)$ is independent of spatial variable $x$.

######  Theorem 4.2.

Assume that _(N1)-(N4)_ hold for system _( 4.2)_. Then the set of initial
condition $u_{0}\in X$, whose solution $u(t,x)$ converges to a spatially
homogeneous $\tau$-periodic solution, is prevalent in $X$.

###### Proof.

Since $F$ satisfies (H1)-(H3) in Section 2, it follow from Theorem 4.1 that
the set of initial conditions whose solutions are asymptotic to linearly
stable periodic solutions, is prevalent in $X$. Recall that Hess [10] has
shown that $u$ is spatially homogeneous if it is a linearly stable periodic
solution of (4.2). And Poláčik and Tereščák [29] further showed that such $u$
is a fixed point of $F$, which means $u$ is $\tau$-periodic. Thus, we obtain
that the set of initial condition $u_{0}$ of (4.2), whose solution $u(x,t)$
converges toward a spatially homogeneous $\tau$-periodic solution, is
prevalent in $X$. ∎

Example 3 (Prevalent dynamics of parabolic equations on $SO(N)$-invariant
domains) Let $G$ be a compact connected subgroup of the special orthogonal
group $SO(N)$. The domain $\Omega$ is said to be $G$-invariant if $a\cdot
x\in\Omega$ for all $x\in\Omega,a\in G$. Now we consider the equation (4) with
the open bounded $G$-invariant domain $\Omega\subset\mathbb{R}^{m}$ with a
$C^{2+\mu}$-boundary $\partial\Omega$, for some $\mu\in(0,1)$,

$\displaystyle\frac{\partial u}{\partial t}-\triangle u$ $\displaystyle=$
$\displaystyle f(t,x,u,\nabla u),\quad\quad\quad\ \ x\in\Omega,\ t>0,$
$\displaystyle\mathcal{B}u$ $\displaystyle=$ $\displaystyle
0,\quad\quad\quad\quad\quad\quad\quad\quad\ x\in\partial\Omega,\ t>0,$ (4.3)
$\displaystyle u(0,x)$ $\displaystyle=$ $\displaystyle u_{0}(x),$

where $\mathcal{B}u=u$ or $\mathcal{B}u=\frac{\partial u}{\partial v}$.

We assume the nonlinearity $f$ satisfies (N1)-(N4). Moreover, we assume that
$f$ satisfies:

(N5) $f(t,a\cdot x,u,a\cdot\xi)=f(t,x,u,\xi)$ for all $a\in G$ and
$(t,x,u,\xi)\in\mathbb{R}_{+}^{1}\times\bar{\Omega}\times\mathbb{R}\times\mathbb{R}^{m}$.

Take $X=C(\bar{\Omega},\mathbb{R})$. The action of $G$ on $X$ is defined by
$a\cdot u=u\circ a$, i.e., $\gamma(a,u)(x)=u(a\cdot x)$ for all $a\in G$,
$u\in X$ and $x\in\bar{\Omega}$. Let $F$ be the $\tau$-periodic Poincaré map
as $F:X\rightarrow X,u(t)\mapsto u(t+\tau)$, which is $C^{1}$. Then $\gamma$
commutes with $F$ (see Takáč [34]). We call a solution $u(t,x)$ is _$G$
-symmetric_ if $u(t,a\cdot x)=u(t,x)$ for any $a\in G$ and $t\geq 0$.
According to the Theorem B, the prevalent solutions of equation (4) that
converge to symmetric functions, i.e., we can obtain that

######  Theorem 4.3.

Assume that _(N1)-(N5)_ hold for system _( 4)_. Then the set of initial
condition $u_{0}\in X$ corresponding to solution $u(t,x)$ converges to a
$G$-symmetric linearly stable $k\tau$-periodic solution $p(x,t)$ _(_ for some
integer $k>0$_)_ , is also prevalent in $X$.

Example 4 (Prevalent dynamics of reaction-diffusion equations on a ball)
Consider the prevalent dynamics of the following Dirichlet initial-boundary
value problem on an $N$-dimensional ball $B=\\{x\in\mathbb{R}^{N}:\|x\|<1\\}$:

$\displaystyle\frac{\partial u}{\partial t}-\triangle u$ $\displaystyle=$
$\displaystyle f(t,u),\quad\quad t>0,\ x\in B,$ $\displaystyle u(x,t)$
$\displaystyle=$ $\displaystyle 0,\quad\quad\quad\quad\quad\quad\quad t>0,\
x\in\partial B,$ (4.4) $\displaystyle u(0,x)$ $\displaystyle=$ $\displaystyle
u_{0}(x),\quad\quad\quad\quad\quad x\in\bar{B}.$

Again, we assume that (N1)-(N4) hold. Let $V$, $X$ and $F$ be defined as in
Example 4. Take the group $G=SO(N)$. Then, we have the following theorem.

######  Theorem 4.4.

There exists a prevalent set $S\subset X$ such that for any solution $u(x,t)$
of equation _( 4)_ with initial value $u_{0}\in S$, there exists a linearly
stable $\tau$-periodic solution $p(x,t)$ such that

_(i)_ $p(\cdot,t)$ is radially symmetric, i.e., $p(x,t)=p(|x|,t)$;

_(ii)_ the solution $\|u(\cdot,t)-p(\cdot,t)\|_{X}\rightarrow 0$ as
$t\rightarrow\infty$.

###### Proof.

Let $V_{\rm{rad}}$ be the space of all radially symmetric functions in $V$ and
let $X_{\rm{rad}}=X\cap V_{\rm{rad}}$. We indentify a function $u\in
X_{\rm{rad}}$ with its radial version $U\in X^{\alpha}\subset C^{1}[0,1]$ with

$U(\|x\|,t)=u(x,t),\quad U^{\prime}(0)=U(1)=0.$

Define by $F_{\rm{rad}}$ be the Poincaré-map associated with the following 1-D
equation on $X_{\rm{rad}}$:

$\displaystyle U_{t}$ $\displaystyle=$ $\displaystyle
U_{rr}+\frac{N-1}{r}U_{r}+F(t,U),\quad\quad\quad t>0,\ 0<r<1,$ $\displaystyle
U_{r}(0,t)$ $\displaystyle=$ $\displaystyle
U(1,t)=0,\quad\quad\quad\quad\quad\quad\quad\quad\quad t>0,$ (4.5)
$\displaystyle U(r,0)$ $\displaystyle=$ $\displaystyle
U_{0}(r),\quad\quad\quad\quad\quad\quad\ \quad\quad\quad\quad\quad 0\leq r\leq
1.$

where $r=\|x\|$ and $F(t,U)=f(t,u)$.

By virtue of Theorem B, one can find a prevalent set $S\subset X$ such that,
for any $u(x,t)$ with initial value $u_{0}\in S$, $u$ converges in $X$ to a
linearly stable periodic solution $p(r,t)$ of the (4) in $X_{\rm{rad}}$. It
follows from Chen and Poláčik [1, Theorem 3.5] that any chain recurrent set of
$F_{\rm{rad}}$ consists of fixed points of $F_{\rm{rad}}$. Then $p\in
X_{\rm{rad}}$ and $p(\cdot,t)=p(\cdot,t+\tau)$. Thus, we obtain that
$\|u(\cdot,t)-p(\cdot,t)\|_{X}\rightarrow 0$ as $t\rightarrow\infty$, which
completed the proof. ∎

######  Remark 4.5.

Chen and Poláčik [1] have proved the asymptotic $\tau$-periodicity of
_positive_ solutions. Our Theorem 4.4 showed that prevalence of the asymptotic
$\tau$-periodicity for solutions of (4).

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