Chủ Nhật, 26 tháng 8, 2018

Waching daily Aug 26 2018

In the last episode of this video series, I spent the vast majority of my time praising

season 2, and, fortunately, my praise of it continues with episode 8, "Titan Rising."

However, I still think that there are certain ways that this episode could have improved,

although the main source of those improvements would come from slight alterations to previous

episodes.

Basically, episode 8 focuses on Terra's and Raven's relationship, and on how Raven goes

from not trusting Terra upon her return to trusting her.

While it is reasonable for Raven to distrust Terra, I think that her position could have

been strengthened if, at some point in one of the previous episodes, there had been an

indication that she was frustrated with Terra or worried about what they would do if she

returned.

Having little moments like that would have helped the individual, episodic plots of the

story tie in with the season's over-arcing plot, and would create a more natural sense

of pacing.

This isn't to say that any of the previous episodes needed to spend any large amount

of time focusing on this - in fact, that would likely be a bad idea, since that could take

away from the focus of those episodes.

I just feel that these side stories could be integrated into the plot in subtle ways

that would lead to a greater sense of narrative cohesion.

The reason that the writers didn't do this, however, is likely because of how the show

was presented to its audience: If a child was watching individual episodes here and

there when they could catch them on TV, having references to a larger plot could be off-putting

or confusing.

But, if that was the reason why the side stories and main plot are so thoroughly separated

and I'm really assuming here, I don't think it quite makes sense: After all, viewers would

be likely to catch separate episodes of the main plot anyway, which would be just as potentially

confusing, if not more-so.

Still, it is possible that my theory for why the episodes are structured this way is completely

incorrect and that there is some other reason for this--but, regardless of the reasons,

I would have preferred if some reference was made to Terra in those episodes or even just

in one of those episodes.

The reasons aren't really what matter to me here.

The end product is.

But, beyond that, this is a strong episode.

It was a good choice to have someone in the Titans distrust Terra, since that makes her

betrayal even more upsetting later on, and it also forces Terra into a pretty interesting

situation.

See, she only left because she mistakenly thought Beast Boy had betrayed her trust and

because of her unruly powers.

And yet now, when her powers are under her control and when the Titans trust her, when

her main problem has seemingly been fixed, she uses her control to manipulate the Titans.

But this has gone beyond being an issue that pertains to her powers and spreads outward

from there: This is a personality issue.

Even now--even when she has gained the ability to carve her own path and consciously do the

right thing, even when the thing that has caused her self-loathing is no longer there,

she continues to act in a way that is in line with her previous self.

Because the only reason she would have to do something like this would be that she believes

that the Titans truly betrayed her when, really, the fact that they accept her now makes it

abundantly clear that they never did.

In fact, it seems as though she only went through with this plan because she believed

that it would never work--because she believed that the Teen Titans would never trust her.

As such, at the end of the episode, when she says, "I don't believe it.

They actually trust me," these are simultaneously the words that confirm that she has gone even

further down her path of self destruction and the words that confirm that she can in

fact escape from this cycle.

That option has been provided to her.

This is what makes Terra compelling to me.

When she has finally received what she's always wanted, it doesn't matter anymore because,

in the end, she needs to trust herself; nobody else can do the right thing for her; they

can only put her in situations that facilitates her doing the right thing.

No matter how her outer world may look, her self-loathing will lead to her hurting others

and destroying her relationships.

She can lift her new home out of the rubble, but if she lifts it out for the wrong reasons

she will never be able to rest easy there.

Unfortunately, that lack of narrative cohesion I mentioned that is caused by the episodic

episodes having so little to do with the over-arcing plot before becomes more problematic the following

episode and episode eleven, both of which take place after Terra-centric episodes but

ignore her existence almost entirely.

This is especially strange in episode nine, where there's no real reason to ignore her.

Sure, she shows up very briefly, but beyond that she's just . . . gone.

And why wouldn't she be playing cards with the other Titans?

Having her there would be a solid way to build up their relationship more before bringing

it crashing down.

Sure, we've seen her fight to become a Titan, but we also need to see her being a Titan,

and the more settings we see that in the better.

So, sure, we need to see her fighting bad guys with them--but we should also see her

just relaxing with them, having fun.

I really can't think of any negatives to not including her here, or at least mentioning

why she isn't involved.

I'm not sure why this decision was made.

Other than that, however, this is one of those episodes I think I would have enjoyed a lot

when I was younger but that I just don't find that interesting now.

The main theme, as explored in this tournament, is that winning isn't everything--which, to

be fair, is a good lesson for children to learn--but at this point that feels like a

very simple lesson that I've already learned.

I do like how the tournament itself is constructed though, with some familiar faces returning

and the natural introduction of some new people.

This is a really good way to introduce them and having previously established boys appear

helps bring the story together and makes this episode feel a little bit less separate from

the rest of the series in a way that actually adds to the episode itself.

I do like how the tournament itself is constructed though, with some familiar faces returning

and the natural introduction of some new people.

This is a really good way to introduce them and having previously established boys appear

helps bring the story together and makes this episode feel a little bit less separate from

the rest of the series in a way that actually adds to the episode itself.

Another good aspect of this episode is that the directions in the fight scenes is, overall,

very well done and some of the fights are really well done.

Honestly though, it's hard to dive too deep into this episode since it's just a simple

but fast tournament that manages to effectively convey its simple message.

It does end in a way I'm not fond of though, and episode eleven ends similarly.

They both leave us with an open end, where the plot isn't resolved and we just have to

assume that it gets resolved before the main plot of the season picks up in the next episode.

Part of what frustrates me about this is that, oftentimes, I'm actually more interested in

seeing how they characters will get themselves out of the situation they find themselves

in in the final moments of the episode.

This is particularly pertinant at the end of this episode because, well, it actually

may have been a better idea to have this tournament be the tournament of heroines.

That way, we could have seen more of Terra and, if she were to be the last one standing

in the tournament, we would also get even more of a sense of how strong she is.

Regardless of the specifics though, seeing more of her at this point would have been

a good thing, and I don't think seeing this tournament with the female characters instead

of the male characters would be any less entertaining.

Fortunately, I consider episode ten a large improvement over episode nine, and it adds

quite a lot to Terra's character and fleshes out her struggle.

In particular, I enjoy these moments where the show slows down and we see Terra struggling

with her decision to betray the Titans, where the show lingers just a bit too long on Terra

and, without anything explicitly being said, communicates to the viewer that she isn't

entirely sure if she wants to do what she's doing now.

And it makes sense that she questions herself here, as this episode puts a lot of emphasis

on Beast Boy's trust in her and on her trust in Beast Boy.

All of their scenes have a fantastic tension to them, with the show periodically cutting

back to Titans tower being attacked.

This reminds us that, although this is the closest we've ever seen Terra and Beast Boy,

every moment it's getting harder for her to turn back.

No matter what happens on this night, she can't take back what she's already done, what's

happening right at that moment, despite the fact that it's natural for the audience to

hope that she'll change her mind here and choose to be good.

And she's also reminded that she already had what she wanted most of all--a home full of

people who accept her.

But even as the strongest reminders of that are slapping her right in the face, the reminder

that it may be too late is slapping the audience right in the face.

Part of what's great about this is how much it adds to Beast Boy's character, too.

His fight with Slade on the Ferris wheel--in fact, the whole fight between Slade, Terra,

and him in the amusement part--is easily the most emotional one in this whole series.

The fact that Beast Boy fights so hard for her is just crushing and it adds so much to

his character.

Sure, he may be an overall happy-go-lucky guy, but here we see his care for others can

turn to intense rage and sadness.

This episode also has some neat imagery, mainly in the house of mirrors, where Beast Boy is

surrounded by Slade, just as the truth becomes inescapable and there's no way to ignore Terra's

strange actions and potential betrayal.

Similarly, Terra is forced to confront her fractured self, to look in the mirror and

see what she's become, how all of her fractured identities have come together to allow this

moment to be possible, where she loses everything she ever wanted.

I could go on but I think if you're at all invested in Terra's struggle this episode

is a great one and very well executed.

Unfortunately, I really don't like season 2's eleventh episode.

In it, Robin makes a mistake and breaks his arm and is forced to let his friends take

care of things while he heals; however, things get stranger in a different version of himself

from a different dimension appears and starts accidentally causing trouble.

While my usual problem with these sorts of episodes breaking up the flow of the story

still stands, my main problem with it is how it handles Robin's character and his progression.

Up until now, Robin's major flaw has been his constant reliance on himself and his inability

to just let his friends get the job done.

Yet here, when Robin finally lets his friends get things done for him, he's told that he

needs to keep trying no matter what, and how it's a good thing for him to keep trying.

Now, I think the idea here is that he's gone too far in the other direction, that he's

gone from being unable to trust his friends to giving up on himself.

So in some ways this is making sense so far.

Meanwhile, this other version of Robin also gets involved, causing trouble because now

he won't stop and let other people get the job done.

So he's a very extreme version of the problem Robin had before, but with a little bit of

a difference in that he's just trying to help and he's not shutting anyone else out.

The strange part of this is, now that Robin's finally chosen to trust his friends, they

. . . can't do this without him?

In fact, it's so bad that they completely fail and he has to essentially fix it all

on his own.

This seems to strengthen his original position and uncertainty with trusting his friends.

This episode makes it seem like the most important thing is that Robin always tries, no matter

how broken or unprepared for battle he is, because he's the only one who can save the

day in the end.

I think it's pretty clear how this goes against the previously established themes and lessons

that surround Robin.

But the show seems to also agree with this, considering how Robin gets transported to

this empty white space and now needs to escape at the end of the episode, and the reason

for this is because he just told this doppelganger of his to give it a shot when the doppelganger

was trying to fix his arm.

So the lesson then is that people shouldn't always just try no matter how prepared they

are.

I know I'm digging into this comedic episode pretty deep and that, overall, it's just meant

to be a funny and fun episode to break up the more serious parts of a story--but I think

that fun episodes like this can still remain thematically consistent with the rest of the

series and build on those established ideas.

There's no reason that development and fun need to be separated here.

But this episode isn't all bad--the chase scenes are interesting enough and I like this

colouring book aesthetic for example, but, ultimately, I don't think these are the most

important parts of the episode, so overall my feelings about this one are still pretty

negative about this one and I think the show would be better without this episode.

This is even more-so the case because the next episode isn't going to show more of the

aftermath of Terra leaving the group.

What I mean is that we never see the Titans thinking about Terra when she's not there

and this could be a good chance to spend more time doing that.

Instead, we got this.

With how this episode plays out, it seems like the Titans only really think of Terra

when she's there.

Even though I'm sure that isn't meant to be the case.

I'm sure that they're supposed to be upset that she's left them and that they think about

that.

I'd be shocked if they didn't.

Next up is the first part of Aftershock which, while it is by no means a bad episode, is

pretty messy and the weakest part of Terra's journey so far.

What I mean is that this episode just doesn't make that much sense.

After all, how do the Titans survive each of their encounters with Terra?

While some of their fights ended in ways that could have resulted in the Titans living,

other fights with her ended pretty conclusively with their deaths.

And yet they just somehow got out of it.

It's a real shame that this is never explained in any way because , when they return at the

end of the episode, it's very perplexing how they even got there.

However, that isn't to say that there's nothing good about the episode--seeing the Titans

struggle to fight Terra brings home just how close they were, having these previous bad

guys return to fight them connects this season with the first season and is a way better

decision than bringing in new baddies, Terra's negative rhetoric continues to expose her

fundamental flaws and shows how her negative self-image has caused her to view everyone

around her in an unfair and harmful light.

Still, the logical problems stuck out to me in a way that was impossible to ignore and

that made this episode far more rocky than any of the other Terra-centric episode.

It's here that there's a very real sense that the writers had to rush through this section

of her story, for some reason or another.

Fortunately, the second part of Aftershock is much stronger.

The way its structured is extremely good.

I especially like this beginning section where, as Terra goes through this decimated city,

she thinks about her time with the Titans, about the good times she's had and that she

can never go back to.

She's surrounded by a shell of the life she could have had, and now the colours have been

drained from the world and even her friends are mere shadows of what they could have been

and what they were.

The music in this episode is also particularly great.

Every piece flows into the next naturally and it makes the pacing of this episode feel

near perfect.

The increased connectivity between scenes it brings makes the whole episode feel like

one extended scene even though it's obviously not that.

There's nothing fractured here.

It's very smooth storytelling.

Then there's one of my favourite parts of Terra's struggle, where escaping Slade becomes

nearly impossible, where she's taken things so far that there's no way that she can just

turn back based on her own power.

Slade has literally taken over her body; as he says, he's "become a part of her" and she

"no longer has control in the matter."

While this works on a literal level very well, it also work on a metaphorical level: the

longer someone stays with an abuser, the harder it is to escape from them; the more they allow

that horrible person's idelogy to affect them, the more likely that they will take in that

idelogy in a way that is part of their self, in a way that canot be simply escaped from

through sheer willpower or thinking it through.

She gave up her free will to control her powers and now she's even lost control of that.

And as Beast Boy is sure to remind her, she chose all of this; she can try to blame someone

else, try to say that they're making her do this, but she ultimately let it happen.

The show does a really good job of placing some blame on Terra here, while still managing

to paint her decidedly as Slade's victim.

It's a pretty nuanced take on how someone who is abused can become an abuser, on how

self-hatred can become a hatred of others, and not once did I feel that Terra as a character

was being vilified or blamed in a way that was unfair to her.

There's also this moment where Terra manages to gain some form of redemption, when she

sacrifice herself in order to fix the mess she helped create and stops everything from

getting destroyed.

This redemption is by no means a perfect one and it's not meant to be: what she does her

doesn't stop what she did before from being wrong, but it does show a complexity of character

and a change that I appreciate.

Ultimately, we're each going to make our own judgement call on how much we think Terra

redeemed herself or on how good of a person she was--but I think that's part of what makes

this arc great.

Terra wasn't just a black-and-white character and, even if she does the right thing at the

end, she's ultimately just fixing her own mistakes.

She's complex and hard to pin down.

As I said last time, I know that some people find Terra annoying, that they find her inability

to make a choice and stick with it obnoxious and perplexing, but I really enjoyed her arc.

I feel that the complexity of her character and the weight of her journey brought far

more positives than negatives to the series, and I hope that the show will continue on

this path of becoming more complex in its themes and characterization in its upcoming

seasons.

As always, thank you for watching this series and engaging with it.

It's amazing to me how well it's doing and how well my channel is doing now because of

this and because of your support.

I also want to specifically thank all my patrons over on patreon.com.

The fact that I see that somebody sees so much value in this, uh, helps me to keep going

with it, and helps me to keep making these.

So, thank you so much to all those people for what they do.

Thank you to all of you for watching, and I hope that you have a fantastic day.

Bye bye.

For more infomation >> Is Teen Titans 2003 As Great As We Remember? - Part 4 | A Complete Review of OG Teen Titans - Duration: 15:05.

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Just test this is not true - Duration: 0:44.

For more infomation >> Just test this is not true - Duration: 0:44.

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What is League of Legends? Beginners Tutorial!! (ENG subbed) - Duration: 2:09.

Bonjour!

You just landed in the gaming world, and now you need to think out of the box?

We present, League of Legends!

But we don't have money to buy it!

I got you, League it's free to play!

I'll play only if it's in my native language!

Welcome then, League it's available in a lot of languages!

But I'm a noob, I really suck at playing games!

I got you, there's a lot of crazy dudes for every level of players!

First Steps.

After kicking some IA ass on the tutorials,

you'll be able to play with up to 4 more friends!

Look! What a beautiful girl!

'Champion', not 'girl'!

And with more than 130 champions to unlock, you will have a lot of options to wreck your enemies!

"Prof", but I like to attack from distance... Is there any one like this on the game?

A lot of them! There's Ashe for example, an ice archer!

Cool! Look at these arrows bro!

But Fiora, what about me, that likes tanking everything for my team?

Vlad, we have Zac for you, which turns into a jello instead of dying.

And from the color looks like it's green apple flavored! By the way Ahri, I really like it!

And what about punching faces? I like to punch everybody, teacher!

Darius, for you we have Tryndamere, that is a troublemaker.

Oh, cool. He even liked the beautiful ice archer!

Oh! and looks like her heart it's also made of ice!

TIME TO WRECK IT ALL!

LoL it's a strategy 5 VS 5 game.

1 buddy goes top,

1 mid,

2 bottom, while 1 buddy goes on the jungle.

Cool, I'll play in the jungle then!

No Way! I'll do it!!

The whole idea is to wreck their entire base, starting with the towers then the inhibitors and finally the Nexus.

Oh man! This makes a looooot of 'nexus'!

Shut up, Darius!

There's MUCH MORE!

The champions can change their skins!

Yeah pal, there's one for every taste:

There's a bull dressed as a cow, 8- Bits champions,

DJs that plays rad music for your teammates, and MUCH MORE!

COOL!

All right kiddo, install it RIGHT NOW, and join us on the Summoners Rift!

For more infomation >> What is League of Legends? Beginners Tutorial!! (ENG subbed) - Duration: 2:09.

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Hanebado Is INTENSE! (or, why it's so fun to watch) - Duration: 5:17.

So, Hanebado, or, the Badminton Play of Ayano Hanesaki, is an anime series produced by Liden

Films and directed by Shinpei Izaki, based on the manga of the same name by Kōsuke Hamada,

about the trials and tribulations faced by the various members of the women's badminton

club of Kitakomachi High School, more specifically those faced by its team captain, Aragaki Nagisa,

a determined athlete with impossibly high standards, and its newest member, Ayano Hanesaki,

a badminton prodigy reluctant to return to the sport.

It's a pretty basic premise, and is one I admittedly was only drawn to for the simple

fact that Badminton was one of the only sports I was interested in growing up, but it's

very quickly hooked me in with just how F&*%IN' HYPE IT IS!

LOOK AT THAT ANIMATION!

LOOK AT THOSE MUSCLES!

LOOK AT THAT SOUL CRUSHING DISAPPOINTMENT!

IT'S JUST LIKE MY CHILDHOOD AW YEAAA-

But seriously, the matches in this show are its biggest draw for me, not just for how great they

look, but also because of the mind games surrounding them.

There's a strong focus on how the characters hone in on and exploit each others' physical

and mental weaknesses to their advantage, how their opponents counter these with their

own sly tactics, and vise versa.

This is often emphasized by their inner monologues to show us the logic and thought processes

that go into each characters' approach to the sport.

The match between Aragaki and the new coach, Tachibana, in the second episode is a great

example of this.

In it, we get to hear both Aragaki's obsessive thoughts straining to keep up with the coach's

pace and the way she beats herself up for not being able to control the match, and Tachibana's

contemplative assessment of Aragaki's playstyle.

This simple trick gives us a better understand of each character and adds a layer of tension

to each match as we hear the characters' thoughts racing, making us feel their desperation

and wondering if they're strategies will be enough for them to win.

This is all heightened by the show's incredible animation and choreography, whose fluidity

makes every move flow faster and hit harder, sometimes brutally so.

As you'd expect from a show about badminton, the level of detail that goes into its badminton

play sweeps you into the tension and uncertainty of every match.

But the drama found in this sport is nothing without the drama of the show's main cast.

I don't want to spoil too much here, but suffice it to say, the character's struggles

centre around not only the conflict between those with 'natural talent' and those

who've had to work hard for their skill, and the frustration that builds between these

groups, but also the idea of those same people pushing themselves beyond their limits to

win in this sport which they've dedicated their lives to, and always falling short of

their goals, and how that affects them, whether the emotional weight of their failure breaks

them, or their refusal to stand down slowly turns them into an a&%$£le, which, not gonna

lie, I'm kinda loving right now.

It highlights the intense emotional strain this dedication puts on a person and the dangers

that can come from it consuming their lives, and while I'm sure the ending of the series

will probably put a positive spin on this theme, about being able to find joy in simply

playing the sport instead of losing your mind trying to win every match, so far this through

line helps the series to work as both an intense sports show and an emotionally raw character drama

Some might even say its, too emotional, over dramatic even, which is a sentiment that I

can't help but agree with.

The series can be quite heavy handed with a lot of things, but especially in how it

tries to make you feel bad for its character, to the point that their struggles and backstories

feel less sympathetic and relatable, and more, sad for the sake of being sad.

Hanesaki's mam abandoning her for losing a badminton game either means she is literally

the worst piece of scum in the world or the writer needed something real strong to justify

Hanesaki's reluctance to join the badminton club.

This tendency to go overboard is also present in the aforementioned inner monologues and

general exposition of the series, as well as in how it specifically characterizes Hanesaki's

skill as being almost supernaturally good, as if she made a deal with the devil or something,

which, honestly would explain a lot . . .

AH, F$%K ME, CALL AN EXORCIST ALREADY!

As well, for as great as the show looks, there's a noticeable inconsistency in style

and quality animation-wise, which, though passable most of the time, can be quite jarring

when you notice it.

And what list of criticisms wouldn't be complete without pointing out the needlessly

overt sexual fanservice passed off as comedy, from high school students wearing uncomfortably

form fitting uniforms to jokes about touching someone without their consent,

because that's funny, right?

Right?

. . . hey, uh, who uh, who're you calling?

(knock knock)

FBI, OPEN UP!

For as much as I enjoy the more over the top aspects of this series, they often work to

the show's detriment as much as they work to its benefit.

However, if like me, you have a fondness for ridiculousness, and can deal with the fanservice,

I think you'll find Hanebado to be a nail bitingly tense experience.

It's filled with amazing animation that compliments and amplifies the intensity of

its action, and whose character drama and earnest emotional honesty add to the strengths of it's story.

It's a series that does everything it can to keep you on the edge of your seat, and

for me, that's what makes it so fun to watch.

And yeah, those are my thoughts.

Tell me what yous think, if yous agree, disagree, what your favourite sports anime is, what

your favourite, well, sport is, etc, and thanks for watching!

If yous enjoyed this and wanna see more, than check out my last video, where I talk about

how My Hero Academia's focus on skill over power helps its story.

Or, check out my video on Hinamatsuri, and why it's so fun to watch.

And don't forget to like, comment, share and of course, subscribe, to Come Fly With Me!

Yous can also follow me on Twitter for more updates about this channel and other stuff,

and hopefully, I'll see yous later!

For more infomation >> Hanebado Is INTENSE! (or, why it's so fun to watch) - Duration: 5:17.

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5.1 What is a Dynamical System? - Duration: 16:30.

Welcome to the fifth module of the MOOC Algorithmic Information Dynamics.

As probably, you know by now my name is Narsis and today I will talk to you about dynamical

systems.

At the end of the 17th century, Leibniz (1646-1716) and Newton (1643-1727), independently one

from the other, invented a brilliant mathematical tool: infinitesimal calculus or differential

and integral calculus.

This is an incredibly efficient crystal ball to predict the future, provided the system

in question is governed by a differential equation.

Using it , Poincaré's work on celestial mechanics (Poincaré 1899), and specifically in a 270-page,

prize-winning, and initially flawed paper (Poincaré 1890) start the qualitative theory

of dynamical systems.

The methods developed therein laid the basis for the local and global analysis of nonlinear

differential equations, including the use of first-return (Poincaré) maps, stability

theory for fixed points and periodic orbits, stable and unstable manifolds, and the Poincaré

recurrence theorem.

In this module, the idea is to make you a very generic introduction to dynamical systems

. we talk about how systems typically only occupy a small subset of the overall space

as they cycle through some set of states.

We will be talking about attractors and the fundamental role they play within dynamics

of a system.

We will discuss chaotic and complex regime consisting of multiple attractors and equilibria

very briefly.

The concept of Boolean network as a discrete dynamical system will be introduced and we

learn how we can analyse them.

Ok let see what we mean with dynamical system.

Within science and mathematics, dynamics is the study of how things change with respect

to time, as opposed to describing things simply in terms of their static properties.

The patterns we observe all around us in how the state of things change overtime is an

alternative ways through which we can describe the phenomena we see in our world.

A dynamical system is a set of possible states, together with a rule that determines the present

state in terms of past states.

As examples for dynamical systems, you can think of any system that is evolving in time.

For example, the pendulum, or whether evolution, or the evolution of population of bacteria

or any kind of season that evolves through time.

A dynamical system have two parts State space and function, and we describe such a system

by them.

So lets see what they are.

Dynamical system is study of the thing, which are changing overtime!

Those things are states and a state space is a model used within dynamic systems to

capture this change in a system's state overtime.

Formally, State space is the set of all possible states of a dynamical system.

Each state of the system corresponds to a unique point in the state space.

For example, the state of an idealized pendulum is uniquely defined by its angle and angular

velocity, so the state space is the set of all possible pairs "(angle, velocity)", which

form the cylinder.

In general, any abstract set could be a state space of some dynamical system.

It could be finite, consisting of just a few points or consisting of an infinite number

of points forming a smooth manifold, as usually the case in ordinary differential equations

and mappings.

Such a state space is often called a phase space.

A state space could be infinite-dimensional, as in partial differential equations and delay

differential equations.

In symbolic dynamics, it is a Cantor set, which is zero-dimensional.

Moreover, the second part of dynamical system, Function tells us, given the current state,

what the state of the system will be in the next instant of time.

For investigating dynamical systems, it is necessary to specify some characteristics

that provide a subdivision into special classes of dynamical systems.

Specific methods are available for some of these classes, thus such a classification

can help to simplify the analysis.

A dynamical system is deterministic If the present state can be determined uniquely from

the past states (no randomness is allowed).

Stochastic models possess some inherent randomness.

Chaotic model is a deterministic model with a behaviour that cannot be entirely predicted.

They are predictable in the very short term, but appears random for longer periods.

An important characteristic of a dynamical system is whether it is continuous or discrete.

In a discrete system, the state variables change only at a countable number of points

in time.

These points in time are the ones at which the event occurs/change in state.

In a continuous dynamical system, the state variables change in a continuous way, and

not abruptly from one state to another (infinite number of states).

When the reals are acting, the system is called a continuous dynamical system, and when the

integers are acting, the system is called a discrete dynamical system.

Continuous systems are given by differential equations whereas discrete dynamical systems

(often called maps) are specified by difference equations.

Let's start by discrete, We denote time by k or n, and the system can be solved by

iteration calculation called iterative maps.

Iterative maps give us less information but are much simpler and better suited to dealing

with very many entities, where feedback is important.

Typical example is annual progress of a bank account.

If the initial deposit is 100000 euros and annual interest is 3%, then we can describe

the system by:

In a continuous system, the time interval between our measurements is

negligibly small making it appear as one long continuum and this is done through the language

of calculus and using differential equation or a set of them.

For example, vertical throw is described by initial conditions h(0), v(0) and equations:

where h is height and v is velocity of a body.

Calculus and differential equations have formed a key part of the language of modern science

since the days of Newton and Leibniz.

Even though an analytical treatment of dynamical systems is usually very complicated, obtaining

a numerical solution is (often) straightforward.

Solving differential equations numerically can be done by a number of schemes.

The techniques for solving differential equations based on numerical approximations were developed

before programmable computers existed.

During WorldWarII, it was common to find rooms of people (usually women) working on mechanical

calculators to numerically solve systems of differential equations for military calculations.

The easiest way is by the 1st order Euler's Method which uses the idea of local linearity

or linear approximation, where we use small tangent lines over a short distance to approximate

the solution to an initial-value problem.

If we zoom in small enough, every curve looks like a straight line, and therefore, the tangent

Line is a great way for us to calculate what is happening over a period.

Today we have many numerical methods to solve differential equations.

And there is no "best way" or "best method", since the method to be chosen heavily depends

on the problem (stiff or non stiff, equation or system, smooth or non smooth right hand

side, etc.).

For a "general" or "standard", a 4th order Runge-Kutta is good, it's easy to add an error

estimator to it with no or little additional cost.

A predictor-corrector Adams-method also does the job, they are also quite popular.

Differential equations are great for few elements they give us lots of information but they

also become very complicated very quickly.

Where as differential equations are central to modern science iterative maps are central

to the study of nonlinear systems and their dynamics as they allow us to take the output

to the previous state of the system and feed it back into the next iteration, thus making

them well designed to capture the feedback characteristic of nonlinear systems.

Another important classification of dynamical system is based on linearity.

Let start by defining a Nonlinear System.

Nonlinear System is a set of (one or more)nonlinear equations.

Nonlinear equations are equations where the unknown quantity that we want to solve for

appears in a nonlinear fashion.

For example, if the quantity in question is a function y(t),then terms such as y^2or sin

y would be nonlinear.

More precisely, a nonlinear equation is one where a linear combination of solutions is

not a new solution.

In a linear system, function that is describing the system behaviour must satisfy two basic

properties

additivity homogeneity .

Which one is linear?

g(x) = 3x; g(y) = 3y or f(x) = x2; f(y) = y2?

additivity g(x+y) = 3x + 3y = g(x) + g(y)

homogeneity 5 * g(x) = 5* 3x = 15x = g(5x)

If a system of ordinary differential equations do not depend on the independent variable,

it called an Autonomous system.

If the independent variable is time, we call it time-invariant system.

In an autonomous system If the input signal x(t) produces an output y(t) then any time

shifted input, x(t + δ), results in a time-shifted output y(t + δ).

Consider these two systems,

System A: System B:

Are they Autonomous?

We examine system A and u can do the same for system B. lets Start with a delay of the

input

Now we delay the output by δ

Therefore the system

is not time-invariant or is nonautonomous.

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