# 0.9 Dynamical systems

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Systems with memory

## "what is a dynamical system?"

When we talk about systems in the most general sense, we are talking about anything that takes in a certain number of inputsand produces a certain number of outputs based on those inputs.

In the figure above, the $u(t)$ inputs could be the jets on asatellite and the $y(t)$ outputs could be the gyros describing the"bearing" of the satellite.

There are two basic divisions of systems: static and dynamic. In a static system, the current outputs are based solely on the instantaneous values of the current inputs.An example of a static system is a resistor hooked up to a current source:

$V(t)=Ri(t)$

At any given moment, the voltage across the resistor (the output) depends only on the value of the current runningthrough it (the input). The current at any time $t$ is simply multiplied by the constant value describing the resistance $R$ to give the voltage $V$ . Now, let's see what happens if we replace the resistorwith a capacitor.

$I(t)=C\frac{d v(t)}{d t}}$

Solving for the voltage in the current voltage relationship above, we have:

$v(t)-v({t}_{0})=\frac{1}{C}\int_{{t}_{0}}^{t} i(t)\,d t$

So in the case of the capacitor, the output voltage depends on the history of the current flowing through it. In a sense, thissystem has memory. When a system depends on the present and past input, it is said to be a dynamical system.

## "describing dynamical systems"

As seen in voltage-current relationship of a capacitor, differential equations have memory and can thus be used todescribe dynamical systems. Take the following RLC circuit as an example:

In circuits (as well as in other applications), memory elements can be thought of as energy storage elements. In this circuitdiagram, there are two energy-storing components: the capacitor and the inductor. Since there are two memory elements, it makessense that the differential equation describing this system is second order.

$\frac{d^{2}y(t)}{dt^{2}}+\frac{7}{2}\frac{d^{1}y(t)}{dt^{1}}+9y(t)=6u(t)$

In the most general case of describing a system with differential equations, higher order derivatives of outputvariables can be described as functions of lower order derivatives of the output variables and some derivatives of theinput variables. Note that by saying "function" we make no assumptions about linearity or time-invariance.

By simply rearranging the equation for the RLC circuit above, we can show that that system is in fact covered by this general relationship.

Of course, dynamical systems are not limited to electrical circuits. Any system whose output depends on current and pastinputs is a valid dynamical system. Take for example, the following scenario of relating a satellite's position to itsinputs thrusters.

## "planar orbit satellite"

Using a simple model of a satellite, we can say that its position is controlled by a radial thruster ${u}_{r}$ , which contributes to its vertical motion, and a tangential thruster ${u}_{}$ which contributes to its motion tangential to its orbit. To simplify the analysis, let's assume that the satellite circles the earth in a planar orbit, and thatits position is described by the distance r from the satellite to the center of the Earth and theangleas shown in the figure.

Using the laws of motion, the following set of differential equations can be deduced:

$\frac{d^{2}r(t)}{dt^{2}}-\frac{d^{1}r(t)}{dt^{1}}^{2}={u}_{r}-\frac{k}{r^{2}}$
$2\frac{d^{1}r(t)}{dt^{1}}\frac{d^{1}(t)}{dt^{1}}+r\frac{d^{1}(t)}{dt^{1}}={u}_{}$

why they named it psychology
telll me
ABDUL
the word "psychology" comes from the Greek words "psyche" meaning life and "logos" meaning "explanation".
Soma
Psyche, in Greek mythology was a goddess who comes to represent the human soul's triumph over the misfortunes of life. The term psychology was coined. Psyche, meaning soul or mind and the root "ology" meaning the study of. I hope this helped.
JC
well done
ABDUL
what is chunking
Chunking is a process by which individual pieces of an information set are broken down and then grouped together in a meaningful whole.
Azquan
chunking is to information as syllabus is to word
ABDUL
what dose it means, operant and classic conditioning in short words?
scientific method, critical thinking.
the dealings with the thought process and human behavior.
jeddy
for acquiring knowledge by experimentation.
jeddy
study scientific study of mind and behavior.
jeddy
scientific study of the human mind and behavior.
jeddy
hey jeddy I have question s for you
Yussuf
hi what is behavioralism ?
jusem
what is psychology..?
Ania
the science of mental life.
Mikiyas
it focuses on understanding and modifying human behaviour
ABDUL
because emotion can't be measured.
only by the person having the emotion.
Jason
it is subjective feeling
suman
only placed within levels of itself
Teena
and accepted as extended or beginning versions of emotions that are completely free of restrictions
Teena
anyone know, (2+2)/2
start
Emotions can't be measured, but they can be recognized. To be able to recognize certain characteristics that'll give me an idea about an individual's state of mind.
jeddy
any one knows (2+2)/2
start
it is 6
suman
anyone (2+2)/2
start
2
😂
suman
then more then just recognize but to comprehend the emotions being observed separating your own emotions from the situation currently in.. and rationally gaging the environment.. while then pulling each possible emotional vibe or energy exerted from the human being observed
Teena
How does learning influence behaviour
our choice of study molds us and encourages us to practice what we learn. social structure also changes behavior based on the environment and peers. how we learn can be experimental as not everyone lesrns the same. some are more of a hands-on absorber, whereas others may learn better from listening
A confidence from personal knowledge. Not from having been taugbt. knowledge that you learned what was taught.
jeddy
life is choice, be ready for the best and be prepare for what yet to come in your way(Destiny)
Wayne
what is development
In what Context are you referring to
Hassan
the progression/degression/growth of something. it is interpreted most commonly in psychology as how something is created or formed. ex: When did you develop an interest in psychology?
when refering to physical, cognitive, and social development, these types of development are determined by many circumstances. such as demographics, biology, and personality/character
it is the apprehensions fluency based by the process of configurations to a design based on product/witness series noticeboards to note the stats on order to retail agreements on stated series order.
Kay
Evolving
jeddy
I'm on my first year of a psychology course, is there anything I should know? anything you wished you knew when you just started psychology ? because I feel like I'm walking into all of this blindly
hello
GOPAL
I feel exactly the same. Just about to start Psychology and Criminology. feeling slightly nervous 🙈
Sarah
and everything is online
Lia
only for a few weeks I believe. Unless I have read it wrong. I believe that they will try and filter everyone in.
Sarah
it's different for every country
Lia
hi
GOPAL
I start 9/14. Online as well.
LaQuisha
I know very little about Psychology
GOPAL
which area are you specialising in
Elle
yeah, everthing your just know, that psychology is everthing come out with, mind, behavior and emotion. and believe, once you study it, you will understand, wjy people act like that.
EDWARD
Don't be in the class with pre notions.. yh
ABDUL
after a successful job interview,Mr.Abhishek felt relaxed and calm,he stopped sweating and felt hungry,which part of his nervous system was activated
Hypothalamus
Words
what is the Borysenko model and why is it important?
Sarah
It explains the link between stress and diseases ie.. How long term stress on the mind or body can lead to chronic migraines, ulcers, depression...
Jennifer
I understand there is a link between stress and illness and disease. that when you are in high-stress situations for prolonged periods of time that the brain releases an abundance of stress hormones such as epinephrine, norepinephrine, cortisol, and aldosterone. idk anymore than that.
Sarah
what is the strees anger
Divya
what is difference between illness and disease
Umair
anyone knows (2+2)/2
start
Hy everybody any one who can help me to understand the concept of Structuralism in easy words?
If you really considered looking into yourself find the reason of the possibilities or the environment and atmosphere in which you're placed with or without choice in voluntary or involuntary from birth to now, and also consider all existential extensions of yourself including spiritual recognition.
Darling
? and compare them to social,judicial, learned events,experiences, influences and dive into there branches? somehow reflects our mechanic motivations and has an effect on who we are and what we represent. ?😲
Darling
critical evaluation on attitude change and formation those that even exist
COMORDOR
who is considered of a father of psychology
Freud
Lithemba
Immanuel Kant: Critique of Pure Reason.
David
wundt
Namrata
Wundt
Disha
wundt
NishaOfficial
wundt
Naheed
wundt
Jasmine
id say freud but wundt continued his work. let's face it without freud there wouldn't be a psychology practices
Michelle
Freud is considered as the father of Psychoanalysis not psychology
Indefatigable
technically Freud would be the father of psychology because he discovered the id ego and super ego
Bryce
Its kind of a loaded question. There are many early greats in psychology each having their own theories and views. I think it is more of a collective of ideals that make up our current view and understanding of psychology.
Jason
I agree with you Jason. there are Amazing psychologist back then that made the foundation for the field of psychology
Bryce
Wilhelm Winston ryt
Faazi
Wundt
Waris
i agree to you Jason
Welli
is there any name for the fear of dying painfully? is it the same as thanatophobia?
hello matthew i am new
Barbara
who is reinhart
hello, why are you asking me this?
She
hello, how are you?
Corey
hey Corey, all fine, how are you doing?
She
Hello everyone
Indefatigable
hello
Hannah
Which specialization in psychology interest you? why?
Ailyn
hello, hannah, I'm fine thanks and how are you?
Corey
or she
Corey
hello hannah and corey
Barbara
What makes a personal quality part of someone's personality?
that's a great question 🤔
Love
what do you want to learn in psychogy?
I have been observing An incredible amount of people, who are using some kind of stimulant. I'd like to help give advice into their drug dramas. Help Repair their faulty thinking?
Shaun
Shaun
Rules *
Shaun
hello
shruti
where from i can learn the rules of unconscious behavior
shruti
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