Discrete and Continuous Dynamical Systems | Malone Math

The Central Idea

A dynamical system combines two basic ingredients:

a description of the system's current state;
a rule describing how that state changes through time.

In its most compact form, the idea is

\[ \boxed{\text{current state}+\text{evolution rule}\longrightarrow \text{future state}}. \]

The evolution may happen in separate steps, such as one year at a time, or continuously, with change occurring at every instant.

What Is a Dynamical System?

A dynamical system is a mathematical system whose state changes according to a specified rule as time progresses.

For example, suppose \(P_n\) represents a population after \(n\) years. A simple model might be

\[ P_{n+1}=1.05P_n. \]

The next population is determined from the current population by multiplying by \(1.05\).

If instead population is treated as changing continuously, we might write

\[ \frac{dP}{dt}=0.05P. \]

Both equations describe change, but they organize time differently.

The State of a System

The state is the collection of information needed to determine the system's future evolution under the model.

A very simple system may require only one number.

\[ x_n=12. \]

A mechanical system may require several quantities, such as position and velocity:

\[ \mathbf{x}(t)=\begin{pmatrix}x(t)\\v(t)\end{pmatrix}. \]

The state does not necessarily contain every imaginable fact about the physical object. It contains the variables that the mathematical model uses to predict future behavior.

How Time Enters the Model

Time is the organizing variable of a dynamical system.

The two most common possibilities are:

discrete time, where time advances in separate steps;
continuous time, where time can take every value in an interval.

This distinction gives us the two major introductory classes of dynamical systems.

Discrete Dynamical Systems

A discrete dynamical system updates its state at separated time values.

We commonly label these times

\[ n=0,1,2,3,\ldots \]

and represent the state by

\[ x_0,x_1,x_2,x_3,\ldots. \]

The evolution rule tells us how to obtain one state from another.

The Discrete Update Rule

A common one-dimensional discrete system has the form

\[ \boxed{x_{n+1}=f(x_n)}. \]

The function \(f\) is called an update map, iteration function, or simply the rule of evolution.

Once \(x_0\) is known, the rule produces

\[x_1=f(x_0),\] \[x_2=f(x_1),\] \[x_3=f(x_2),\]

and so on.

Iteration

Repeatedly applying the same rule is called iteration.

\[x_{n+1}=f(x_n),\]

then

\[x_1=f(x_0),\] \[x_2=f(f(x_0)),\] \[x_3=f(f(f(x_0))).\]

Dynamical systems are therefore closely connected to sequences, recursive relationships, functions, algorithms, and repeated processes.

Example: A Simple Discrete System

Consider

\[x_{n+1}=2x_n\]

with

\[x_0=3.\]

The first state is

\[x_1=2(3)=6.\]

Then

\[x_2=2(6)=12,\] \[x_3=2(12)=24,\] \[x_4=2(24)=48.\]

The trajectory begins

\[\boxed{3,\ 6,\ 12,\ 24,\ 48,\ldots}.\]

An Iterated Rule May Have a Direct Formula

The system

\[x_{n+1}=2x_n\]

can be solved directly.

Since each step multiplies the previous value by \(2\),

\[x_n=x_0 2^n.\]

With \(x_0=3\),

\[\boxed{x_n=3\cdot2^n}.\]

The recursive form describes the evolution step by step. The explicit form gives the state directly at step \(n\).

Discrete Exponential Growth and Decay

A fundamental discrete system is

\[x_{n+1}=ax_n.\]

Repeated iteration gives

\[\boxed{x_n=x_0a^n}.\]

The behavior depends strongly on \(a\).

If \(a>1\), positive states grow in magnitude.
If \(0<a<1\), positive states decay toward zero.
If \(a=1\), every state remains unchanged.
If \(a<0\), signs alternate as the system evolves.

Example: Discrete Decay

Suppose

\[x_{n+1}=0.8x_n\]

with

\[x_0=100.\]

Then

\[x_1=80,\] \[x_2=64,\] \[x_3=51.2.\]

The explicit formula is

\[\boxed{x_n=100(0.8)^n}.\]

Each step retains \(80\%\) of the previous state.

Discrete Systems Can Include Inputs

Not every update simply multiplies the previous value.

Consider

\[x_{n+1}=0.9x_n+10.\]

The term \(0.9x_n\) represents retention of part of the current state, while the constant \(10\) represents an addition during every time step.

Models like this can describe inventories, savings balances, resource stores, populations with migration, and many other systems.

Example: Iterating an Affine Rule

Let

\[x_{n+1}=0.9x_n+10\]

and suppose

\[x_0=50.\]

Then

\[x_1=0.9(50)+10=55.\]

Next,

\[x_2=0.9(55)+10=59.5.\]

Then

\[x_3=0.9(59.5)+10=63.55.\]

The system continues changing because each new value becomes the input for the next update.

Discrete Systems with Several State Variables

A system may need several variables to describe its state.

We can collect them into a vector:

\[ \mathbf{x}_n=\begin{pmatrix}x_n\\y_n\end{pmatrix}. \]

An update rule might be

\[ \begin{pmatrix}x_{n+1}\\y_{n+1}\end{pmatrix} = \begin{pmatrix}0.8x_n+0.1y_n\\0.2x_n+0.9y_n\end{pmatrix}. \]

Now the two components influence each other's future values.

Matrix Form of a Linear Discrete System

A linear discrete system can often be written compactly as

\[\boxed{\mathbf{x}_{n+1}=A\mathbf{x}_n}.\]

For example,

\[ \begin{pmatrix}x_{n+1}\\y_{n+1}\end{pmatrix} = \begin{pmatrix}0.8 & 0.1\\0.2 & 0.9\end{pmatrix} \begin{pmatrix}x_n\\y_n\end{pmatrix}. \]

This connects dynamical systems directly to matrices, eigenvalues, eigenvectors, and linear algebra.

Continuous Dynamical Systems

A continuous dynamical system treats the state as evolving continuously through time.

Instead of

\[n=0,1,2,3,\ldots,\]

the time variable may take values such as

\[t=0,\quad0.1,\quad0.25,\quad1,\quad\pi,\quad10.4,\]

together with every other value in the interval being studied.

Differential Equations as Evolution Rules

Continuous dynamical systems are commonly described by differential equations.

A basic example is

\[\boxed{\frac{dx}{dt}=f(x)}.\]

The derivative tells us the instantaneous rate at which the state is changing.

Instead of directly specifying the next state, as a discrete system does, the differential equation specifies the velocity of the state through its possible values.

Reading the Differential Equation

Consider

\[\frac{dx}{dt}=3x.\]

This says that the instantaneous rate of change of \(x\) is three times its current value.

If \(x=2\), then

\[\frac{dx}{dt}=6.\]

If \(x=10\), then

\[\frac{dx}{dt}=30.\]

The system's current state determines how quickly it changes.

Continuous Exponential Growth

Consider

\[\frac{dx}{dt}=kx.\]

Its solution is

\[\boxed{x(t)=x_0e^{kt}}.\]

Here \(x_0\) is the initial state.

If \(k>0\), the system grows exponentially. If \(k<0\), it decays exponentially.

Example: Continuous Growth

Suppose

\[\frac{dP}{dt}=0.04P\]

and

\[P(0)=500.\]

Then

\[\boxed{P(t)=500e^{0.04t}}.\]

At \(t=10\),

\[P(10)=500e^{0.4}.\]

The continuous model specifies the population at every modeled time, not only at integer steps.

Example: Continuous Decay

Suppose a quantity satisfies

\[\frac{dx}{dt}=-0.2x\]

with

\[x(0)=80.\]

Then

\[\boxed{x(t)=80e^{-0.2t}}.\]

Because

\[e^{-0.2t}\to0\qquad\text{as}\qquad t\to\infty,\]

the state approaches zero over the long term.

Continuous Systems with Several Variables

Continuous systems may also require several state variables.

\[ \mathbf{x}(t)=\begin{pmatrix}x_1(t)\\x_2(t)\\\vdots\\x_m(t)\end{pmatrix}. \]

The evolution equation becomes

\[\boxed{\frac{d\mathbf{x}}{dt}=\mathbf{F}(\mathbf{x})}.\]

Each component may depend on several or all of the other state variables.

Example: Position and Velocity

A moving object's position alone may not be enough to determine its future. Its velocity matters too.

Let \(x(t)=\text{position}\) and \(v(t)=\text{velocity}\).

\[\frac{dx}{dt}=v.\]

If acceleration depends on position, we might also have

\[\frac{dv}{dt}=-kx.\]

Together,

\[ \boxed{\begin{aligned}\frac{dx}{dt}&=v,\\\frac{dv}{dt}&=-kx.\end{aligned}} \]

This is a two-dimensional continuous dynamical system.

The Harmonic Oscillator as a Dynamical System

Combining

\[\frac{dx}{dt}=v\]

and

\[\frac{dv}{dt}=-kx\]

gives

\[\frac{d^2x}{dt^2}=-kx.\]

This is the familiar equation behind ideal oscillatory motion.

Dynamical-systems language allows us to study the same motion by examining the changing state \((x,v)\).

That change in viewpoint will become important when we study phase spaces and phase portraits.

Discrete versus Continuous Evolution

Feature	Discrete System	Continuous System
Time	\(n=0,1,2,\ldots\)	\(t\in\mathbb{R}\) over an interval
Typical state notation	\(x_n\)	\(x(t)\)
Evolution rule	\(x_{n+1}=f(x_n)\)	\(\dfrac{dx}{dt}=f(x)\)
Primary operation	Iteration	Integration of a differential equation
Result	A sequence of states	A continuous trajectory through time

The Same Phenomenon Can Be Modeled Both Ways

Whether a model should be discrete or continuous depends partly on the phenomenon and partly on the level of approximation we choose.

Population growth could be modeled by

\[P_{n+1}=1.03P_n\]

if we update the population once per year.

It could instead be approximated by

\[\frac{dP}{dt}=0.03P\]

if we treat growth as occurring continuously.

These are related models, but they are not mathematically identical.

The Meaning of a Time Step Matters

In a discrete model, \(x_{n+1}=1.05x_n\), the meaning of the multiplier depends on what one step represents.

one second;
one day;
one month;
one year;
one generation;
one production cycle.

A dynamical model is incomplete unless the units and interpretation of its time variable are understood.

Initial Conditions

An evolution rule alone usually does not determine one particular motion. We also need an initial state.

For a discrete system we might specify

\[x_0=4.\]

For a continuous system we might specify

\[x(0)=4.\]

This information is called an initial condition.

The model plus the initial condition determines the trajectory we want to study.

Different Initial Conditions Can Produce Different Futures

Consider \(x_{n+1}=2x_n\).

If \(x_0=1\), then \(1,2,4,8,\ldots\) results.

If instead \(x_0=3\), then \(3,6,12,24,\ldots\) results.

The evolution law is identical, but the trajectories are different because the starting states are different.

What Is a Trajectory?

A trajectory records how a state evolves through time.

For a discrete system, the trajectory may appear as

\[x_0,\ x_1,\ x_2,\ x_3,\ldots.\]

For a continuous system, the trajectory is represented by \(x(t)\).

With several state variables, a trajectory becomes a path through a multidimensional state space.

Orbits

In dynamical-systems language, the sequence of states produced by repeated iteration is often called an orbit.

\[\boxed{x_0,\ f(x_0),\ f(f(x_0)),\ldots}.\]

The terms orbit and trajectory are closely related. Their exact usage can vary slightly by context, but both concern the evolution of states under the system.

Autonomous Systems

A system is called autonomous when the evolution rule does not depend explicitly on time.

For example, \(x_{n+1}=x_n(1-x_n)\) is autonomous because the same rule is applied at every step.

Likewise, \(\frac{dx}{dt}=x(1-x)\) is autonomous because the right side depends on \(x\), not explicitly on \(t\).

Nonautonomous Systems

A system is nonautonomous when the evolution rule depends explicitly on time.

For example, \(x_{n+1}=0.9x_n+\sin n\) contains the current time step \(n\).

Similarly, \(\frac{dx}{dt}=-x+\sin t\) contains an explicit time-dependent forcing term.

Such equations are useful when external influences vary through time.

Linear Dynamical Systems

Some dynamical systems are linear.

A discrete linear system can take the form \(\mathbf{x}_{n+1}=A\mathbf{x}_n\).

A continuous linear system can take the form

\[\boxed{\frac{d\mathbf{x}}{dt}=A\mathbf{x}}.\]

Linear systems are especially important because matrix methods allow their behavior to be studied systematically.

Nonlinear Dynamical Systems

A system is nonlinear when its evolution rule contains nonlinear relationships among the state variables.

Examples include

\[x_{n+1}=rx_n(1-x_n)\]

and

\[\frac{dx}{dt}=x-x^3.\]

Nonlinearity can produce behavior that does not occur in the simplest linear models, including multiple equilibria, changing stability, periodic behavior, bifurcations, and chaos.

A Famous Discrete Nonlinear System

One of the most studied discrete dynamical systems is the logistic map:

\[\boxed{x_{n+1}=rx_n(1-x_n)}.\]

Here \(r\) is a parameter controlling the update rule.

Despite its simple algebraic form, changing \(r\) can produce dramatically different long-term behavior.

Depending on the parameter, the system can settle toward a fixed state, oscillate among several states, or exhibit chaotic behavior.

Example: Iterating the Logistic Map

Let \(x_{n+1}=2x_n(1-x_n)\) with \(x_0=0.2\).

\[x_1=2(0.2)(0.8)=0.32.\] \[x_2=2(0.32)(0.68)=0.4352.\] \[x_3=2(0.4352)(0.5648)\approx0.4916.\]

Each output becomes the next input.

Parameters Control Families of Systems

A parameter does more than determine an initial state. It can change the evolution law itself.

In \(x_{n+1}=rx_n(1-x_n)\), different choices of \(r\) create different dynamical systems.

state variables, which evolve through time;
parameters, which specify the structure or regime of the model.

Later, bifurcation analysis will examine what happens when parameters are changed.

A First Look at Equilibrium

Some states do not change under the evolution rule.

In a discrete system \(x_{n+1}=f(x_n)\), an equilibrium or fixed point \(x^\ast\) satisfies

\[\boxed{f(x^\ast)=x^\ast}.\]

If the system begins at \(x^\ast\), it remains there.

Equilibrium in Continuous Time

For \(\frac{dx}{dt}=f(x)\), an equilibrium \(x^\ast\) satisfies

\[\boxed{f(x^\ast)=0}.\]

At that state, \(\frac{dx}{dt}=0\), so the state does not move.

Fixed points and equilibria will receive detailed treatment later in this section.

Example: Finding a Discrete Fixed Point

Consider \(x_{n+1}=0.5x_n+4\).

A fixed point satisfies \(x^\ast=0.5x^\ast+4\).

Subtract \(0.5x^\ast\):

\[0.5x^\ast=4.\]

Therefore,

\[\boxed{x^\ast=8}.\]

Indeed, \(0.5(8)+4=8\).

Long-Term Behavior

Dynamical-systems analysis often asks less about the exact state after a small number of steps and more about what happens after a very long time.

Does the state approach an equilibrium?
Does it grow without bound?
Does it decay toward zero?
Does it oscillate?
Does it repeat periodically?
Does it approach a more complicated attractor?
Does the behavior depend sensitively on the initial condition?

Qualitative Analysis

An exact formula is valuable, but dynamical-systems analysis does not always require one.

the direction of motion;
the location of equilibria;
whether equilibria attract or repel nearby states;
whether solutions remain bounded;
whether trajectories cycle;
how long-term behavior changes with parameters.

These questions allow us to understand systems even when explicit solutions are difficult or impossible to obtain.

Exact Solutions versus Dynamical Understanding

Suppose a differential equation has a complicated solution formula.

A dynamical analysis might still reveal that every positive solution approaches the same equilibrium.

For many applications, that conclusion may be more informative than a long explicit expression.

Dynamical systems therefore shifts some emphasis from

\[\text{What is the exact formula?}\]

toward

\[\boxed{\text{How does the system behave?}}\]

The Sign of the Evolution Rule Gives Information

Consider the continuous system

\[\frac{dx}{dt}=x(1-x).\]

If \(0<x<1\), then \(x(1-x)>0\), so \(\frac{dx}{dt}>0\) and \(x\) increases.

If \(x>1\), then \(x(1-x)<0\), so \(x\) decreases.

We can learn a great deal about the system without explicitly solving the differential equation.

Continuous Logistic Growth

A continuous logistic model has the form

\[\boxed{\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)}.\]

Here:

\(P\) is the population state;
\(r\) is a growth parameter;
\(K\) is a carrying-capacity parameter.

When \(P\) is small compared with \(K\), growth resembles exponential growth. As \(P\) approaches \(K\), the factor \(1-\frac{P}{K}\) becomes smaller and slows the growth.

Do Not Confuse the Two Logistic Systems

The discrete logistic map \(x_{n+1}=rx_n(1-x_n)\) and the continuous logistic differential equation \(\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)\) contain related nonlinear ideas, but they are different dynamical systems.

Discrete and continuous models can behave differently even when their equations appear structurally similar.

This distinction becomes especially important when studying oscillations, bifurcations, and chaos.

Continuous Systems Can Be Approximated Discretely

Suppose \(\frac{dx}{dt}=f(x)\). Over a small time interval \(\Delta t\), we can approximate

\[\frac{\Delta x}{\Delta t}\approx f(x_n).\]

Therefore,

\[\Delta x\approx \Delta t\,f(x_n).\]

This gives the discrete update

\[\boxed{x_{n+1}=x_n+\Delta t\,f(x_n)}.\]

This is the basic idea behind Euler's method.

Example: Turning Continuous Change into Steps

Suppose \(\frac{dx}{dt}=2x\) and choose \(\Delta t=0.1\).

Euler's update becomes

\[x_{n+1}=x_n+0.1(2x_n).\]

Simplify:

\[\boxed{x_{n+1}=1.2x_n}.\]

The continuous differential equation has been approximated by a discrete dynamical system.

A Bridge Between Discrete and Continuous Mathematics

Dynamical systems brings together several mathematical subjects.

Discrete models connect naturally to:

sequences;
recurrence relations;
difference equations;
iteration;
algorithms;
matrix powers.

Continuous models connect naturally to:

derivatives;
differential equations;
vector fields;
integration;
linear algebra;
numerical differential-equation methods.

Deterministic Dynamical Systems

The systems introduced in this lesson are primarily deterministic.

Once the current state and evolution rule are known, the next state is determined by the model.

For \(x_{n+1}=f(x_n)\), a particular value of \(x_n\) determines one corresponding value of \(x_{n+1}\).

Deterministic does not mean that long-term behavior must be simple.

Deterministic Does Not Mean Predictably Simple

One of the most important lessons of dynamical-systems theory is that a completely deterministic rule can still produce extremely complicated behavior.

A system can obey a precise equation while nearby starting states eventually produce very different trajectories.

This phenomenon is central to the mathematical study of chaos.

Chaos therefore does not simply mean randomness. A chaotic deterministic system still follows its evolution rule at every step.

What About Randomness?

Some evolving systems include genuinely random components.

A discrete model might have the form

\[x_{n+1}=f(x_n)+\varepsilon_n,\]

where \(\varepsilon_n\) represents a random disturbance.

Such systems lead toward stochastic processes and stochastic dynamical systems.

The present lesson concentrates on deterministic evolution so that the foundational geometry and logic of dynamical systems remain clear.

Where Dynamical Systems Appear

Dynamical systems provide a common mathematical language for change across many disciplines.

population and ecological models;
epidemic models;
chemical reaction networks;
mechanical systems;
planetary and orbital motion;
electrical circuits;
climate and atmospheric models;
control systems;
financial accumulation models;
economic adjustment models;
traffic and transportation models;
neural and biological systems;
industrial process models.

A Dynamical Modeling Workflow

identify the system being studied;
choose the state variables;
choose the time scale;
decide whether time should be discrete or continuous;
identify parameters and external inputs;
construct the evolution rule;
specify initial conditions;
compute or approximate trajectories;
study long-term behavior;
compare the model with the phenomenon it represents.

When Is a Discrete Model Natural?

Discrete time is especially natural when changes occur or are observed at separate events.

annual population counts;
generations of an organism;
monthly financial balances;
daily inventory updates;
production cycles;
repeated algorithmic calculations;
year-to-year economic data.

The model's mathematics should reflect the structure of the process being modeled.

When Is a Continuous Model Natural?

Continuous time is especially useful when the system is treated as changing at every instant.

motion through space;
fluid flow;
temperature change;
electrical current;
chemical concentration;
continuous population approximations;
orbital dynamics.

Continuous modeling often leads naturally to differential equations.

The Choice Is a Modeling Decision

A real process does not arrive labeled as "discrete" or "continuous."

The modeler decides how the process should be represented based on:

the mechanism being studied;
the available data;
the desired time resolution;
the required accuracy;
the mathematical methods available;
the purpose of the model.

The same physical process may therefore support several useful dynamical models at different scales.

Simulation

When repeated updates are cumbersome, computers can simulate the trajectory of a dynamical system.

For a discrete system, \(x_{n+1}=f(x_n)\), an algorithm can repeatedly:

store the current value \(x_n\);
calculate \(f(x_n)\);
assign the result to \(x_{n+1}\);
repeat.

Simulation makes it possible to investigate systems whose long-term behavior is difficult to determine by hand.

Discrete Systems in a Spreadsheet

A recurrence relation is naturally suited to spreadsheet calculation.

For \(x_{n+1}=0.8x_n+5\), one row can contain \(x_n\), while the next row applies the update formula.

Copying the formula downward performs the iteration automatically.

This makes dynamical systems particularly accessible for numerical experimentation.

Computational Experiments

Dynamical systems invite experimentation.

the initial condition;
a parameter;
the size of a numerical time step;
the number of iterations;
the evolution rule itself.

We can then compare the resulting trajectories and look for patterns.

This combination of mathematical analysis and computation is a central feature of modern dynamical-systems work.

Common Error: Treating Discrete and Continuous Time as Identical

The equations \(x_{n+1}=1.1x_n\) and \(\frac{dx}{dt}=0.1x\) do not produce exactly the same evolution.

The first has solution \(x_n=x_0(1.1)^n\), while the second has solution \(x(t)=x_0e^{0.1t}\).

Similar interpretations do not make the mathematical systems identical.

Common Error: Confusing States and Parameters

In \(x_{n+1}=rx_n(1-x_n)\), \(x_n\) is a state variable because its value changes as \(n\) changes.

The quantity \(r\) is normally treated as a parameter because it specifies which update rule is being used.

Changing \(x_0\) changes the initial state.

Changing \(r\) changes the system itself.

Common Error: Ignoring the Initial Condition

The equation \(\frac{dx}{dt}=2x\) does not specify one unique trajectory.

Its general solution is \(x(t)=Ce^{2t}\), where the constant \(C\) depends on the initial condition.

If \(x(0)=5\), then \(C=5\) and \(\boxed{x(t)=5e^{2t}}\).

Common Error: Confusing the Rule with a Trajectory

The equation \(x_{n+1}=f(x_n)\) describes the dynamical system.

A sequence such as \(1,\ 2,\ 4,\ 8,\ldots\) is one trajectory generated by that system from a particular initial condition.

The rule defines a family of possible trajectories. The initial state selects the particular trajectory being followed.

Common Error: Equating Chaos with Randomness

Chaotic behavior can arise from a deterministic equation.

The next state may still be completely determined by the current state.

What makes chaotic systems difficult to predict over long periods is their strong sensitivity to initial conditions and the resulting amplification of small differences.

Chaos and randomness are therefore related ideas in some applications, but they are not mathematical synonyms.

A Reliable Procedure for a Discrete Dynamical System

Identify the state variable or state vector.
Identify what one time step represents.
Write the update rule.
Identify all parameters.
Specify the initial state.
Calculate the next state.
Repeat the update to generate a trajectory.
Look for fixed points, cycles, growth, decay, or other patterns.
Investigate the long-term behavior.
Check whether the conclusions make sense for the modeled system.

A Reliable Procedure for a Continuous Dynamical System

Identify the state variable or state vector.
Identify the time variable and its units.
Write the differential equation or system.
Identify parameters and external inputs.
Specify initial conditions.
Determine the direction and rate of change.
Solve analytically when practical.
Approximate numerically when necessary.
Locate equilibria and investigate their behavior.
Analyze the long-term trajectory.

Practice 1: Discrete Iteration

Let

\[x_{n+1}=3x_n\]

with

\[x_0=2.\]

Find \(x_1,\qquad x_2,\qquad x_3\).

Practice 2: Discrete Decay

Let

\[x_{n+1}=0.5x_n\]

and

\[x_0=64.\]

Find the first five states beginning with \(x_0\).

Practice 3: Identify the Type of System

Classify

\[y_{n+1}=y_n+4\]

as discrete or continuous.

Practice 4: Identify the Type of System

Classify

\[\frac{dP}{dt}=0.08P\]

as discrete or continuous.

Practice 5: Continuous Growth

Solve

\[\frac{dx}{dt}=0.5x\]

with

\[x(0)=10.\]

Practice 6: Find a Fixed Point

Find the fixed point of

\[x_{n+1}=0.75x_n+5.\]

Practice 7: Interpret the State

Suppose

\[\mathbf{x}(t)=\begin{pmatrix}p(t)\\v(t)\end{pmatrix},\]

where \(p(t)\) is position and \(v(t)\) is velocity.

Explain what information the state vector contains at a particular time.

Practice 8: Euler Update

Suppose

\[\frac{dx}{dt}=-2x.\]

Using Euler's method with \(\Delta t=0.1\), construct the corresponding update rule \(x_{n+1}=x_n+\Delta t\,f(x_n)\).

Practice 9: State or Parameter?

Consider

\[P_{n+1}=rP_n.\]

If \(P_n\) is the population at generation \(n\) and \(r\) is a fixed reproductive multiplier, identify the state variable and the parameter.

Practice 10: Compare Models

Compare

\[P_{n+1}=1.02P_n\]

with

\[\frac{dP}{dt}=0.02P.\]

State one important mathematical difference between the two models.

Practice Answers

Practice 1

Starting with \(x_0=2\), we obtain \(x_1=6\), \(x_2=18\), and \(x_3=54\).

\[\boxed{x_1=6,\qquad x_2=18,\qquad x_3=54}.\]

Practice 2

\[\boxed{64,\ 32,\ 16,\ 8,\ 4}.\]

Practice 3

It is a \(\boxed{\text{discrete dynamical system}}\).

Practice 4

It describes a continuous dynamical system.

Practice 5

\[\boxed{x(t)=10e^{0.5t}}.\]

Practice 6

\[x^\ast=0.75x^\ast+5\] \[0.25x^\ast=5\] \[\boxed{x^\ast=20}.\]

Practice 7

The state vector records both the object's current position and its current velocity. Together these variables provide the state information used by the model to determine subsequent motion.

Practice 8

\[x_{n+1}=x_n+0.1(-2x_n)\] \[\boxed{x_{n+1}=0.8x_n}.\]

Practice 9

\(P_n\) is the state variable and \(r\) is the parameter.

Practice 10

The first model advances through discrete time steps, while the second describes instantaneous change throughout continuous time.

\[P_n=P_0(1.02)^n\] \[P(t)=P_0e^{0.02t}.\]

What to Remember

A dynamical system combines a state with an evolution rule.

\[\boxed{\text{state}+\text{evolution rule}\longrightarrow\text{trajectory}}.\]

For a discrete system, the basic form is

\[\boxed{x_{n+1}=f(x_n)}.\]

For a continuous system, the basic form is

\[\boxed{\frac{dx}{dt}=f(x)}.\]

the state describes the system at a particular time;
parameters help define the evolution law;
initial conditions select a particular trajectory;
equilibria are states that remain unchanged;
long-term behavior is often more important than a single calculation;
numerical methods can approximate trajectories when exact solutions are difficult.

The central question of dynamical-systems analysis is:

\[\boxed{\text{How does the state evolve through time?}}\]

Where We Go Next

We have described a dynamical system in terms of states and evolution rules, but we have not yet developed the geometric setting in which those states live.

When a system contains several variables, every possible combination of their values can be regarded as a point in a larger mathematical space. As the system evolves, its state moves through that space and traces a trajectory.

The next lesson, State Spaces and Trajectories, develops this geometric viewpoint and provides the foundation for phase portraits, equilibria, stability, periodic orbits, attractors, bifurcations, and chaos.