The Central Idea
A state tells us where the system is now. A state space tells us all the places the system could be. A trajectory tells us where the system goes.
\[ \boxed{ \text{state space}+\text{evolution rule}+\text{initial state} \longrightarrow \text{trajectory} } \]The state-space viewpoint is especially valuable when a system contains several interacting variables.
What Is a State?
The state of a dynamical system is the collection of variables needed by the model to determine its future evolution.
A one-variable system may have the state
\[ x. \]A two-variable system may have the state
\[ \mathbf{x}= \begin{pmatrix} x\\ y \end{pmatrix}. \]A mechanical system may use position and velocity:
\[ \mathbf{x}= \begin{pmatrix} q\\ v \end{pmatrix}. \]What Is a State Space?
A state space is the set of all states allowed by the mathematical model.
If the state variable is one real number,
\[ x\in\mathbb{R}, \]then the state space may be the real line.
If the state consists of two real variables,
\[ (x,y)\in\mathbb{R}^2, \]then the state space may be a plane.
State Space Is Not Physical Space
A state space may resemble ordinary geometric space, but its coordinates do not have to represent physical position.
For example,
\[ (P,R) \]might represent the populations of two species.
The horizontal coordinate could represent prey population and the vertical coordinate predator population. A point in that plane is then a population state, not a physical location.
One-Dimensional State Spaces
Suppose the entire state of a system is described by one scalar \(x\).
If every real value is allowed, the state space is
\[ \mathbb{R}. \]If only nonnegative values make sense, the state space might instead be
\[ [0,\infty). \]A population model, for example, usually excludes negative population states.
Two-Dimensional State Spaces
If two variables are needed to describe the state, then each state is an ordered pair:
\[ (x,y). \]The state space may be
\[ \mathbb{R}^2, \]or some restricted region of the plane.
The point
\[ (2,5) \]represents one possible state of the system.
Higher-Dimensional State Spaces
A system with \(m\) state variables may use a state vector
\[ \mathbf{x}= \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_m \end{pmatrix}. \]The corresponding state space may be
\[ \mathbb{R}^m. \]Although dimensions greater than three cannot be drawn directly, the same mathematical ideas still apply.
Constraints Shape the State Space
A model often restricts which mathematical states are meaningful.
If \(x\) and \(y\) represent populations, then
\[ x\ge 0,\qquad y\ge 0. \]The relevant state space is therefore the first quadrant:
\[ \boxed{ \{(x,y)\in\mathbb{R}^2:x\ge0,\ y\ge0\} }. \]Physical, biological, financial, or geometric constraints can all restrict the state space.
State Variables versus Parameters
State variables change as the system evolves. Parameters normally remain fixed during one particular run of the model.
In
\[ x_{n+1}=rx_n(1-x_n), \]\(x_n\) is the state variable and \(r\) is a parameter.
Different values of \(r\) define different members of a family of dynamical systems.
Initial States
A trajectory begins from an initial state.
In a one-dimensional discrete system we may write
\[ x_0=2. \]In a two-dimensional continuous system we may write
\[ \mathbf{x}(0)= \begin{pmatrix} 2\\ -1 \end{pmatrix}. \]Changing the initial state may produce a different trajectory even when the evolution law remains unchanged.
What Is a Trajectory?
A trajectory is the sequence or path of states generated as a dynamical system evolves.
For a discrete system,
\[ x_0,\ x_1,\ x_2,\ x_3,\ldots \]gives a trajectory through discrete time.
For a continuous system,
\[ \mathbf{x}(t) \]describes a continuously evolving path through state space.
Discrete Trajectories
Consider
\[ x_{n+1}=\frac12x_n \]with
\[ x_0=8. \]The trajectory is
\[ 8,\ 4,\ 2,\ 1,\ \frac12,\ldots. \]In the state space \(\mathbb{R}\), the system visits one point after another.
Direction Matters
A trajectory is not merely a set of points. It has an order determined by time.
For
\[ 8,\ 4,\ 2,\ 1,\ldots, \]the evolution moves from \(8\) toward \(0\).
Reversing the list would represent a different direction of evolution.
Arrows are therefore commonly used when trajectories are drawn.
Orbits in Discrete Systems
In discrete dynamical systems, the trajectory generated by repeated iteration is often called an orbit.
If
\[ x_{n+1}=f(x_n), \]then the orbit beginning at \(x_0\) is
\[ \boxed{ x_0,\ f(x_0),\ f^2(x_0),\ f^3(x_0),\ldots }. \]Here \(f^2\) means composition of \(f\) with itself, not squaring the function value.
Example: A Discrete Orbit
Let
\[ f(x)=x+2 \]and
\[ x_0=1. \]Then
\[ x_1=3,\qquad x_2=5,\qquad x_3=7. \]The orbit begins
\[ \boxed{1,\ 3,\ 5,\ 7,\ldots}. \]Two-Dimensional Discrete Trajectories
Suppose
\[ \mathbf{x}_{n+1} = \begin{pmatrix} 0.5 & 0\\ 0 & 0.25 \end{pmatrix} \mathbf{x}_n \]with
\[ \mathbf{x}_0= \begin{pmatrix} 4\\ 8 \end{pmatrix}. \]Then
\[ \mathbf{x}_1= \begin{pmatrix} 2\\ 2 \end{pmatrix}, \] \[ \mathbf{x}_2= \begin{pmatrix} 1\\ 0.5 \end{pmatrix}. \]Each vector is a point in the plane, so the orbit is a sequence of points moving toward the origin.
Continuous Trajectories
In continuous time, a trajectory is not merely a collection of isolated states.
The state changes through every intermediate time.
If
\[ \frac{dx}{dt}=-x \]and
\[ x(0)=4, \]then
\[ \boxed{x(t)=4e^{-t}}. \]The state continuously approaches zero as \(t\) increases.
A Parametric Curve in State Space
Suppose a two-dimensional state is
\[ \mathbf{x}(t)= \begin{pmatrix} x(t)\\ y(t) \end{pmatrix}. \]As \(t\) changes, the point
\[ (x(t),y(t)) \]traces a parametric curve in the state plane.
This curve is the system's trajectory in state space.
Example: Circular Trajectory
Consider
\[ x(t)=\cos t, \qquad y(t)=\sin t. \]Since
\[ x^2+y^2=1, \]the trajectory lies on the unit circle.
As \(t\) increases, the state moves counterclockwise around the circle.
The equation \(x^2+y^2=1\) describes the geometric path, while the parametric equations also describe the direction and timing of the motion.
Path versus Time Parameterization
Two trajectories can trace the same geometric curve while moving along it differently.
For example,
\[ x(t)=\cos t,\qquad y(t)=\sin t \]and
\[ x(t)=\cos(2t),\qquad y(t)=\sin(2t) \]both trace the unit circle.
The second system travels around the circle twice as quickly with respect to the parameter \(t\).
State Space Removes Explicit Time from the Picture
A standard graph of \(x(t)\) places time on one axis.
A state-space graph instead places state variables on the axes.
For example, a graph of
\[ x(t) \]might use \(t\) horizontally and \(x\) vertically.
A state-space plot of a mechanical system might instead use position \(x\) horizontally and velocity \(v\) vertically.
Time is then represented implicitly by movement along the trajectory.
Time-Series Plot versus State-Space Plot
| Feature | Time-Series Plot | State-Space Plot |
|---|---|---|
| Horizontal axis | Usually time | A state variable |
| Vertical axis | A state variable | Another state variable |
| Main purpose | Show how a variable changes with time | Show how the full state moves |
| Direction of time | Explicit | Shown by arrows or ordering |
State Space for Position and Velocity
For a moving object, position alone may not determine what happens next.
A useful state is
\[ \mathbf{x}(t)= \begin{pmatrix} x(t)\\ v(t) \end{pmatrix}. \]The corresponding state space has coordinates
\[ (x,v). \]One point tells us both where the object is and how it is moving.
Example: Harmonic Motion in State Space
Consider
\[ \frac{dx}{dt}=v, \] \[ \frac{dv}{dt}=-x. \]One solution is
\[ x(t)=A\cos t, \] \[ v(t)=-A\sin t. \]Therefore,
\[ x^2+v^2=A^2. \]The trajectory is a circle in the \((x,v)\)-state space.
Different Initial States Produce Different Trajectories
In the harmonic system,
\[ x^2+v^2=A^2, \]the value of \(A\) depends on the initial state.
A larger initial displacement or velocity produces a larger circle in state space.
The evolution law remains unchanged, but the initial condition selects which trajectory is followed.
A Family of Trajectories
A dynamical system usually has many possible trajectories.
Each initial state selects one of them.
The full collection of trajectories reveals the structure of the flow through state space.
In two dimensions, this collection is closely related to a phase portrait, which we will study in a later lesson.
Velocity in State Space
For a continuous autonomous system
\[ \frac{d\mathbf{x}}{dt}=\mathbf{F}(\mathbf{x}), \]the vector
\[ \mathbf{F}(\mathbf{x}) \]tells us the instantaneous direction and rate of motion from the state \(\mathbf{x}\).
Thus the differential equation defines a vector field over the state space.
Example: A Simple Vector Field
Consider
\[ \frac{dx}{dt}=x, \qquad \frac{dy}{dt}=-y. \]At the state
\[ (x,y)=(2,3), \]the state-space velocity is
\[ \left( \frac{dx}{dt}, \frac{dy}{dt} \right) = (2,-3). \]The system is moving toward increasing \(x\) and decreasing \(y\) at that instant.
The Evolution Rule Acts Everywhere in State Space
A dynamical law is not merely a rule for one particular trajectory.
It tells us how the system would evolve from every allowed state.
For
\[ \frac{d\mathbf{x}}{dt}=\mathbf{F}(\mathbf{x}), \]each point \(\mathbf{x}\) in the state space is assigned a velocity vector.
The trajectories follow this field.
Invariant Regions
An invariant region is a region of state space that trajectories cannot leave once they begin inside it.
For example, a population model may preserve
\[ P(t)\ge0. \]If every trajectory beginning with a nonnegative population remains nonnegative, then the nonnegative region is invariant under the model.
Why Invariant Regions Matter
Invariant regions help us determine which parts of the mathematical state space are dynamically relevant.
They can represent:
- physically possible states;
- biologically meaningful states;
- bounded operating regions;
- conserved constraints;
- safe regions in control systems.
They are especially useful when exact trajectory formulas are unavailable.
Trajectories and Conservation Laws
A conserved quantity can restrict trajectories to particular subsets of the state space.
Suppose
\[ H(x,y)=C \]remains constant along every trajectory.
Then a trajectory that begins on the level set
\[ H(x,y)=C \]remains on that level set.
Example: Conserved Radius
For the system
\[ \frac{dx}{dt}=y, \qquad \frac{dy}{dt}=-x, \]consider
\[ H(x,y)=x^2+y^2. \]Differentiate along a trajectory:
\[ \frac{dH}{dt} = 2x\frac{dx}{dt} + 2y\frac{dy}{dt}. \]Substitute the system:
\[ \frac{dH}{dt} = 2xy+2y(-x)=0. \]Therefore \(x^2+y^2\) remains constant.
Trajectories Can Approach a Point
Consider
\[ \frac{dx}{dt}=-x, \qquad \frac{dy}{dt}=-y. \]The solutions are
\[ x(t)=x_0e^{-t}, \qquad y(t)=y_0e^{-t}. \]As
\[ t\to\infty, \]both coordinates approach zero.
Every trajectory approaches the state
\[ (0,0). \]Trajectories Can Move Away from a Point
Consider instead
\[ \frac{dx}{dt}=x, \qquad \frac{dy}{dt}=y. \]Then
\[ x(t)=x_0e^t, \qquad y(t)=y_0e^t. \]Unless the system starts at the origin, the magnitude of the state grows exponentially.
Nearby trajectories move away from
\[ (0,0). \]This distinction previews the later study of stability and instability.
Trajectories Can Be Periodic
A trajectory is periodic if the state repeats after a fixed amount of time.
In continuous time,
\[ \mathbf{x}(t+T)=\mathbf{x}(t) \]for some
\[ T>0. \]The smallest positive such \(T\), when it exists, is the period.
Periodic trajectories often appear as closed curves in state space.
Periodic Orbits in Discrete Time
For a discrete map, a point may return to itself after several iterations.
If
\[ f^k(x)=x, \]then \(x\) belongs to a periodic orbit whose period divides \(k\).
A fixed point is the special case
\[ k=1. \]Example: A Two-Cycle
Consider the discrete map
\[ f(x)=1-x. \]Starting with
\[ x_0=0.2, \]we obtain
\[ x_1=0.8, \] \[ x_2=0.2. \]The orbit repeats:
\[ \boxed{0.2,\ 0.8,\ 0.2,\ 0.8,\ldots}. \]This is a period-two orbit.
Trajectories Can Be Bounded
A trajectory is bounded if it remains within some finite region of the state space.
The circular trajectory
\[ x^2+y^2=A^2 \]is bounded because the state always remains a distance \(A\) from the origin.
Boundedness is often an important first question in long-term analysis.
Trajectories Can Be Unbounded
For
\[ \frac{dx}{dt}=x, \]a positive initial state gives
\[ x(t)=x_0e^t. \]Since
\[ x(t)\to\infty, \]the trajectory is unbounded in forward time.
Forward and Backward Time
Dynamical systems are often studied in the direction of increasing time, called forward time.
When the equations permit it, we can also ask how trajectories behave as time decreases.
A point that attracts trajectories in forward time may repel them in backward time.
This distinction becomes important in stability and invariant-manifold analysis.
Can Trajectories Cross?
For many ordinary differential equations satisfying the usual uniqueness conditions, two distinct solution trajectories cannot cross at the same state and time direction.
If they did, the same state would have two different future evolutions, contradicting uniqueness.
This gives phase portraits an organized geometric structure.
Discrete Orbits Can Share States
In a discrete system, two different initial states may eventually arrive at the same later state.
Once they do, their future evolution is identical under a deterministic update rule.
For example, if
\[ f(a)=c \qquad\text{and}\qquad f(b)=c, \]then both orbits have the same future after reaching \(c\).
The Geometry of Long-Term Behavior
State-space analysis allows long-term behavior to be described geometrically.
A trajectory may:
- approach a point;
- approach a closed curve;
- remain on a conserved level set;
- escape to infinity;
- move among several regions;
- approach a more complicated invariant set.
These possibilities lead naturally to equilibria, periodic orbits, attractors, and chaos.
State Spaces in Population Models
Suppose \(R(t)\) and \(F(t)\) represent two interacting populations.
The state is
\[ \mathbf{x}(t)= \begin{pmatrix} R(t)\\ F(t) \end{pmatrix}. \]A point
\[ (R,F)=(1000,50) \]represents one simultaneous population configuration.
A trajectory records how both populations change together.
State Spaces in Economics
A macroeconomic dynamical model might use variables such as
\[ (K,D), \]where \(K\) is capital and \(D\) is debt.
The state-space trajectory then records how these quantities evolve jointly rather than examining either variable in isolation.
State Spaces in Control Systems
Control engineering often uses a state vector
\[ \mathbf{x}(t) \]together with an input
\[ \mathbf{u}(t). \]A common model is
\[ \frac{d\mathbf{x}}{dt} = A\mathbf{x}+B\mathbf{u}. \]The state-space representation makes it possible to analyze how inputs move the system from one state to another.
State Spaces in Orbital Motion
The position of an orbiting object is not enough by itself to determine the future orbit.
Velocity is also required.
A three-dimensional mechanical state may therefore be
\[ \mathbf{x} = \begin{pmatrix} x\\y\\z\\v_x\\v_y\\v_z \end{pmatrix}. \]The corresponding state space is six-dimensional.
Dimension of a State Space
The dimension of the state space is determined by the number of independent quantities required to specify a state.
One scalar state variable gives dimension one.
Two independent state variables give dimension two.
A mechanical system involving three spatial coordinates and three velocity coordinates has a six-dimensional state space.
Second-Order Equations and State-Space Dimension
A second-order differential equation can often be rewritten as a first-order system with two state variables.
Consider
\[ \frac{d^2x}{dt^2}=-x. \]Define
\[ v=\frac{dx}{dt}. \]Then
\[ \frac{dx}{dt}=v, \qquad \frac{dv}{dt}=-x. \]The natural state is therefore \((x,v)\).
Why First-Order Form Is Important
State-space methods typically represent continuous systems as first-order equations:
\[ \boxed{ \frac{d\mathbf{x}}{dt} = \mathbf{F}(\mathbf{x},t) }. \]Higher-order equations can usually be converted into this form by introducing additional state variables.
This provides a unified language for dynamical systems of many kinds.
State-Space Representation of a Discrete System
A multidimensional discrete system can be written
\[ \boxed{ \mathbf{x}_{n+1} = \mathbf{F}(\mathbf{x}_n) }. \]Each iteration maps one point of the state space to another.
The resulting sequence of points forms the orbit.
State-Space Representation of a Continuous System
A multidimensional continuous system can be written
\[ \boxed{ \frac{d\mathbf{x}}{dt} = \mathbf{F}(\mathbf{x}) }. \]The vector field \(\mathbf{F}\) specifies the instantaneous state-space velocity at every state.
Integrating the system produces trajectories through state space.
Trajectory Data
A trajectory may be represented numerically as a table.
| \(n\) | \(x_n\) | \(y_n\) |
|---|---|---|
| 0 | 4 | 8 |
| 1 | 2 | 2 |
| 2 | 1 | 0.5 |
The same data can be plotted as the state-space points
\[ (4,8),\quad(2,2),\quad(1,0.5). \]Numerical Trajectories
Many dynamical systems do not have convenient closed-form solutions.
In those cases, numerical methods can approximate the trajectory.
For
\[ \frac{d\mathbf{x}}{dt} = \mathbf{F}(\mathbf{x}), \]Euler's method gives
\[ \boxed{ \mathbf{x}_{n+1} = \mathbf{x}_n + \Delta t\,\mathbf{F}(\mathbf{x}_n) }. \]The approximation itself becomes a discrete trajectory through the state space.
Common Error: Confusing State Space with the Graph of a Function
A state-space plot does not necessarily show a function
\[ y=f(x). \]Its axes represent components of the state.
A trajectory may loop, circle, or have geometric features that would fail the vertical-line test.
This causes no problem because a state-space trajectory is not required to represent \(y\) as a single-valued function of \(x\).
Common Error: Putting Time on a State-Space Axis Automatically
In a time-series graph, time is usually an explicit axis.
In state space, the axes normally represent state variables.
For a position-velocity system, the axes are
\[ x \qquad\text{and}\qquad v, \]while time determines the direction in which the trajectory is followed.
Common Error: Omitting Necessary State Variables
A chosen state must contain enough information to determine future evolution under the model.
For a second-order mechanical system, position alone is generally insufficient.
Two objects can occupy the same position while having different velocities and therefore different futures.
Including velocity resolves the ambiguity.
Common Error: Treating Parameters as Coordinates
A parameter may influence every trajectory without being part of the state space used for a single fixed model.
In
\[ x_{n+1}=rx_n(1-x_n), \]\(x_n\) is the evolving state.
If \(r\) is fixed, it is not another state coordinate.
We may later build parameter diagrams that display \(r\), but that is a different representation.
Common Error: Ignoring Direction Along a Trajectory
A geometric curve alone does not completely describe the dynamics.
Two systems could trace the same curve in opposite directions.
Direction arrows, time labels, or the underlying equations are needed to distinguish the dynamics.
A Reliable Procedure for Building a State Space
- Identify the variables needed to determine future evolution.
- Collect those variables into a state vector.
- Determine the allowed values of each variable.
- Apply physical or mathematical constraints.
- Determine the dimension of the resulting state space.
- Write the evolution rule in terms of the state variables.
- Specify an initial state.
- Generate or solve for the resulting trajectory.
- Track the direction of time along that trajectory.
- Study its long-term geometric behavior.
A Reliable Procedure for Reading a Trajectory
- Identify the coordinates represented by the axes.
- Locate the initial state.
- Determine the direction of increasing time.
- Observe whether the trajectory approaches or leaves particular states.
- Check whether it remains bounded.
- Look for repeated or closed motion.
- Identify regions the trajectory cannot leave.
- Compare nearby trajectories when studying stability.
- Relate the geometric behavior back to the original model.
Practice 1: Identify the State Space
A model uses one population variable
\[ P\ge0. \]Describe a natural state space.
Practice 2: Two-State System
A model tracks two nonnegative populations \(x\) and \(y\).
Write the natural state space using set notation.
Practice 3: Discrete Trajectory
Let
\[ x_{n+1}=2x_n \]with
\[ x_0=1. \]List the first five states.
Practice 4: Two-Dimensional Orbit
Suppose
\[ x_{n+1}=\frac12x_n, \qquad y_{n+1}=\frac12y_n \]with
\[ (x_0,y_0)=(8,4). \]Find the next three states.
Practice 5: Position and Velocity
A moving object is modeled by the second-order equation
\[ \frac{d^2x}{dt^2}=-4x. \]Introduce a velocity variable and rewrite the equation as a first-order system.
Practice 6: State-Space Velocity
Consider
\[ \frac{dx}{dt}=2x, \qquad \frac{dy}{dt}=-3y. \]Find the state-space velocity at
\[ (x,y)=(1,2). \]Practice 7: Periodic Trajectory
Suppose
\[ x(t)=2\cos t, \qquad y(t)=2\sin t. \]Find an equation for the geometric trajectory in the \(xy\)-plane.
Practice 8: Long-Term Behavior
Consider
\[ x(t)=5e^{-t}, \qquad y(t)=-2e^{-t}. \]Determine the state approached as
\[ t\to\infty. \]Practice 9: State Variable or Parameter?
In
\[ x_{n+1}=ax_n, \]suppose \(a\) remains fixed while \(x_n\) changes with \(n\).
Identify the state variable and the parameter.
Practice 10: Interpret a State
A mechanical state is
\[ (x,v)=(3,-5). \]Interpret the two coordinates.
Practice Answers
Practice 1
Since only nonnegative populations are allowed, a natural state space is
\[ \boxed{[0,\infty)}. \]Practice 2
Both coordinates must be nonnegative:
\[ \boxed{ \{(x,y)\in\mathbb{R}^2:x\ge0,\ y\ge0\} }. \]Practice 3
Starting with \(x_0=1\),
\[ x_1=2,\qquad x_2=4,\qquad x_3=8,\qquad x_4=16. \]Therefore the first five states are
\[ \boxed{1,\ 2,\ 4,\ 8,\ 16}. \]Practice 4
Starting at
\[ (8,4), \]we obtain
\[ (x_1,y_1)=(4,2), \] \[ (x_2,y_2)=(2,1), \] \[ (x_3,y_3)= \left(1,\frac12\right). \]Practice 5
Define
\[ v=\frac{dx}{dt}. \]Then
\[ \boxed{ \begin{aligned} \frac{dx}{dt}&=v,\\ \frac{dv}{dt}&=-4x. \end{aligned} } \]Practice 6
At \((1,2)\),
\[ \frac{dx}{dt}=2(1)=2, \] \[ \frac{dy}{dt}=-3(2)=-6. \]Therefore the state-space velocity is
\[ \boxed{(2,-6)}. \]Practice 7
Since
\[ x^2=4\cos^2t \]and
\[ y^2=4\sin^2t, \]adding gives
\[ \boxed{x^2+y^2=4}. \]The trajectory is a circle of radius \(2\).
Practice 8
Since
\[ e^{-t}\to0, \]both coordinates approach zero:
\[ \boxed{(x(t),y(t))\to(0,0)}. \]Practice 9
\(x_n\) changes with the iteration, so it is the state variable.
\(a\) remains fixed and controls the update law, so it is the parameter.
Practice 10
The coordinate
\[ x=3 \]says that the object is at position \(3\), while
\[ v=-5 \]says that its velocity is \(5\) units per unit time in the negative direction.
What to Remember
A state contains the variables needed to determine the modeled future of a system.
A state space is the set of all states allowed by the model.
A trajectory records the system's evolution through that state space.
\[ \boxed{ \text{initial state} \longrightarrow \text{trajectory through state space} } \]- one state variable often gives a one-dimensional state space;
- two state variables often give a two-dimensional state space;
- higher-dimensional systems use state vectors;
- constraints may restrict the allowable region;
- discrete trajectories are sequences of states;
- continuous trajectories are paths through state space;
- direction along a trajectory represents increasing time;
- families of trajectories reveal the qualitative structure of a system.
The state-space viewpoint transforms the study of change from a collection of formulas into a geometry of evolving states.
Where We Go Next
Once states and trajectories are represented geometrically, one question becomes especially important: are there states that do not move at all?
In a discrete system, such a state maps back to itself. In a continuous system, its state-space velocity is zero.
The next lesson, Fixed Points and Equilibria, develops these stationary states, shows how to find them, and explains why they organize much of the long-term behavior of dynamical systems.