Note

If you come across any errors, please let me know. I will be happy to fix it.
Happy reading!! :smiley:

References

• Andrew Ng, Machine Learning, Coursera. [coursera][youtube]

Fixed point

Just to reiterate, if the ordinary differential equation (ODE) in $\eqref{eq:1}$ represents a dynamical system:

Fixed points of this system are given by the roots of the equation $\eqref{eq:2}$:

Fixed points of Multi-dimensional system

My previous post only explained the definition of fixed point and provided an example with a scalar-valued dynamical system. Now, lets discuss a case of multi-dimensional ODE.

We will start with the system given by equation $\eqref{eq:3}$:

where $\mathbf{f}$ is a vector-valued function, $\mathbf{x}$ and $\mathbf{\dot x}$ are $n$-dimensional vectors:

We find the fixed points (a.k.a. equilibrium states) of the system by following $\eqref{eq:2}$:

The roots of $\eqref{eq:5}$ will give us the value of $\mathbf{x_{eq}}$, i.e., fixed points of our multi-dimensional system.

Stable and Unstable Fixed Points

We analyzed the system in a one-dimensional case (here) using a small perturbation $\delta$ at the equilibrium condition of the system. We will follow the similar procedure here as well.

We evaluated $\mathbf{f}^{\prime}\mathbf{(x)}$ at $\mathbf{x_{eq}}$ to see if our fixed point is stable or unstable. In case of one-dimensional system, it was easy since $f^{\prime}(x_{eq})>0$ is unstable fixed point $x_{eq}$ while it is stable when $f^{\prime}(x_{eq})<0$. In case of high-dimensional system, we cannot do this.

To analyze the behavior of our $n$-dimensional system at $\mathbf{x_{eq}}$, we will introduce the perturbation $\mathbf{\delta x}$ at $\mathbf{x_{eq}}$. Thus, we end up with the following:

Using Taylor expansion on $\eqref{eq:6}$:

But, we know at fixed points, equation $\eqref{eq:5}$ holds and thus, $\eqref{eq:7}$ reduces to $\eqref{eq:8}$.

We can ignore the higher order terms $\mathbf{H.O.T.}$ for values of $\mathbf{\delta{x}}$ close to $\mathbf{0}$, resulting in equation $\eqref{eq:9}$.

$\mathbf{f}^{\prime}\mathbf{(x)}$ is the Jacobian of $\mathbf{f(x)}$ at $\mathbf{x_{eq}}$, i.e., a linear approximation of our dynamical system $\mathbf{f(x)}$ near $\mathbf{x_{eq}}$ (you can refer this for further details on Jacobian).

Using this Jacobian, equation $\eqref{eq:12}$, at our fixed point $\mathbf{x_{eq}}$ for the dynamical system under consideration, we can calculate its eigenvalues and interpret the results of the fixed point.

Therefore, we find the eigenvalues for equation $\eqref{eq:13}$,

Here, $\lambda$ denotes the eigenvalue of the system. The roots of $\eqref{eq:13}$ are the eigenvalues the dynamical system at the fixed point $\mathbf{x}=\mathbf{x_{eq}}$.

Eigenvalue interpretation

For a continuous-time nonlinear dynamical system given by $\eqref{eq:3}$, the eigenvalues $\lambda$ that are found as roots of equation $\eqref{eq:13}$ can be interpreted as:

• If any of the eigenvalues have a real part $Re(\lambda)>0$: $\mathbf{x_{eq}}$ is an unstable fixed point.
• If all $Re(\lambda)<0$: $\mathbf{x_{eq}}$ is a stable fixed point.
• If $\lambda=0$: $\mathbf{x_{eq}}$ is a neutral fixed point.
• If eigenvalues $\lambda$ are complex conjugates, i.e., $Im(\lambda) \ne 0$: The dynamical system has oscillatory behavior around the fixed point.

Important points to note regarding this article

In this post, we discussed a general case of interpreting the fixed points of a dynamical system. By general, I mean $\mathbf{f(x)}$ is a non-linear, continuous-time vector-valued function representing a dynamical system. Below are certain points one should note about any non-linear dynamical system:

• We assumed that the system is non-linear and linearized it using Taylor series expansion near its fixed point (a.k.a. equilibrium).
• We evaluated the stability of a fixed point near the equilibrium condition by perturbing the system ($\mathbf{x_{eq}}+\mathbf{\delta x}$).
• This approach of interpreting the stability of the system by linearizing it near the equilibrium does not tell much about a system’s asymptotic behavior at large.
  • We only understand how the system behaves locally or in the neighborhood of the fixed points.
• In practical or real-world systems, it may not be possible to interpret the global stability characteristics of the system. Thus, the stability analysis around the neighborhood of the fixed point is useful for many practical applications such as sustaining a non-linear system’s state near or at the fixed point.
• In general, global asymptotic behaviors of any non-linear dynamical system can be complex and there are no systematic methods to predict and analyze such behaviors.

References and Further Readings:

• Deshpande, A. M. Stablility of Fixed Point of a Dynamical System. [web]
• Strogatz, Steven H. Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering. CRC press, 2018.
• Khalil, Hassan K. “Lyapunov stability.” Control Systems, Robotics and AutomatioN–Volume XII: Nonlinear, Distributed, and Time Delay Systems-I (2009): 115.
• Bomze, Immanuel M., and Jörgen W. Weibull. “Does neutral stability imply Lyapunov stability?.” Games and Economic Behavior 11.2 (1995): 173-192.
• Fixed point. [web]
• Jacobian matrix [video]
• Stability Theory. [web]
• Lyapunov Stability. [web]

Vector Space

A vector space is a collection of objects called vectors, which may be added together and multiplied/scaled by scalars. Scalars are often taken to be real numbers.

\(\mathbf{x}\) is a vector of \(n\) dimensions.

Function

A function is relationship between two sets. It associates an element from frist set to exactly one element of second set.

In equation \eqref{eq1}, the set represented by \(\mathit{X}\) is called domain of function \(f\) and \(\mathit{Y}\) is called codomain of function \(f\). This notation can be read as ‘the function \(f\) mapping elements of set \(\mathit{X}\) to elements of set \(\mathit{Y}\)’. Similarly, \eqref{eq2} can be read as ‘\(f\) maps \(\mathbf{x}\) to \(f(\mathbf{x})\)’.

Scalar-valued function or Scalar field

The function which maps a vector to a scalar value.

Equation \eqref{eq3} maps an \(n\)-dimensional vector to a scalar value. It is a scalar-valued function. \(y\) is a scalar and \(\mathbf{x}\) is a vector of \(n\)-dimensions in \eqref{scalar-func}.

Vector-valued function

A vector-valued function maps one vector space to another vector space.

Equation \eqref{eq4} maps an \(n\)-dimensional vector to a vector of \(m\)-dimensions. It is a vector-valued function. The output value \(\mathbf{y}\) in \eqref{eq5} is of \(m\)-dimensions and the corresponding input value \(\mathbf{x}\) is of \(n\)-dimensions.

Gradient

The gradient of a scalar-valued differentiable function is also referred to as vector field or vector-valued function.

Gradient of vector-valued function at a point \(\mathbf{x}\) in domain \(\mathit{X} \in \Re^n\) is:

At each point of a scalar-valued function, a gradient is a tangent vector representing an infinitesimal change in vector input. Notice that here a column vector is used to represent the gradient of the function at point \(x\).

Derivative

Derivative at each point of the scalar-valued function is a co-tangent vector, a linear form that expresses how much the scalar output of a function changes for a given infinitesimal change in the input vector. Notice, we represent the derivative of a scalar-valued function as a row vector. This is unlike the gradient vector (that used column vector).

\(\label{eq-derivative1}\)

Note: derivative is just a transpose of gradient.

Linear approximation of a scalar-valued function

Linear approximation of a function \(f(\mathbf{x})\) at a point \(\mathbf{x_{0}} \in \Re^{n}\):

Jacobian (a.k.a. derivative of Vector-valued Function)

Derivative of \(\mathbf{f}\) in equation \eqref{eq5} linearly maps tangent space \(T_{\mathbf(x)}\) to tanget space \(T_{\mathbf(y)}\).

First order-partial derivative of vector-valued function forms the Jacobian matrix. We will denote the Jacobian by the notation \(\mathbf{J}\)

Note: Jacobian has the dimensions of \(m \times n\).

A Jacobian is the vertical stack of derivative vectors corresponding to each output element of vector \(\mathbf{y}\) (i.e., row of the Jacobian matrix). This definition makes sense, and we can relate it to the derivative of a scalar-valued function defined above. The derivative of a scalar-valued function, \(m=1\), is a row vector.

Hessian matrix

Hessian matrix for scalar-valued function or scalar field given by \eqref{scalar-func} is a square matrix of second order partial derivatives of this scalar-valued function.

Rotation

Rotational matrices are special orthogonal matrices. I am not going to discuss any property of these matrices over here. But this post is a quick reference for rotation using z-y-x Euler angles. For further details, you can refer to this.

Euler Angle Transformation

The most important thing you must remember before reading further about transformations using Euler angles is:
The order of matrix multiplication of rotational matrices is of extreme importance.

1. Euler angles are used to specify the orientation of one reference frame relative to another reference frame.
2. Euler angles are specified by the three angles, viz., $\psi$, $\theta$, $\phi$.
3. Euler angles represent three consecutive rotations in the order of $\psi$, $\theta$, $\phi$ so that one coordinate axes system is made to coincide with another system.
4. Again, the order of rotation, i.e., $\psi$, $\theta$, $\phi$, is very very important.
5. Angle $\psi$ - Angle to be rotated about the current frame's Z-axis.
6. Angle $\theta$ - Angle to be rotated about the current frame's Y-axis.
7. Angle $\phi$ - Angle to be rotated about the current frame's X-axis.

Sequence of rotation of three Euler angles:

Let’s say one has to go from frame 1 to frame 2 using Euler angle \(\psi\), \(\theta\), \(\phi\). Lets start with the current frame, i.e., frame 1.

1. Original Frame: $Ox_{1}y_{1}z_{1}$
2. Target Frame: $Ox_{2}y_{2}z_{2}$
3. Angle $\psi$ is the rotation about $Oz_{1}$.
   • This transformation takes frame $Ox_{1}y_{1}z_{1}$ to frame $Ox_{1}^{\prime}y_{1}^{\prime}z_{1}^{\prime}$
   • $Ox_{1}$ is rotated to $Ox_{1}^{\prime}$
   • $Oy_{1}$ is rotated to $Oy_{1}^{\prime}$
   • $Oz_{1}$ is rotated to $Oz_{1}^{\prime}$. The axes $Oz_{1}$ and $Oz_{1}^{\prime}$ overlap with each other.
   • One should note, $Oy_{1}^{\prime}$ now falls in the plane $Oy_{2}z_{2}$ and $Ox_{1}^{\prime}$ now falls in the plane $Ox_{2}z_{2}$. These two planes are in the target frame of reference.
4. Angle $\theta$ is rotation about $Oy_{1}^{\prime}$.
   • This transformation takes frame $Ox_{1}^{\prime}y_{1}^{\prime}z_{1}^{\prime}$ to frame $Ox_{1}^{\prime\prime}y_{1}^{\prime\prime}z_{1}^{\prime\prime}$.
   • $Ox_{1}^{\prime}$ is rotated to $Ox_{1}^{\prime\prime}$.
   • $Oy_{1}^{\prime}$ is rotated to $Oy_{1}^{\prime\prime}$. The axes $Oy_{1}^{\prime}$ and $Oy_{1}^{\prime\prime}$ overlap with each other.
   • $Oz_{1}^{\prime}$ is rotated to $Oz_{1}^{\prime\prime}$.
   • $Ox_{1}^{\prime\prime}$ now coincides with the axes $Ox_{2}$ and $Oz_{1}^{\prime\prime}$ now falls in the plane $Oy_{2}z_{2}$. The axis $Ox_{2}$ and the plane $Oy_{2}z_{2}$ are in the target frame of reference.
5. Angle $\phi$ is rotation about $Ox_{1}^{\prime\prime}$.
   • This transformation takes frame $Ox_{1}^{\prime\prime}y_{1}^{\prime\prime}z_{1}^{\prime\prime}$ to frame $Ox_{1}^{\prime\prime\prime}y_{1}^{\prime\prime\prime}z_{1}^{\prime\prime\prime}$. The resulting frame $Ox_{1}^{\prime\prime\prime}y_{1}^{\prime\prime\prime}z_{1}^{\prime\prime\prime}$ is also the target frame $Ox_{2}y_{2}z_{2}$.
   • $Ox_{1}^{\prime\prime}$ is rotated to $Ox_{1}^{\prime\prime\prime}$ ($Ox_{2}$).
   • $Oy_{1}^{\prime\prime}$ is rotated to $Oy_{1}^{\prime\prime\prime}$ ($Oy_{2}$).
   • $Oz_{1}^{\prime\prime}$ is rotated to $Oz_{1}^{\prime\prime\prime}$ ($Oz_{2}$).
   • $Oy_{1}^{\prime\prime\prime}$ now coincides with the axes $Oy_{2}$ and $Oz_{1}^{\prime\prime\prime}$ coincides with the axis $Oz_{2}$. The axes $Oy_{2}$ and $Oz_{2}$ are in the target frame of reference.

Constrains on three Euler angles:

Euler angle rotation using rotation matrices faces the issue of singularity or gimbal lock. These angle rotations are constraint to avoid ambiguities. The limits are as follows:

Rotational Transformations

Let there be two frames defined as start frame and target frame.

• Start Frame: \(Ox_{1}y_{1}z_{1}\)
• Target Frame: \(Ox_{2}y_{2}z_{2}\)
• A point \(p\) is defined in the start and target frames as \(p^{1}\) and \(p^{2}\) respectively.

Now, with the help of Euler angles, transform the point given in the start frame to the corresponding point in the target frame.

Note: Order of transformation is: (1) Angle \(\psi\) along \(Oz_{1}\); (2) Angle \(\theta\) along \(Oy_{1}^{\prime}\); (3) Angle \(\phi\) along \(Ox_{1}^{\prime\prime}\).

• Rotation by \(\psi\) along \(Oz_{1}\): This transforms point \(p^{1}\) in frame \(Ox_{1}y_{1}z_{1}\) to point \(p^{1^{\prime}}\) in frame $Ox_{1}^{\prime}y_{1}^{\prime}z_{1}^{\prime}$.

• The above transformation is followed by \(\theta\) around \(Oy_{1}^{\prime}\) and then by \(\phi\) around \(Ox_{1}^{\prime\prime}\).

Therefore, the resultant transformation matrix for going from Frame \(Ox_{1}y_{1}z_{1}\) to Frame \(Ox_{2}y_{2}z_{2}\) is:

Code

The code for euler angle transformation is available on Google Colab:

You can also find the corresponding notebook on github:

References

• Pamadi, B. N. (2004). Performance, stability, dynamics, and control of airplanes. American Institute of aeronautics and astronautics.

Fixed Point

Consider a dynamical system given by the following ordinary differential equation (ODE):

A fixed point of this system is given by:

Therefore, \(f(x) = 0\) or roots of the function \(f(x)\) form the fixed points of the dynamical system.

Stable and Unstable Fixed Points

In layman’s terms, you can say the following about stable and unstable fixed points.

Stable Fixed Point: Put a system to an initial value that is “close” to its fixed point. The trajectory of the solution of the differential equation \(\dot x = f(x)\) will stay close to this fixed point.

Unstable Fixed Point: Again, start the system with initial value “close” to its fixed point. If the fixed point is unstable, there exists a solution that starts at this initial value but the trajectory of the solution will move away from this fixed point.

In other words, one can also think of a stable fixed point as the attractor and unstable fixed point as the repeller. A particle governed by \(\dot x = f(x)\), a stable fixed point will force this particle to move towards itself. On the other hand, an unstable fixed point will force a particle away from itself.

Mathematical Intuition:

For a dynamical system, \(\dot x = f(x)\), a fixed point is \(f(x) = 0\).

If \(f^{\prime}(x) \gt 0\), we have magnitude of \(f(x)\) increasing at x. This can be represented as \(f(x - \delta) \lt 0 \lt f(x + \delta)\) for a sufficiently small value of \(\delta \gt 0\).

Thus, if we start from \(x+\delta\) which is “close” \(x\), the ODE from equation (1) will keep on increasing the value of $x$. And it we start from \(x-\delta\) which is “close” \(x\), equation (1) will move the particle way from \(x\) by decreasing the value of \(x\).

Therefore, if \(f^{\prime} (x) \gt 0\), we have an unstable fixed point and vice versa.

Note: The condition of \(f^{\prime} (x) \lt 0\) or \(f^{\prime} (x) \gt 0\) are sufficient conditions to guarantee fixed point stability or unstability respectively. These are not the necessary conditions, i.e., it is possible to have stable and unstable fixed points where \(f^{\prime} (x) = 0\).

Intuitive Example:

For the differential equation \(x^{\prime} = sin(x)\):

Using linear stability analysis, fixed points occurs when \(f(x)=sin(x)=0\) or \(x=kπ\) where \(k\) is integer.

\(f^{\prime}(x)=cos(kπ)=1\) if \(k\) is even. and \(f^{\prime}(x)=cos(kπ)=−1\) if \(k\) is odd.

Therefore, \(x\) is unstable when \(k\) is even, and stable if \(k\) is odd.

High Dimensional Dynamical Systems

This post discussed the definition of a fixed point of a dynamical system. A simple one-dimensional dynamical system is used as an illustration to explain the concept. A more detailed discussion on general nonlinear, continuous-time, multi-dimensional dynamical systems and their fixed points is provided in my next post.

Reference and Further Readings:

• Strogatz, S. H. (2018). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC press. [book]
• Deshpande, A. M. (2021). Stability of Fixed Points of High Dimensional Dynamical Systems.[web]

Let there be two coordinate frames in a 3-dimensional space, \(A\) and \(B\). There is a vector \(r^{A}\) defined in Frame \(A\) as shown in figure below.

Using Quaternions we can rotate from frame \(A\) to \(B\) by angle \(\theta\) around vector \(r^{A}\).

Quaternion describing this orientation is \(q^{A}_{B}\):

\(q^{A}_{B}\) describes the orientation of frame \(B\) relative to frame \(A\).

Quaternion arithmetic requires the quaternion describing orientation to be of unit length.
Therefore, quaternions are first normalized to have a magnitude of 1.

Conjugate quaternion:

Compound orientations using quaternions: Let \(q^{A}_{B}\) and \(q^{B}_{C}\) be two quaternions. \(q^{A}_{B}\) - orientation of \(B\) w.r.t. \(A\).
\(q^{B}_{C}\) - orientation of \(C\) w.r.t. \(B\).

The compound orientation \(q^{A}_{C}\) is defined as:

\(q^{A}_{C}\) - orientation of \(C\) w.r.t. \(A\).
\(\bigotimes\) represents quaternion product.

Quaternion Product:
Let \(p\) and \(q\) be two quaternions. Then the product of these two quaternions is:

Quaternion product is not commutative, i.e., \(p \bigotimes q \ne q \bigotimes p\).

NOTE: I know there may be a lot of confusion in Quaternion mathematics. The blogpost by Fan Zheng provides an excellent overview of conventions used in Quaternion representations. [2]

References

1. Madgwick, S. (2010). An efficient orientation filter for inertial and inertial/magnetic sensor arrays. Report x-io and University of Bristol (UK), 25, 113-118. [link]
2. Zhang, F. (2017). Quaternion Conventions: Hamilton and JPL. [link]

Let us start thinking about it in terms of mathematics. A random variable holds the outcome of a random experiment. For example, several people standing in front of you right now. This number can be anything from the set of natural numbers \(N \equiv \{0, 1, 2, 3, ...\}\) at the instance of observation. But once you observe the true value, this random experiment is over. Let’s say, you observed 3 people standing in from of you. Let’s complicate our experiment. What is the number of people standing in front of you over time in a day? Now, the random variable which holds a value from set \(N\) will take different values at each instance of observation. You will need to observe multiple instances for the entire day. Not just that, the set of observations you make will vary each day. It will not be the same. If you count the people standing in front of you for each instance of today, it will just be a single realization of all the possibilities of observations on different days. To understand this stochasticity over time, one needs to understand the Random Processes.

The rest of the page tries to formalizes the idea of Random Processes.

Random Experiment

An experiment whose outcome cannot be predicted with certainty is called a Random Experiment.

Examples:

1. The lifetime of a light bulb.
2. Rolling a dice.
3. A coin toss.

Random Variable

Example:
A coin toss experiment may result in two outcomes, viz., Heads (\(H\)) or Tails (\(T\)). Mathematically speaking the sample space of this experiment can be defined as \(\{H, T\}\). We can assign a numerical value to these possible outcomes of the sample space. Let’s say, we call getting \(H\) in the coin toss as \(1\) and getting \(T\) as \(0\). Therefore, we can represent the sample space in terms of numerical value as \(\{1, 0\}\).

One should notice that assigning a value of \(1\) to \(H\) is a deterministic function, i.e., whenever one gets an outcome of the coin toss as \(H\), we will always assign it a value of \(1\).

Let \(X\) be the random variable for the random experiment of tossing a coin.
Here, \(X\) is a deterministic function that assigns a real-value to the outcome of an experiment of tossing a coin. Thus, we can write it as follows:
\(X(H) = 1\) and \(X(T)=0\)
\(X(\xi)=x\) where \(x\in\{0, 1\}\) and \(\xi\in\{T, H\}\). Note, how \(\xi\) is a parameter to the function \(X\) and the outcome is the value \(x\).

The random variable in the case of the above example of a random experiment can take only discrete values. Thus, \(X\) is called a discrete random variable. It is also possible for a random variable to take continuous values. For example, it the random experiment is measuring the height of an individual, the height is not a discrete number, it can take any value in the interval \([0, \infty)\).

Once, we observe the outcome of our experiment, there is no more randomness involved. For example, we observe a \(H\) in a coin toss and thus, we have the information of the exact state of the experiment. More formally, this is known as realization of the random variable. In this case, \(X(H)=1\) is one realization of the random variable \(X\) and \(X(T)=0\) is another.

Random Processes

Random Process depends on a random variable as well as an indexing variable.
In many stochastic processes, the indexing variable which will represent some index set will be related to a temporal dimension. For example, while one is observing a random experiment, the observations are done over an interval of time. Multiple observations are made at different time instances which may have been affected by randomness. This randomness does not affect only a single observation, but its effects are seen over the duration of the experiment. Random process can take the time of observation as an input parameter which will be represented as an indexing variable.

Definition:

The random process is an infinite indexed collection of random variables defined over a common probability space.
Random process: \(\{X(t): t\in T\}\)
This can be read as \(X(t)\) is the random variable at index \(t\) and \(t\) is drawn from an index set \(T\). The index set \(T\) can be discrete where it make take values as \(\{1, 2, 3, ...\}\). In continuous case, it may take value in a range, for example, the time of the experiment may fall in the interval \([5, 10]\). Here, the index parameter typically represents temporal domain. But this value can also represent indexing of a spatial domain.

A more intuitive way to understand this definition is as follows -
Random process: \(X(\xi, t)\) or \(X_{t}(\xi)\) or \(X_{t}\)
It is a function of the outcome of a random experiment \(\xi\) at index \(t\). The variable \(\xi\) is omitted and the indexing is represented as the subscript for convenience of presentation in \(X_{t}\). This page follows one of the conventions mentioned above unless otherwise stated.

Example of Random Process with temporal indexing:

1. Noise in an IMU (inertial measurement unit) sensor which is moving at a constant speed from point A to point B on a road.
   • This IMU is measuring acceleration, its outcome at anytime can be written as \(a_{x}(t) = X(t)\). Since the velocity is constant, the experiment should result in perfect zeros. But as the sensor is noisy, the value in the \(x\)-direction at time \(t\) maybe some non-zero random value.
2. The price of stock recorded every day.
   • Stock prices vary every day, and the variation if observed seems random.

Example of Random Process with spatial indexing:

1. Flipping of 4 unbiased coins at 4 different locations.
   • Coins are at 4 different locations. The indices representing these locations are \(T=\{1, 2, 3, 4\}\).
   • All the coins are unbiased and the sample space for each coin toss is \(\{H, T\}\). Each outcome is equally probable.
   • Random process is \(X(\xi, t)\) where \(\xi \in \{H, T\}\) and \(t \in \{1, 2, 3, 4\}\).

As explained before, random process \(X(\xi, t)\) can be understood as a function of two variables, \(\xi \in \Xi\) and \(t \in T\). Here, \(\Xi\) is the sample space of random experiment underlying this random process and \(T\) is the index set.