# Notes on equilibrium and stability of the dynamical systems.
#### Author: [Sharath](https://sharathraparthy.github.io/)
## What is a dynamical system?
It is any system that evolves and changes through time governed by a set of rules. Using dynamical systems we can study the long term behavior of an evolving system.
Formally, it is a triplet $(X, T, \phi)$ where $X$ denotes the state space, $T$ denotes the time space and $\phi: X \times T \rightarrow X$ is the flow (this is the rule that governs the evolution).
There are few properties of flow:
1. $\phi(X, 0) = X$
1. Principle of compositionality: $\phi(\phi(x, t), s) = \phi(x, t+s)$
There are two kinds of system evolution we can think of depending on the temporal domain; discrete or continuous.
**Discrete-time dynamical system:** A discrete-time dynamical system consists of a non-empty set $X$ and a map $f: X \rightarrow X$. For $n \in \mathbb{N}$, the $n^{th}$ iterate of f is the $n$-fold composition $f^n = f \odot \cdot \cdot \cdot \odot f$
**Continuous-time dynamical system:**
In this formulation, we have a density of possible times that is basically given by real numbers $\mathbb{R}$. The solutions are basically solid lines (as between any two points we can find a point or time $t \in \mathbb{R}$).
## Notion of equilibrium
For the rest of the notes, we will only consider the continuous time dynamics but however we can port the results to discrete case as well.
Let's consider the following ODE;
$$
\dot{x} = f(x), \hspace{5mm} x \in \mathbb{R^n}
$$
Solving this equation would give us a dynamical system whose flow is $\phi(x, t)$.
We will define a equilibrium solution or a fixed point as a "root" of the vector field $f(\tilde{x}) = 0$. This implies that the derivative at $\tilde{x}$ is zero which inturn means there is no motion.
## Stability of the system
There is a notion of stability when we talk about the dynamical systems. This in general says that "if we start close enough, we will stay close enough."
There is also a notion of *asymptotic stability* which is quite strong. The intuition is "if wet start close enough then you will be attracted to $\tilde{x}$.
Now we will generalize and formalize these ideas to general solutions $\tilde{x}(t)$.
**Lyapunov stability**: $\tilde{x}(t)$ is said to be stable (or lyapunov stable) if, given $\epsilon > 0$ there exists a $\delta = \delta(\epsilon) > 0$ such that for any other solution $y(t)$ satisfying $|\tilde{x}(t_0) - y(t_0)| < \delta$, then $|\tilde{x}(t) - y(t)| < \epsilon$ for $t > t_0 \in \mathbb{R}$
This definition basically says that if we start within the $\delta$ distance of the initial condition of $t$ at $t_0$ ($|\tilde{x}(t_0) - y(t_0)| < \delta$), then we will remain, throughout the $t > t_0$, in the $\epsilon$ tube or $\epsilon$ neighborhood ($|\tilde{x}(t) - y(t)| < \epsilon$) of the solution. This is shown in the following picture.
![](https://i.imgur.com/vngtn4r.png)
Now let's also try to define asymptotic stability formally.
**Asymptotic stability**: $\tilde{x}(t)$ is said to be asymptotically stable, if it is lyapunov stable and for any other solution $y(t)$, there exists a constant $b > 0$ such that $|\tilde{x}(t_0) - y(t_0)| < b$, then $\lim _{t \rightarrow \infty}|\tilde{x}(t) - y(t)| = 0$
![](https://i.imgur.com/t6yMGui.png)
This intuitively means that, if we start within the $b$ neighborhood, then in the limit the distance between $\tilde{x}(t)$ and $y(t)$ shrinks to $0$. Here, if we observe carefully, we require the solution to be lyapunov stable. The reason is that there are some weird solutions where we can go infinitely far away before actually converging asymptotically. These are called "homoclinic orbits" and for such cases it's difficult to guarantee the boundedness of the solution even though the condition $\lim _{t \rightarrow \infty}|\tilde{x}(t) - y(t)| = 0$ is true. And hence for asymptotic stability, we also require lyapunov stable solutions.
## Orbital stability v/s trajectory stability
In dynamical systems, when we talk about stability, we often talk either in terms of orbits or trajectories. The distinction is subtle but important. So, let's try to define these:
1. **Trajectory:** A trajectory is a solution to a differential equation and is "parameterized" by time $t$.
1. **Orbit:** An orbit is the set of all points that are visited by a trajectory. $O(x_0) = \{x \in X \mid x = x(x_0, t) \forall \in T \}$. This is not parameterized.
Whatever notion of stabilities discussed above are defined keeping trajectories in mind. This is because comparision is made at given time $t$ and we check if the norm $|\tilde{x}(t) - y(t)|$ either lies in the epsilon neighborhood or asymptotically converges to zero. The notion of orbital stability is widely used in the scenarios where the time is "stretched" or "squeezed". The notion of orbital stability is more applicable here are we remove the dependence on $t$ in this case. In other words, if we don't case about when we reach particular point in time, then orbital stability makes more sense. Let's formally define that.
For defining the orbital stability, we need to defined the "distance" with respect to a set.
**Distance to a set:** We define the distance between a point $p \in X$ and a set $S \subset X$ by $d(p, S) = \inf_{x \in S} |p - x|$
This is basically the smallest distance between the point $p$ and any point in the set $S$
**Orbital stability:** Given a trajectory $x(t)$, it is said to be orbitally stable if, given $\epsilon > 0$, there exists $\delta = \delta(\epsilon) > 0$ such that for any other solution $y(t)$ satisfying $|\tilde{x}(t_0) - y(t_0)| < \delta$, then $d(y(t), O^{+}(x_0, t_0)) < \epsilon$ for $t > t_0$
We can define asymptotic orbital stability similar to the asymptotic stability where the "set" distance asymptotically reaches zero.
Trajectory stability is a stronger requirement whereas the orbital stability is a weak requirement for stability.
<!-- ## Determining the stability -->

Author: Sharath Paper Link Overview Curriculum learning is inspired by the way human learns, where the examples are shown in the increasing order of the difficulty. More sepcifically te network is exposed to the easier examples in the early stages of training and then gradually to the tougher ones. This paper studies the benifits of showing this sequential ordered eamples to the network and comments about when it works and when it doesn't. Contributions: This paper introduces a phenomenon called implicit curricula. One of the claims they make is that the the ordered learning (curriculum, anti-curriculum and random) almost performs same in the standard settings. Curricula is benificial when there is a limited time budget and in noisy regime

7/4/2021Author: Sharath Overview This paper discusses how the BNNs behave as we scale to the large models. This paper discusses the "soap-bubble" issue in case of high dimensional probability spaces and how MFVI suffers from this. As a way to tackle this issue, the authors propose a new variational posterior approximation in hyperspherical coordinate system and show that this overcomes the soap-bubble issue when we sample from this posterior. The geometry of high dimensional spaces One of the properties of high dimensional spaces is that there is much more volume outside any given neighbourhood than inside of it. Betacount et al explained this behaviour visually with two intuitive examples. For first example let us consider partitioning our parameter space in equal rectangular intervals as shown below. We can see that as we increase the dimensions the distribution of volume around the center decreases. This becomes almost negligible as compared to the its neighbourhood in high dimensional cases where $D$ is very large. We can observe a similar behaviour if we consider spherical view of parameter space, where the exterior volume grows even larger than the interior in high dimensional spaces as shown in the figure. . How this intuition of volumes in high dimensions explain soap bubble phenomenon?

7/4/2021Author: Sharath What is a convex function? Let's try to define a convex function formally and geometrically. Formally, a function $f$ is said to be a convex function if the domain of $f$ is a convex set and if it satisfies the following $\forall x \ \text{and} \ y \in \text{dom} f$; \begin{equation} f(\theta x + (1 - \theta)y) \leq \theta f(x) + (1 - \theta)f(y) \end{equation} Geometrically it means that the value of a function at the convex combination of two points of the function always lies below the convex combination of the values at the corresponding points. It means that if we draw a line at any two points $(x, y) \in \text{dom} f$, then this line/chord always lies above the function $f$.

7/4/2021Author: Sharath Chandra Paper Link tags: simulation, interaction-networks, robotics In this paper the authors proposed a hybrid dynamics model, Simulation-Augemented Interaction Networks, where they incorporated Interaction Networks into a physics engine for solving real world complex robotics control tasks. Brief Outline: Most of the physics based simulators serves as a good platform for carrying out robot planning and control tasks. But no simulator is a perfect because it has it's own modelling errors. So, most of the physics engines (mujoco, bullet, gazebo etc.,) demonstrate some descrepencies between their predictions and actual real world predictions. To decrease these errors, many methods have been propsed in the literature. Some of the methods include randomizing the simulation environments, famously known as Domain Randomization. In this paper, model errors are tackled by learning a residual model between the real world and simulator. In other words, instead of adding pertubations to the environment parameters, here we utilize some real world data to correct the simulator. Even though this method uses some real world data, this method is shown to be sample efficient and have better generalization capabilities. Interaction Networks

7/4/2021
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