This document discusses nonlinear dynamical systems and modeling techniques. Nonlinear dynamical systems have multiple inputs, feedback loops, and sensitivity to initial conditions. They can be modeled using techniques like state space models, principal component analysis, neural networks, and chaos theory. Modeling nonlinear dynamical systems involves accounting for their emergent behavior from component interactions, distributed nature, and potential to evolve into chaotic states.

1. NON LINEAR
DYNAMICAL SYSTEMS
VIKRAM SINGH SANKHALA

2. ISOLATED SYSTEM
• Zero interaction with environment.
• In Real World it is difficult to isolate a system from its Environment

3. NON LINEAR SYSTEMS
• Multiple Inputs
• Feedback Loops
• Error grow Exponentially
• Sensitivity to Initial Conditions

4. DISTRIBUTED SYSTEMS
• Multiple Interacting Non Linear Dynamic Systems as Components

5. MODELLING NON LINEAR DYNAMICAL SYSTEMS
• Stochastic noise-driven linear and nonlinear dynamical systems.
• Another approach involves a state space where time series observations are
transformed to the phase space vectors. They model the dynamics of a system by
modeling the dynamics of the corresponding points in the phase space using a
Mapping Function
• The Mapping Function can be used for Prediction of future states.

6. DIMENSION REDUCTION
• Principal Components Analysis – Linear Method
• Non Linear Techniques are also there

7. ALGORITHMIC METHODS
• As soon as you introduce self-loops in your model structure, you introduce time,
and therefore the system becomes dynamical.
• In many statistical models, we don't have a closed form solution for inference on
particular variables, so we must use iterative/algorithmic methods to approximate
/ sample values.
• Neural Networks play an important role for modelling Dynamic non linear
systems.

8. ARTIFICIAL NEURAL NETWORKS
• Consist of Connected local processing elements (neurons)
• These accept weighted inputs from other such elements
• They use these weighted inputs to give a single output

9. FREEMAN 1991
• Artificial neural systems were designed to capture some of the useful brain
functions by modeling the features of the brain.
• Chaos "may be the chief property that makes the brain different from an artificial-
intelligence machine“.

10. CHAOS
• Chaos is statistically indistinguishable from randomness, and yet it is deterministic
and not random at all.
• A chaotic system will produce the same results if given the same inputs
• But you can not predict in what way the system's behavior will change for any
change in the input to that system
• It is random appearing, and yet has a large degree of underlying order.

11. BUTTERFLY EFFECT
• Affects Initial Conditions
• Initial Condition inaccuracy grows exponentially.
• We get extra ordinary and Counter Intuitive Results

12. CHAOS ENGINEERING
• How Emergent behavior from Component Interactions can cause the system to
devove into Chaotic state,
• Controlled Experiments on Distributed Systems by introducing failure.