Non linear dynamical systems

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.