1. General Considerations
Nonlinear
not linear (x)
not necessarily linear (o)
Why study nonlinear system ?
All physical systems are nonlinear in nature.
Nonlinearities may be introduced intentionally into a system in order to compensate for the effect of other undesirable nonlinearities, or to obtain better performance (on-off controller).
Linear system  closed form solution Nonlinear system  closed form solution X
(Some predictions – qualitative analysis)

2. 1. Phenomena of Nonlinear Dynamics
Linear vs. Nonlinear
Input
Output
System
state,
Definitions : Linear : when the superposition holds
Nonlinear : otherwise

3. Stability & Output of systems
Stability depends on the system’s parameter (linear)
Stability depends on the initial conditions, input signals as well as the system parameters (nonlinear).
Output of a linear system has the same frequency as the input although its amplitude and phase may differ.
Output of a nonlinear system usually contains additional frequency components and may, in fact, not contain the input frequency.

4. Superposition * Superposition Sys.  Sys. = Sys. Is (1) linear ? +
So is it linear?  No, under zero initial conditions only.

5. Linearity What is the linearity when ? +
A mnemonic rule: All functions in RHS of a differential equation are linear. System is linear atleast at zero input or zero initial condition
Ex:

6. Time invariant vs. Time varying
System (1) is time invariant  parameters are constant
- Linear time varying system
System (2) is time invariant  no function has t as its argument.
- Nonlinear time varying system

7. Autonomous & Non - Autonomous
Time invariant system are called autonomous and time varying are called non - autonomous. In this course, ‘autonomous’ is reserved for systems with no external input, i.e.,
Thus autonomous are time invariant systems with no external input. This course will address nonlinear system, both time invariant and time varying, but mostly autonomous.
Ex:

8. Equilibrium Point Equilibrium Point
We start with an autonomous system.
Definition: is an equilibrium point (or a steady state,
or a singular point) 
If det(A)0,(1)has a unique equilibrium point, (Linear System).
Nonlinear system ? × × × × × ×
multiple equilibrium points

9. Linear Autonomous Systems
What can a linear autonomous system do?
where
For 1-dim sys.
For 2-dim sys.

10. Linear Autonomous Systems (Contd.)

11. Solution of Linear systems
For linear sys, the following facts are true
Solution always exists locally.
Solution always exists globally.
Solution is unique each initial condition produces a different trajectory.
Solution is continuously dependent on initial conditions for every finite t,
Equilibrium point is unique (when det A0).

12. Periodic Solution
If there is one periodic solution, there is an infinite set of periodic solutions. (There is no isolated closed solution.) Ex:
( many periodic solutions, w.r.t. I.C.)

13. Non - linear Autonomous System
What can a nonlinear autonomous system do ?
A solution may not exist, even locally.
Basically everything.
Here the solution is chattering, because Therefore, no differential function satisfying the equation exists.

14. Solutions Solution may not exist globally. Solution may not be unique.
Assume
finite escape time
(= : linear system)

15. Equilibrium point
Equilibrium point doesn’t have to be unique.
Ex:
Ex:

16. Periodic Solutions
Nonlinear system may have isolated closed (periodic) solutions.
Ex:

17. Isolated closed solution
Chaotic regimes  non periodic, bounded behavior
Isolated closed solution ( only one periodic solution.)  Isolated attractive periodic solution
Ex:
( lightly damped structure with large
elastic deflections )

18. 2. Second Order Systems Isoclines
called “vector field”
 Set of all trajectories on plane  Phase portrait

19. Isocline(contd.)
Curve c is called an isocline: when a trajectory intersect the isocline, it has slope c, connecting isoclines, we can obtain a solution.
Ex:

20. Linearization Linearization
A nonlinear system can be represented as a bunch of linear systems - each valid in a small neighborhood of using linearization.
Specifically, assume that is continuously, differentiable,
Take one of the equilibrium, say Introduce,
where =0

21. Linearization(contd.)
Consider a sufficiently small ball around
The linearization of at is defined by
Ex:

22. Linearization(contd.)
Then the two linearizations are

23. Singular Points
Nature of singular points
(a)

24. Phase Portraits

25. Phase Portraits(contd.)
(b)

26. Phase Portraits(contd.)
Let
stable focus
unstable focus
center

27. Nonlinear system Nonlinear system
Assume that the nature of this singular point in the linear system is
What is the nature of the singular point in the nonlinear system ?
Ans) Same, except for center.Center for the linear system doesn’t mean center in the nonlinear system.
Equilibrium of a nonlinear system such that the linearization has no eigenvalues on the imaginary axis is called hyperbolic.
Thus, for hyperbolic equilibria, the nature is the same as the linearization.

28. Nonlinear system(contd.)
Ex: