HOMEWORK 1. EE 5329

Kristian Hengster Movric, 1000676470

-6 -4 -2 0 2 4 6-5

-4

-3

-2

-1

0

1

2

3

4

5a=0.05

Figure 1. Phase portrait of Van der Pol oscillator, a=0.05

-5 -4 -3 -2 -1 0 1 2 3 4 5-6

-4

-2

0

2

4

6a=0.9

Figure 2. Phase portrait of Van der Pol oscillator, a=0.9

0 5 10 15 20 25 30 35 40-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5a=0.05,y(0)=0.2,y'(0)=0

Figure 3. Time response of Van der Pol oscillator a=0.05, y(0)=0.2, y’(0)=0

0 5 10 15 20 25 30 35 40-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5a=0.9,y(0)=0.2,y'(0)=0

Figure 4. Time response of Van der Pol oscillator a=0.9, y(0)=0.2, y’(0)=0

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6a=0.05,y(0)=0.2,y'(0)=0

Figure 5. Phase portrait of Van der Pol oscillator a=0.05, y(0)=0.2, y’(0)=0

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-3

-2

-1

0

1

2

3Phase portrait a=0.9,y(0)=0'.2,y'(0)=0

Figure 6. Phase portrait of Van der Pol oscillator a=0.9, y(0)=0.2, y’(0)=0

0 50 100 1500

50

100x3

0 50 100 150-20

0

20x1

0 50 100 150-50

0

50x2

Figure 7. Time response of the Lorentz system for x1(0)=x2(0)=x3(0)=0.3

-20-10

010

20

-40

-20

0

20

400

10

20

30

40

50

The Strange Attractor

Figure 8. The strange (chaotic) attractot of the Lorentz system

-5

0

5

-5

0

5

0

2

4

6

8

s

Magnitude of H(s)

w

abs(

H(s

))

Figure 9. Magnitude 3D plot of the given transfer function

-5

0

5

-5

0

5

-2

-1

0

1

2

s

Phase of H(s)

w

arg(

H(s

))

Figure 10. Phase 3D plot of the given transfer function

-40

-30

-20

-10

0Bode diagram of H(s)

Mag

nitu

de (

dB)

10-1

100

101

102

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 11. Bode amplitude and phase frequency characteristics