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1. Prove that map R₁ ×R¹ → R¹ given by (t, x) → x(t²+1) is not a dynamical system on R. Prove that map R₁ × R² → R² given by (t, x) → - sint cos t cos t sin t is a dynamical system on R2. Hint: one easy way is to recall the geometric meaning of this X, X = X1 X2 2

1. Prove that map R₁ ×R¹ → R¹ given by (t, x) → x(t²+1) is not a dynamical system on R. Prove that map R₁ × R² → R² given by (t, x) → - sint cos t cos t sin t is a dynamical system on R2. Hint: one easy way is to recall the geometric meaning of this X, X = X1 X2 2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Prove that map R₁×R¹ → R¹ given by (t, x) → x(t²+1) is not a dynamical system on R. Prove that map R₁ × R² → R² given by (t, x) → is a dynamical system on R2. Hint: one easy way is to recall the geometric meaning of this matrix transformation. Another approach is to use the complex numbers and relate to eit (x1 + ix2). For the latter dynamical system o fix any SC R². Draw (S). cos t sin t sin cos t X, X = X1 x2 2
Transcribed Image Text:1. Prove that map R₁×R¹ → R¹ given by (t, x) → x(t²+1) is not a dynamical system on R. Prove that map R₁ × R² → R² given by (t, x) → is a dynamical system on R2. Hint: one easy way is to recall the geometric meaning of this matrix transformation. Another approach is to use the complex numbers and relate to eit (x1 + ix2). For the latter dynamical system o fix any SC R². Draw (S). cos t sin t sin cos t X, X = X1 x2 2
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