By Sagan H.

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Dynamical Systems where C is a constant. Thus, if f = 0, energy is conserved; if f (x) > 0, then energy is taken out of the system; and, if f (x) < 0, then energy is put into the system. Integrating the diﬀerential equation once results in an equivalent ﬁrstorder system: dx dt dy dt = y − F (x) = −x, where x f (z)dz. F (x) = 0 Hypothesis H. 1. F (−x) = −F (x) for all x. 2. There is a number α > 0 such that F (x) is negative for 0 < x < α. 3. There is a number β ≥ α such that F (x) is positive and strictly increasing for x > β.

Show by example that the resulting system having constant coeﬃcients can have a positive eigenvalue even though both the eigenvalues of C have negative real parts. Conclude that eigenvalues of a periodic coeﬃcient matrix do not determine stability properties of the linear system. 4. 1) are bounded for −∞ < t < ∞. Also, show that if |∆| > 1, then all solutions (excluding the one that is zero everywhere) are unbounded on −∞ < t < ∞. 5. 2 using numerical simulation. 6. Verify the weak ergodic theorem for√the function F (s1 , s2 ) = 1 + cos s1 + cos s2 and ω1 = 1 and ω2 = 2 by direct substitution.

10 1. 2 (with inductance L = 0). Using the notation of that section, we have V = H(p)W, where H(p) = (RCp + 1)−1 . This formula should be interpreted as one for the transforms of V and W : V ∗ = H(p)W ∗ , and so V (t) is the inverse Laplace transform of the function H(p)W ∗ (p). It follows that V (t) = exp − t RC t V (0) + 0 exp − (t − s) RC W (s) ds . RC Finally, we note that another useful representation of the matrix exp(At) can be found using Laplace transforms. Namely, we can deﬁne exp(At) = 1 2πi (pI − A)−1 exp(pt)dp.

### Advanced calculus by Sagan H.

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