viii Contents Exercises 4.1 114 4.2. Stability of Fixed Points 114 Exercises 4.2 119 4.3. Scalar Equations 119 Exercises 4.3 124 4.4. Two Dimensions and Nullclines 126 Exercises 4.4 133 4.5. Linearized Stability of Fixed Points 134 Exercises 4.5 143 4.6. Competitive Populations 145 Exercises 4.6 150 4.7. Applications 152 Exercises 4.7 158 4.8. Theory and Proofs 159 Chapter 5. Phase Portraits Using Scalar Functions 169 5.1. Predator–Prey Systems 169 Exercises 5.1 172 5.2. Undamped Forces 173 Exercises 5.2 182 5.3. Lyapunov Functions for Damped Systems 183 Exercises 5.3 190 5.4. Bounding Functions 191 Exercises 5.4 195 5.5. Gradient Systems 195 Exercises 5.5 199 5.6. Applications 199 Exercises 5.6 210 5.7. Theory and Proofs 210 Chapter 6. Periodic Orbits 213 6.1. Introduction to Periodic Orbits 214 Exercises 6.1 218 6.2. Poincar´ e–Bendixson Theorem 219 Exercises 6.2 226 6.3. Self-Excited Oscillator 229 Exercises 6.3 232 6.4. Andronov–Hopf Bifurcation 232 Exercises 6.4 240 6.5. Homoclinic Bifurcation 242

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