Differential Equations: A Dynamical Systems Approach

of Part II.- Systems of Ordinary Differential Equations The Higher-Dimensional Theory x? = f(t,x).- 6 Systems of Differential Equations.- 6.1 Graphical Representation of Systems.- 6.2 Theorems for Systems of Differential Equations.- 6.3 Example: Sharks and Sardines.- 6.4 Higher Order Equations.- 6.5 Mechanical Systems with One Degree of Freedom.- 6.6 Essential Size, Conservative Laws.- 6.7 The Two-Body Problem.- 6.8 Flows.- 6 Exercises.- 7 Systems of Linear Differential Equations.- 7.1 Linear Differential Equations in General.- 7.2 Linearity and Superposition Principles.- 7.3 Linear Differential Equations with Constant Coefficients: Eigenvectors and Decoupling.- 7.4 Linear Differential Equations with Constant Coefficients: Exponentials of Matrices.- 7.5 Two by Two Matrices and the Bifurcation Diagram.- 7.6 Eigenvalues and Global Behavior.- 7.7 Nonhomogeneous Linear Equations.- 7 Exorcises.- 8 Systems of Nonlinear Differential Equations.- 8.1 Zeroes of Vector Fields and Their Linearization.- 8.2 Sources are Sources and Sinks are Sinks.- 8.3 Saddles.- 8.4 Limit Cycles.- 8.5 The Poincaré-Bendixson Theorem.- 8.6 Symmetries and Volume-Preserving Equations.- 8.7 Chaos in Higher Dimensions.- 8.8 Structural Stability.- 8 Exercises.- 8* Structural Stability.- 8*.1 Preliminaries for Structural Stability.- 8*.2 Structural Stability of Sinks and Sources.- 8*.3 Time to Pass by a Saddle.- 8*.4 Structural Stability of Limit Cycles.- 8*.5 Why Poincaré-Bendixson Rules Out “Chaos” in the Plane.- 8*.6 Structural Stability in the Plane.- 8* Exercises.- 9 Bifurcations.- 9.1 Saddle-Node Bifurcation.- 9.2 Andronov-Hopf Bifurcations.- 9.3 Saddle Connections.- 9.4 Semistable Limit Cycles.- 9.5 Bifurcation in One-Parameter Families.- 9.6 Bifurcation in Two-Parameter Families.- 9.7 Grand Example.- 9 Exercises.- Appendix L: Linear Algebra.- L1 Theory of Linear Equations: In Practice.- L2 Theory of Linear Equations: Vocabulary.- L3 Vector Spaces and Inner Products.- L4 Linear Transformations and Inner Products.- L5 Determinants and Volumes.- L6 Eigenvalues and Eigenvectors.- L7 Finding Eigenvalues: The QR Method.- L8 Finding Eigenvalues: Jacobi’s Method.- Appendix L Exercises.- Appendix L Summary.- Appendix T: Key Theorems From Parts I and III.- References.- Answers to Selected Problems.