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.
This is a continuation of the subject matter discussed in the first book, with an emphasis on systems of ordinary differential equations and will be most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, and applied mathematics, as well as in the life sciences, physics, and economics.
After an introduction, there follow chapters on systems of differential equations, of linear differential equations, and of nonlinear differential equations. The book continues with structural stability, bifurcations, and an appendix on linear algebra. The whole is rounded off with an appendix containing important theorems from parts I and II, as well as answers to selected problems.
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