Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Differential Equations → Ordinary Differential Equations ↓


Systems of Differential Equations


Systems of differential equations are a vital part of understanding complex dynamic systems in mathematics, engineering, physics, and many other fields. While a single differential equation can describe the rate of change of a variable, systems of difference equations can model the change of several interconnected variables simultaneously. This lesson will introduce what these systems are, how they can be solved, and provide simple examples and explanations with visual aids where possible.

What is a system of differential equations?

A system of differential equations consists of two or more interrelated differential equations that describe how unknown functions and their derivatives relate to each other. These systems are generally applicable to dealing with multiple interacting quantities. The most common types of systems are linear and non-linear systems.

More formally, a system of ordinary differential equations can be written as:

    x' = f(t, x, y, z, ...) y' = g(t, x, y, z, ...) z' = h(t, x, y, z, ...) ...

Here, x', y' and z' are derivatives of unknown functions with respect to time t. The functions f, g, h, etc. represent the relationship between these functions and time.

Linear systems

Linear systems are the simplest form of systems of differential equations, and they can often be solved using matrix methods. A linear system can be expressed in matrix form as follows:

    X' = AX + B

where X is a vector of unknown functions, A is a matrix of coefficients, and B is a constant vector.

For example, consider a simple system:

    x' = 3x + 4y y' = 2x + y

This system can be rewritten in matrix form:

    | x' | = | 3 4 | | x | | y' | | 2 1 | | y |
System in Matrix Form:|x' | = | 3 4 | | x ||y' || 2 1 | | y |

Solutions of linear systems

The solution to a linear system can often be found using the eigenvectors and eigenvalues of the matrix A. The general solution to the homogeneous system X' = AX can be expressed in terms of these eigenvalues and eigenvectors.

Eigenvectors and eigenvalues provide information about the behavior of the system, such as stability or oscillation. For a 2x2 system:

    A = | a11 a12 | | a21 a22 |

Calculate the eigenvalue λ by solving the characteristic equation:

    det(A - λI) = 0

Then find the eigenvectors for each eigenvalue.

Example: predator-prey model

A classic example demonstrating systems of differential equations is the predator-prey model, also known as the Lotka-Volterra equations. This model describes the interaction of two species: a predator and a prey.

The system of equations is:

    x' = αx - βxy y' = δxy - γy

where x is the prey population, y is the predator population, and α, β, δ, γ are positive constants describing the interaction rates.

Population of prey (x)Predator population (years)interaction

Analyzing this system might reveal a periodic solution, where both populations oscillate over time, indicating dynamic challenges of predator-prey interactions.

Nonlinear systems

Non-linear systems are more complex and challenging to solve due to their non-linear nature. Non-linear systems can exhibit complex behavior such as chaos, bifurcations, and limit cycles.

Consider the following nonlinear system:

    x' = x(1 - x) - xy y' = -y + xy

Solutions in nonlinear systems often require numerical methods or qualitative analysis.

Qualitative analysis

An important part of the study of systems of differential equations is to understand the qualitative behavior of the solutions without finding explicit solutions. Phase plane analysis is a powerful tool for visualizing the trajectories of systems in a state plane.

steady

Equilibrium points and their stability can significantly affect the dynamics of the system. Linearization around equilibrium points can aid in analyzing stability via eigenvalues.

Numerical solution

Many systems of differential equations, especially non-linear systems, do not have closed-form solutions, requiring numerical methods for approximation. Methods such as Euler's method, Runge-Kutta method and others are essential tools.

For example, the Euler method approximates the solution through a simple iterative process:

    x_(n+1) = x_n + h*f(t_n, x_n, y_n) y_(n+1) = y_n + h*g(t_n, x_n, y_n)

where h is the time step size.

Conclusion

Systems of differential equations are a powerful mathematical tool that enables the modeling of complex systems with many interacting variables. By understanding the fundamentals of both linear and nonlinear systems, using matrix methods, eigenvectors, qualitative and numerical analysis, one can describe, analyze and predict the behavior of dynamic systems in various scientific fields.


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