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 ↓


Stability Analysis


Stability analysis is an important part of understanding ordinary differential equations (ODEs) in mathematics, especially at the undergraduate level. It involves determining how solutions to ODEs behave over time, especially in response to small disturbances or changes in initial conditions.

1. Introduction to sustainability

When dealing with differential equations, it is essential to understand whether the solutions remain close to a particular state over time. This concept is known as stability. A stable system will return to its equilibrium state after a small disturbance, while an unstable system will drift away from equilibrium.

2. Equilibrium point

Before considering stability, we need to identify the equilibrium points of the system. The equilibrium point of an ordinary differential equation is the point where the derivative is zero.

Consider the simple ODE: dx/dt = f(x) An equilibrium point x0 satisfies f(x0) = 0.

These points are like the resting states of the dynamic system. If you imagine a ball in a bowl, the bottom of the bowl will be the equilibrium point.

Visual example

equilibrium point

3. Types of stability

3.1 Asymptotic stability

An equilibrium point is asymptotically stable if solutions starting near the equilibrium point not only remain close to equilibrium but also tend toward equilibrium as time approaches infinity.

3.2 Lyapunov stability

An equilibrium point is Lyapunov stable if whenever solutions start close enough to the equilibrium point, they remain close, but as time goes on they may not necessarily converge to the equilibrium point.

3.3 Volatility

An equilibrium point is unstable if solutions that start close together do not remain close; rather, they drift apart from the equilibrium point.

4. Linear stability analysis

For linear systems, determining stability is relatively simple. Consider this system:

dx/dt = Ax

where A is a matrix of constants. The stability of this system can be determined by looking at the eigenvalues of the matrix A.

  • If the real parts of all eigenvalues are negative, the equilibrium is asymptotically stable.
  • If the real part of any eigenvalue is positive, the equilibrium is unstable.
  • If the real part of the eigenvalue is zero, further analysis is needed to determine stability.

Example

Consider the linear system: dx/dt = [[-2, 0], [0, -3]] x The eigenvalues are -2 and -3, both negative. Thus, this system is asymptotically stable.

5. Non-linear stability analysis

For non-linear systems, stability analysis is more complicated. The linearization method allows us to study a non-linear system by approximating it with a linear point near an equilibrium point.

If you have a non-linear system:

dx/dt = f(x)

The system can be linearized around the equilibrium point x0 by computing the Jacobian matrix J of the partial derivatives at x0:

J = [ [∂f₁/∂x₁, ..., ∂f₁/∂xₙ], ..., [∂fₙ/∂x₁, ..., ∂fₙ/∂xₙ] ]

Example

Consider:

dx/dt = y dy/dt = -x + y - x²

Balance point: (0, 0)

Jacobian matrix at (0, 0):

J = [ [0, 1], [-1, 1] ]

The eigenvalues of J determine the stability of the origin.

6. Lyapunov's direct method

Another way to study stability is Lyapunov's direct method. This method does not require solving a differential equation.

Find a scalar function V(x), called the Lyapunov function, that satisfies:

  • V(x) > 0, and V(0) = 0 for x ≠ 0
  • dV/dt ≤ 0 along the trajectory of the system

If such a function exists, then the point is Lyapunov stable. If dV/dt is actually <, then the point is asymptotically stable.

Example

Consider the system:

dx/dt = -x

We choose V(x) = x². Then:

dV/dt = 2x(dx/dt) = 2x(-x) = -2x² ≤ 0

This system is asymptotically stable at x = 0.

7. Visualizing stability with phasor diagrams

The phase diagram is a graphical representation of the trajectories of a dynamic system in the phase plane. Let's look at a simple case:

Example

Stable node

In the above phase diagram, the trajectories converge to a point, indicating an asymptotically stable system.

8. Summary and conclusion

Stability analysis of ordinary differential equations provides insight into the long-term behavior of dynamic systems. By examining equilibrium points and understanding concepts such as asymptotic stability, we can predict whether small perturbations will cause a system to recover or fall apart.

Various methods including linearization, Lyapunov methods, and phase delineation provide us with tools to assess stability for both linear and nonlinear systems.

In practice, these methods are invaluable in fields such as engineering, physics, and economics, where they help design stable systems and understand complex dynamic phenomena.


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Stability Analysis

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