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 ↓


Dynamical Systems


A dynamical system is a mathematical framework used to describe the time-dependent position or state of a point or system in geometric space. In essence, it is a tool for understanding how things evolve over time according to specific rules. Most dynamical systems can be described using differential equations, which helps us track continuous changes.

Overview of dynamical systems

At their core, dynamical systems involve some variables that depend on time, their state, and change according to a set of fixed rules. An elementary example is a simple harmonic oscillator, such as a swinging pendulum or a vibrating spring. In these systems, differential equations describe the various forces acting on the objects.

A dynamical system can usually be written as a system of differential equations:

  ,
  frac{dx}{dt} = f(x, t)
  ,

Here, x represents the state of the system, and t represents time. f(x, t) is a function that describes how the state of the system changes over time.

Types of dynamical systems

Dynamical systems can generally be classified as follows:

  • Continuous dynamical systems: These are described by ordinary differential equations (ODEs) like the form given above. Here time is considered as a continuous variable. They are useful for modelling many physical processes.
  • Discrete dynamical systems: These use differential equations where the system evolves in discrete time steps. Such systems are necessary in situations where changes occur at intervals, such as population generation in biology.

Continuous dynamic system example: Simple pendulum

A simple pendulum, which swings back and forth, can be modeled by the second-order differential equation:

  ,
  frac{d^2theta}{dt^2} + frac{g}{L} sin(theta) = 0
  ,

Where:

  • (theta) is the displacement angle.
  • g is the acceleration due to gravity.
  • L is the length of the pendulum.
l

In this example, the motion of the pendulum is part of a continuous system that can be solved to give an equation of motion, which describes how the pendulum moves over time. By solving this differential equation, you can estimate the position of the pendulum at any time.

Discrete dynamic system example: Population growth

Consider a simple model that describes the growth of a population where the size of each generation depends on the previous generation:

  ,
  x_{n+1} = r x_n (1 - x_n)
  ,

This is known as the logistic map, where:

  • x_n is the population size in the n generation.
  • r is the growth rate constant.

This equation can show a variety of complex behaviors, including chaotic behavior, as the rate r varies. It is famous for its ability to demonstrate chaos, making it a key example in the study of complex systems.

Understanding phase space

In the study of dynamical systems, the concept of phase space is very useful. Phase space is a multidimensional space in which all possible states of a system are represented. Each state corresponds to a unique point in this space.

For a simple pendulum, its position will be completely described by its angle (theta) and angular velocity (omega). Thus, the phase space for a simple pendulum is a two-dimensional space.

State 1 State 2

As a system evolves, its representation in phase space will trace a path known as a trajectory. The shape of these trajectories can provide information about the properties of the system, such as stability and periodicity.

Fixed point and stability

A fixed point in a dynamic system is a point in phase space where the system can remain indefinitely, without any change - essentially a point of equilibrium. For example, consider the pendulum discussed earlier. When it comes to rest in a downward vertical position, it is at a fixed point.

The stability of a fixed point is important for understanding system behavior. A fixed point is said to be stable if nearby points in phase space move toward the fixed point over time and eventually stabilize at it. Conversely, a fixed point is unstable if nearby points move apart as time increases.

The stability of a fixed point can be analyzed by linearizing the system equations near the fixed point and studying the resulting linear system. This process often involves calculating the Jacobian matrix at the fixed point.

Attractor

An attractor is a set of numerical values towards which a system evolves. Systems can have different types of attractors such as point attractors (a single point in space), periodic attractors (a closed loop, indicating oscillation), or strange attractors (which exhibit chaos).

An example of a strange attractor is the Lorenz attractor, which is known for its chaotic and fractal structure. The Lorenz system is defined by the following set of differential equations:

  ,
  frac{dx}{dt} = sigma(y - x)
  ,
  ,
  frac{dy}{dt} = x(rho - z) - y
  ,
  ,
  frac{dz}{dt} = xy - beta z
  ,

where (sigma), (rho), and (beta) are parameters.

Sensitivity to chaos and initial conditions

A fascinating topic within dynamical systems is chaos. A system is said to be chaotic if it exhibits extreme sensitivity to initial conditions, leading to seemingly random behavior. Even a small change in the initial state of the system can lead to very different trajectories.

Illustrating chaos with a logistic map

The logistic map described earlier can show chaotic behavior depending on the value of r. For particular values, the predictions of the system appear to be random, reflecting the essence of chaotic systems.

Applications of dynamical systems

The applications of dynamical systems are in many areas:

  • Physics: The study of chaotic systems such as motion, fluid dynamics, and weather.
  • Biology: Population dynamics, modeling of neural activity in the brain, and genetic networks.
  • Economics: Understanding market dynamics and predicting financial trends.
  • Engineering: Control Systems, Robotics, and Signal Processing.

These systems provide a framework to model, predict, and understand complex phenomena, making them central to scientific research.

Conclusion

Dynamic systems provide a powerful way to analyze time-dependent phenomena using differential equations. By understanding the types of dynamic systems, analyzing phase space, and exploring chaotic behavior, you gain deep insights into the complex world around us. Whether nature or technology, mastering this field opens the door to solving complex problems and pushing the boundaries of innovation.


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