"The change of the output is not proportional to the change of the input."
It studies how complex systems can exhibit nonlinear and chaotic behaviors, including self-organization and self-replication.
Chaos Theory: Chaos theory studies the behavior of dynamic systems that are highly sensitive to initial conditions.
Nonlinear Differential Equations: Nonlinear differential equations involve nonlinear relationships between variables and are essential for modeling nonlinear dynamical systems.
Bifurcation Theory: Bifurcation theory studies how changes in parameters of a system cause qualitative changes in the behavior of solutions to differential equations.
Fractals: Fractals are self-similar patterns that repeat at different scales and are common in many nonlinear dynamical systems.
Attractors: Attractors are states or patterns towards which chaotic systems tend to converge over time.
Phase Space: Phase space is a mathematical construct used to study the behavior of a system by plotting the trajectories of all possible states of the system.
Strange Attractors: Strange attractors are attractors with fractal geometry, and they are characteristic of chaotic systems.
Lyapunov Exponents: Lyapunov exponents are measures of how quickly nearby trajectories in phase space diverge from one another, which is a measure of the level of chaos in a system.
Self-Organization: Self-organization occurs when a complex system spontaneously develops new structures, patterns, or behaviors without being externally driven.
Emergence: Emergence is the phenomenon whereby complex systems exhibit behaviors that cannot be predicted simply by examining the behavior of their individual components.
Resilience: Resilience refers to the capacity of a system to withstand and recover from disturbances or perturbations.
Self-Similarity: Self-similarity is the property whereby a pattern or structure exhibits the same features at different scales or levels of organization.
Feedback Loops: Feedback loops are mechanisms in which the output of a system is fed back as input, causing a change in the system's behavior.
Complex Systems: Complex systems are systems composed of many interconnected and interdependent parts that exhibit emergent behavior.
Network Theory: Network theory is the study of complex systems represented as networks or graphs, where the nodes represent individual components and the edges represent connections between them.
Cellular Automata: Cellular automata are mathematical models of complex systems that simulate the behavior of individual components based on rules governing their interactions.
Neural Networks: Neural networks are models of complex systems inspired by the behavior of biological neural networks, and they are used in machine learning and artificial intelligence.
Game Theory: Game theory is the study of strategic decision-making in complex systems and is used in economics, psychology, and political science.
Information Theory: Information theory is the study of the quantification, storage, and communication of information, and it has applications in computer science and communication engineering.
Complex Adaptive Systems: Complex adaptive systems are systems composed of many interacting agents that adjust their behavior in response to changes in their environment or in the behavior of other agents.
Chaos Theory: The study of systems that are highly sensitive to initial conditions, where small changes can lead to significant differences in outcomes over time.
Fractal Geometry: The study of shapes or patterns that repeat at different scales or magnifications.
Strange Attractors: A pattern in a chaotic system that attracts nearby points towards it, but with no fixed point of rest.
Bifurcation Theory: The study of how small changes in parameters can cause a system to undergo a qualitative change in behavior or structure.
Nonlinear Oscillations: The study of systems that exhibit periodic behavior but in a nonlinear manner, where the amplitude or frequency of the oscillation changes over time.
Pattern Formation: The study of how complex patterns and structures emerge spontaneously in nonlinear systems.
Self-Organization: The study of how systems spontaneously organize themselves into more complex and ordered structures without external intervention.
Self-Similarity: The property of a system where it is similar or identical to itself at different scales or magnifications.
Phase Space Analysis: The study of the behavior of a system's variables over time, mapped as trajectories in a multi-dimensional space.
"Most systems are inherently nonlinear in nature."
"They may appear chaotic, unpredictable, or counterintuitive."
"A set of simultaneous equations in which the unknowns appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one."
"Yes, systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations."
"Nonlinear systems are commonly approximated by linear equations (linearization)."
"Interesting phenomena such as solitons, chaos, and singularities are hidden by linearization."
"Some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable, or even chaotic."
"This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology."
"Some authors use the term nonlinear science for the study of nonlinear systems."
"This term is disputed by others."
"Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals."