A system does not remain in a steady state continuously since it is regularly subjected to varying internal and external events as well as stimuli, which suggests that a dynamical system man-made, physical or biological changes in time. In addition to these properties, the actions caused by a system can change its environment and thus lead to the alteration of the forces kept it in a stable state in the past. Systems, being around us, may be small or large, made up of subsystems or components that dynamically connect with one another to exhibit complex behaviors. While such systems or their inherent components may be living beings like animals or humans, they may also be mechanical entities. In addition, various organizations and institutions formed by humans are systems themselves, which may also be regarded as the components or subsystems of larger systems [Ghosh, A. (2015)]. Within this scope, system science studies the dynamic behavior of systems based on interactions of its components and their interactions with other systems. In a system, there could be the interactions between humans, between animals, between machines, or between organizations as stated. It is also possible that the interactions occur between the components of different types such as between human beings and animals, between human beings and machines as well as between human beings and institutions.
Dynamic nature of the systems and their components that render systems study interesting. While it may seem simple to observe some system inputs and their corresponding outcomes, it may be quite difficult to measure their exact relationships due to the missing or incomplete knowledge of the systems or the behaviors thereof. Systems knowledge, in these regards, can enable better understanding of the interactions concerned, allowing us to optimize the related machines if possible or make them more adaptable in situations which we cannot make alterations.
Dynamic systems theory refers to a set of concepts describing behavior as an emergent product of self-organizing, adaptive and multicomponent system that evolves in time. One foundational dynamic systems theory is multicausality which conveys the convergence of multiple forces to create behavior. Another foundational concept is self- organization, which shows that the behavior of a system is an emergent product of multiple components that interact through time with interactions that being dependent on context [Perone, S., & Simmering, V. R. (2017)]. The key idea of self-organization is that patterned behavior emerges out of the interactions of multiple elements of the system, to put differently, and the patterns of behavior are not specified in advance, but instead, they are softly assembled in the moment.
Systems organize into attractor states, namely the ways in which the components of a system interact reliably. Another point is that systems are historical meaning that their organization in an attractor state in the moment biases them to revisit the attractor states at a future point in time. The states may become increasingly stable through experience, which suggests that they are resistant to perturbations from internal and external forces. Systems, being open to the environment, show that external forces can shift the components of a system into a new way of interacting, and this trend may often be nonlinear. The last key concept that could be mentioned here is the nesting of timescales, for instance, neural firing is nested within the timescale of cognition, nested within the timescale of behavior over learning and development.
The physical measures’ existence in relation with systems and hyperbolicity points the derivation of particular features that may include central limit theory, decay of correlations statistical stability and locality along with the extreme levels of observations with respect to rare events and occurrences thereof. Given that dynamical systems are often modeled by differential equations, and one change in the structure’s solution in these with parameters varying is essential to understand different phenomena. Furthermore, transition between different types of dynamics is another pivotal point to understand and heteroclinic bifurcations may arise with the connection generated between two or more than two sets separated. Dynamical systems endowed with specific types can be unstable having their persisting dynamic properties following the slight changes retaining the type attribute. These are due to the evolution laws and when the system manifests symmetrical properties.
A dynamical system, at any time, on the other hand has a state that represents a point in an appropriate state space. Often given by a tuple of real numbers or by a vector in a geometrical manifold, this system also manifests the evolution rule of the dynamical system as a function describing what future states follow from the current state. The function is deterministic, which means for a given time interval only one future state follows from the current state [Strogatz, S. H. (2001)], [Katok, A., & Hasselblatt, B. (1995)]. Yet, some systems are stochastic, which means random events can also have impacts on the evolution of the state variables. In physics, on the other hand, a dynamical system refers to a particle or ensemble of particles with states varying over time, and obeying differential equations that involve time derivatives. If one wants to make a prediction concerning the system’s future behavior, then an analytical solution of such equations or their integration over time through computer simulation is to be realized.
The evolution principle of change refers to anything that generates the next state out of the current one, and for each dimension in the state space (for each variable or component), one needs to specify a rule of change. The coupling between these dimensions or variables can explain the reason why the whole thing is a system. As a system whose current state generates its successive state by a rule or principle of change, a dynamic system has the current state which refers to the cause of the next state, and so forth, all through the process with the related sequence of states represented. These steps can be discrete steps or points on a continuum. Causality is the explanation of such a process around this scheme. The evolution rule defining a particular dynamic system is the expression of the process causality that governs the system.
In mathematics, a dynamical system refers to a system where a function describes the time dependence of a point in an ambient space (i.e. in a parametric curve). Some examples can be provided as mathematical models describing the swinging of a clock pendulum, the flow of water, random motion of particles in air, and so on. A general definition brings together several concepts in mathematics like ODEs and ergodic theory, with their allowance of different choices of the space and how time is measure. Whereas time can be measured by integers, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, the space may be a set or manifold without the need of a smooth space-time structure defined on it. The study of dynamical systems is involved with a focus on dynamical systems theory that owns multiple applications to a wide spectrum of fields like mathematics, medicine, physics, biology, chemistry, engineering, social sciences, and so forth. It can also be noted that dynamical systems are a fundamental part of chaos theory, bifurcation theory, logistic map dynamics, self-assembly, self-organization processes as well as edge of chaos notion. You may kindly refer to...
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