# Dynamical system

The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system. Overview Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because: The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. History Many people regard Henri Poincaré as the founder of dynamical systems.[3] Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). .

Related:  System Theoryin Cellular biophysics

Feedback "...'feedback' exists between two parts when each affects the other."[1](p53, §4/11) A feedback loop where all outputs of a process are available as causal inputs to that process Cell signaling Traditional work in biology has focused on studying individual parts of cell signaling pathways. Systems biology research helps us to understand the underlying structure of cell signaling networks and how changes in these networks may affect the transmission and flow of information. Such networks are complex systems in their organization and may exhibit a number of emergent properties including bistability and ultrasensitivity.

Calculus History Modern calculus was developed in 17th century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (see the Leibniz–Newton calculus controversy), but elements of it have appeared in ancient Greece, China, medieval Europe, India, and the Middle East. Ancient Biochemical systems theory Biochemical systems theory is a mathematical modelling framework for biochemical systems, based on ordinary differential equations (ODE), in which biochemical processes are represented using power-law expansions in the variables of the system. This framework, which became known as Biochemical Systems Theory, has been developed since the 1960s by Michael Savageau and others for the systems analysis of biochemical processes.[1] According to Cornish-Bowden (2007) they "regarded this as a general theory of metabolic control, which includes both metabolic control analysis and flux-oriented theory as special cases".[2] Representation The dynamics of a species is represented by a differential equation with the structure: where Xi represents one of the nd variables of the model (metabolite concentrations, protein concentrations or levels of gene expression). j represents the nf biochemical processes affecting the dynamics of the species. On the other hand,

State-space representation In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space. Cell migration Cell migration is a central process in the development and maintenance of multicellular organisms. Tissue formation during embryonic development, wound healing and immune responses all require the orchestrated movement of cells in particular directions to specific locations. Errors during this process have serious consequences, including intellectual disability, vascular disease, tumor formation and metastasis. An understanding of the mechanism by which cells migrate may lead to the development of novel therapeutic strategies for controlling, for example, invasive tumour cells. Cells often migrate in response to specific external signals, including chemical signals and mechanical signals.

Conceptual model A conceptual model is a model made of the composition of concepts, which are used to help people know, understand, or simulate a subject the model represents. Some models are physical objects; for example, a toy model which may be assembled, and may be made to work like the object it represents. The term conceptual model may be used to refer to models which are formed after a conceptualization (generalization)[1] process in the mind. Conceptual models represent human intentions or semantics[citation needed][dubious ].

Metabolic control analysis MCA was originally developed to describe the control in metabolic pathways but was subsequently extended to describe signaling and genetic networks. MCA has sometimes also been referred to as Metabolic Control Theory but this terminology was rather strongly opposed by Henrik Kacser, one of the founders[citation needed]. More recent work[4] has shown that MCA can be mapped directly on to classical control theory and are as such equivalent. Control Coefficients A control coefficient[7][8][9] measures the relative steady state change in a system variable, e.g. pathway flux (J) or metabolite concentration (S), in response to a relative change in a parameter, e.g. enzyme activity or the steady-state rate (

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