Piecewise-Linear Economic-Mathematical Models with Regard to Unaccounted Factors Influence on a Plane

2013 ◽  
pp. 173-204
Keyword(s):  
Mathematical Models ◽  
Piecewise Linear

Strange Attractors and Chaos in Nonlinear Mechanics

1983 ◽  
Vol 50 (4b) ◽  
pp. 1021-1032 ◽  
Author(s):  
P. J. Holmes ◽  
F. C. Moon
Keyword(s):  
Mathematical Models ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Nonlinear Differential Equations ◽  
Homoclinic Orbits ◽  
Strange Attractors ◽  
Electrical Systems ◽  
Nonlinear Mechanical ◽  
Sensitive Dependence

We review several examples of nonlinear mechanical and electrical systems and related mathematical models that display chaotic dynamics or strange attractors. Some simple mathematical models — iterated piecewise linear mappings — are introduced to explain and illustrate the concepts of sensitive dependence on initial conditions and chaos. In particular, we describe the role of homoclinic orbits and the horseshoe map in the generation of chaos, and indicate how the existence of such features can be detected in specific nonlinear differential equations.

scholarly journals Simplification and its consequences in biological modelling: conclusions from a study of calcium oscillations in hepatocytes

2005 ◽  
Vol 3 (7) ◽  
pp. 319-331 ◽  
Author(s):  
James P.J Hetherington ◽  
Anne Warner ◽  
Robert M Seymour
Keyword(s):  
Sensitivity Analysis ◽  
Systems Biology ◽  
Mathematical Models ◽  
Piecewise Linear ◽  
Random Perturbation ◽  
Building Blocks ◽  
Specific Model ◽  
Calcium Oscillations ◽  
System Level ◽  
Biological Modelling

Systems Biology requires that biological modelling is scaled up from small components to system level. This can produce exceedingly complex models, which obscure understanding rather than facilitate it. The successful use of highly simplified models would resolve many of the current problems faced in Systems Biology. This paper questions whether the conclusions of simple mathematical models of biological systems are trustworthy. The simplification of a specific model of calcium oscillations in hepatocytes is examined in detail, and the conclusions drawn from this scrutiny generalized. We formalize our choice of simplification approach through the use of functional ‘building blocks’. A collection of models is constructed, each a progressively more simplified version of a well-understood model. The limiting model is a piecewise linear model that can be solved analytically. We find that, as expected, in many cases the simpler models produce incorrect results. However, when we make a sensitivity analysis, examining which aspects of the behaviour of the system are controlled by which parameters, the conclusions of the simple model often agree with those of the richer model. The hypothesis that the simplified model retains no information about the real sensitivities of the unsimplified model can be very strongly ruled out by treating the simplification process as a pseudo-random perturbation on the true sensitivity data. We conclude that sensitivity analysis is, therefore, of great importance to the analysis of simple mathematical models in biology. Our comparisons reveal which results of the sensitivity analysis regarding calcium oscillations in hepatocytes are robust to the simplifications necessarily involved in mathematical modelling. For example, we find that if a treatment is observed to strongly decrease the period of the oscillations while increasing the proportion of the cycle during which cellular calcium concentrations are rising, without affecting the inter-spike or maximum calcium concentrations, then it is likely that the treatment is acting on the plasma membrane calcium pump.

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