Qualitative Upper and Lower Approximations of Complex Nonlinear Chaotic and Nonchaotic Models

Qualitative Upper and Lower Approximations of Complex Nonlinear Chaotic and Nonchaotic Models

Článek WoS

Soft sciences, e.g. economics, ecology, sociology and their various integrations, are often used to develop e.g. forecasting models and optimization constraints. However, highly nonlinear, vague, partially inconsistent and multidimensional systems are prohibitively difficult to study at the quantitative level. Different types of quantitative simplifications are therefore used, e.g. linearization. The resulting models are oversimplified and therefore inapplicable results are often obtained. There are just three values used to quantify qualitative variables and their derivatives: plus/increasing; zero/constant; negative/decreasing. It means that a set of scenarios, i.e. the qualitative solution, is discrete. A qualitative model can be simplified by ignoring some of its equations. The simplified model is less restrictive and gives more scenarios. This set of scenarios is the model’s upper approximation. An additional equation makes the model more complex and the resulting set of fewer scenarios is the lower approximation. A qualitative model of dumped oscillations, upper and lower approximations of well-known Lorenz model and a vaguely known five-dimensional bankruptcy model are presented in detail. The upper approximation of the Lorenz set of differential equations has 213 and its lower approximation has 189 scenarios. No a prioriknowledge of qualitative models theory is required.

Soft sciences, e.g. economics, ecology, sociology and their various integrations, are often used to develop e.g. forecasting models and optimization constraints. However, highly nonlinear, vague, partially inconsistent and multidimensional systems are prohibitively difficult to study at the quantitative level. Different types of quantitative simplifications are therefore used, e.g. linearization. The resulting models are oversimplified and therefore inapplicable results are often obtained. There are just three values used to quantify qualitative variables and their derivatives: plus/increasing; zero/constant; negative/decreasing. It means that a set of scenarios, i.e. the qualitative solution, is discrete. A qualitative model can be simplified by ignoring some of its equations. The simplified model is less restrictive and gives more scenarios. This set of scenarios is the model’s upper approximation. An additional equation makes the model more complex and the resulting set of fewer scenarios is the lower approximation. A qualitative model of dumped oscillations, upper and lower approximations of well-known Lorenz model and a vaguely known five-dimensional bankruptcy model are presented in detail. The upper approximation of the Lorenz set of differential equations has 213 and its lower approximation has 189 scenarios. No a prioriknowledge of qualitative models theory is required.

Qualitative models; complex nonlinear chaotic models; complex nonlinear nonchaotic models; upper approximation; lower approximation.

Qualitative models; complex nonlinear chaotic models; complex nonlinear nonchaotic models; upper approximation; lower approximation.

DOHNAL, M.; DOUBRAVSKÝ, K.

2017

07.01.2016

World Scientific Publishing

Singapur

0218-1274

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS

25

13

Singapurská republika

1

12

12

http://www.worldscientific.com/doi/10.1142/S0218127415501734