We study the relationship between corruption in public procurement and economic growth
within the Solow framework in discrete time, while assuming that the public good is an input in
the productive process and that the State fixes a monitoring level on corruption. The resulting
model is a bidimensional triangular dynamic system able to generate endogenous fluctuations
for certain values of some relevant parameters. We study the model from the analytical point
of view and find that multiple equilibria with nonconnected basins are likely to emerge. We
also perform a stability analysis and prove the existence of a compact global attractor. Finally,
we focus on local and global bifurcations causing the transition to more and more complex
asymptotic dynamics. In particular, as our map is nondifferentiable in a subset of the states
space, we show that border collision bifurcations occur. Several numerical simulations support
the analysis. Our study aims at demonstrating that no long-run equilibria with zero corruption
exist and, furthermore, that periodic or aperiodic fluctuations in economic growth are likely to
emerge. As a consequence, the economic system may be unpredictable or structurally unstable.

Serena Brianzoni

Raffaella Coppier

Elisabetta Michetti

Complex Dynamics in a Growth Model with Corruption in Public Procurement

The stable difference schemes for the approximate solution of the nonlocal
boundary value problem for multidimensional hyperbolic equations with dependent
in space variable coefficients are presented. Stability of these difference
schemes and of the first- and second-order difference derivatives is obtained.
The theoretical statements for the solution of these difference schemes for one-dimensional
hyperbolic equations are supported by numerical examples.

Allaberen Ashyralyev

Necmettin Aggez

Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

An <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-firm production game known as oligopoly will be examined with isoelastic price function and linear cost under al Cournot competition. After the best responses of the firms are determined, a dynamic system with adaptive expectations is introduced. It is first shown that the local asymptotic behavior of the system is identical with that of the adaptive adjustment process in which the firms cautiously determine their outputs. Dynamic analysis is confined to two special cases, one in which 
                <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> is divided into two groups and the other in which 
                <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> is divided into three groups. Then stability conditions will be derived and the global behavior of the equilibria will be illustrated including chaos control. Lastly the two- and three-group models are compared with two-firm (duopoly) and three-firm (triopoly) models to shed light on roles of the number of the firms.

Akio Matsumoto

Ferenc Szidarovszky

Stability, Bifurcation, and Chaos in <svg style="vertical-align:-0.1092pt" height="11.3125" version="1.1" viewBox="0 0 14.9875 11.3125" width="14.9875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153 l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z"/></g></svg>-Firm Nonlinear Cournot Games

By using the topological degree theory and the fixed point index theory, the existence of nontrivial solutions and sign-changing solutions for a class of boundary value problem on time scales is obtained. We should point out that the integral operator corresponding to above boundary value problem is not assumed to be a cone mapping.

Huiye Xu

Existence of Nontrivial Solutions and Sign-Changing Solutions for Nonlinear Dynamic Equations on Time Scales

We propose to study a generalized family of max-type difference equations and then prove the global attractivity of a particular case of it under some
parameter conditions. Through some numerical results of other cases, we finally pose a generic conjecture.

Xiaofan Yang

Wanping Liu

Jiming Liu

Global Attractivity of a Family of Max-Type Difference Equations

A periodic delay single-species difference system with
feedback control is established. With the help of analysis method and Lyapunov
function, a good understanding of the permanence and global attractivity of the
system is gained. Numerical simulations are presented to verify the validity of the
proposed criteria. Our results show that feedback control has no influence on the
permanence while it has influence on the global attractivity of the system.

Yi Yang

Zhijun Liu

Feedback Control in a Periodic Delay Single-Species Difference System

By using Morse theory, the critical point theory, and the character
of <svg style="vertical-align:-0.0pt;width:25.075001px;" id="M1" height="12.3625" version="1.1" viewBox="0 0 25.075001 12.3625" width="25.075001"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,12.3625)">
		<g transform="translate(72,-62.11)">
			<text transform="matrix(1,0,0,-1,-71.95,62.16)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝐾</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-62.51,66.65)">
				<tspan style="font-size: 7.85px; " x="0" y="0">1</tspan>
				<tspan style="font-size: 7.85px; " x="3.9230001" y="0">/</tspan>
				<tspan style="font-size: 7.85px; " x="6.104188" y="0">2</tspan>
			</text>

		</g>
	</g>
</svg>, we consider the existence and multiplicity results of solutions to the following discrete
nonlinear two-point boundary value problem <svg style="vertical-align:-2.10374pt;width:216.2375px;" id="M2" height="15.1375" version="1.1" viewBox="0 0 216.2375 15.1375" width="216.2375"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,15.1375)">
		<g transform="translate(72,-59.89)">
			<text transform="matrix(1,0,0,-1,-71.95,62.03)">
				<tspan style="font-size: 11.21px; " x="0" y="0">−</tspan>
				<tspan style="font-size: 11.21px; " x="7.6774802" y="0">Δ</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-56.18,66.65)">
				<tspan style="font-size: 7.85px; " x="0" y="0">2</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-51.76,62.03)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑥</tspan>
				<tspan style="font-size: 11.21px; " x="6.1644001" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="9.8966637" y="0">𝑘</tspan>
				<tspan style="font-size: 11.21px; " x="18.459576" y="0">−</tspan>
				<tspan style="font-size: 11.21px; " x="28.625233" y="0">1</tspan>
				<tspan style="font-size: 11.21px; " x="34.229233" y="0">)</tspan>
				<tspan style="font-size: 11.21px; " x="41.07732" y="0">=</tspan>
				<tspan style="font-size: 11.21px; " x="51.870625" y="0">𝑓</tspan>
				<tspan style="font-size: 11.21px; " x="58.091064" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="61.823326" y="0">𝑘</tspan>
				<tspan style="font-size: 11.21px; " x="67.898064" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="72.560593" y="0">𝑥</tspan>
				<tspan style="font-size: 11.21px; " x="78.724991" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="82.457253" y="0">𝑘</tspan>
				<tspan style="font-size: 11.21px; " x="88.53199" y="0">)</tspan>
				<tspan style="font-size: 11.21px; " x="92.264259" y="0">)</tspan>
				<tspan style="font-size: 11.21px; " x="95.996521" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="100.67026" y="0">𝑘</tspan>
				<tspan style="font-size: 11.21px; " x="109.86082" y="0">∈</tspan>
				<tspan style="font-size: 11.21px; " x="120.64291" y="0">ℤ</tspan>
				<tspan style="font-size: 11.21px; " x="128.79112" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="132.52339" y="0">1</tspan>
				<tspan style="font-size: 11.21px; " x="138.1274" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="142.80113" y="0">𝑇</tspan>
				<tspan style="font-size: 11.21px; " x="148.96553" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg> subject to <svg style="vertical-align:-1.99126pt;width:100.7875px;" id="M3" height="12.075" version="1.1" viewBox="0 0 100.7875 12.075" width="100.7875"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,12.075)">
		<g transform="translate(72,-62.34)">
			<text transform="matrix(1,0,0,-1,-71.95,64.37)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑥</tspan>
				<tspan style="font-size: 11.21px; " x="6.1644001" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="9.8966637" y="0">0</tspan>
				<tspan style="font-size: 11.21px; " x="15.500664" y="0">)</tspan>
				<tspan style="font-size: 11.21px; " x="22.337543" y="0">=</tspan>
				<tspan style="font-size: 11.21px; " x="33.130848" y="0">0</tspan>
				<tspan style="font-size: 11.21px; " x="41.850674" y="0">=</tspan>
				<tspan style="font-size: 11.21px; " x="52.643974" y="0">Δ</tspan>
				<tspan style="font-size: 11.21px; " x="60.736153" y="0">𝑥</tspan>
				<tspan style="font-size: 11.21px; " x="66.900551" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="70.632813" y="0">𝑇</tspan>
				<tspan style="font-size: 11.21px; " x="76.797218" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>, where <svg style="vertical-align:-0.0pt;width:10.075px;" id="M4" height="9.2749996" version="1.1" viewBox="0 0 10.075 9.2749996" width="10.075"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,9.275)">
		<g transform="translate(72,-64.58)">
			<text transform="matrix(1,0,0,-1,-71.95,64.63)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑇</tspan>
			</text>

		</g>
	</g>
</svg> is a positive integer, <svg style="vertical-align:-2.03625pt;width:142.53751px;" id="M5" height="12.1875" version="1.1" viewBox="0 0 142.53751 12.1875" width="142.53751"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,12.1875)">
		<g transform="translate(72,-62.25)">
			<text transform="matrix(1,0,0,-1,-71.95,64.33)">
				<tspan style="font-size: 11.21px; " x="0" y="0">ℤ</tspan>
				<tspan style="font-size: 11.21px; " x="8.1482162" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="11.88048" y="0">1</tspan>
				<tspan style="font-size: 11.21px; " x="17.48448" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="22.147009" y="0">𝑇</tspan>
				<tspan style="font-size: 11.21px; " x="28.311407" y="0">)</tspan>
				<tspan style="font-size: 11.21px; " x="35.159496" y="0">=</tspan>
				<tspan style="font-size: 11.21px; " x="45.952801" y="0">{</tspan>
				<tspan style="font-size: 11.21px; " x="51.332642" y="0">1</tspan>
				<tspan style="font-size: 11.21px; " x="56.936642" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="61.599167" y="0">2</tspan>
				<tspan style="font-size: 11.21px; " x="67.203171" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="71.876907" y="0">…</tspan>
				<tspan style="font-size: 11.21px; " x="84.956642" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="89.619171" y="0">𝑇</tspan>
				<tspan style="font-size: 11.21px; " x="95.783569" y="0">}</tspan>
				<tspan style="font-size: 11.21px; " x="101.16341" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="105.83714" y="0">Δ</tspan>
			</text>

		</g>
	</g>
</svg> is the forward difference operator defined by <svg style="vertical-align:-1.99126pt;width:140.16251px;" id="M6" height="12.075" version="1.1" viewBox="0 0 140.16251 12.075" width="140.16251"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,12.075)">
		<g transform="translate(72,-62.34)">
			<text transform="matrix(1,0,0,-1,-71.95,64.37)">
				<tspan style="font-size: 11.21px; " x="0" y="0">Δ</tspan>
				<tspan style="font-size: 11.21px; " x="8.0921764" y="0">𝑥</tspan>
				<tspan style="font-size: 11.21px; " x="14.256576" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="17.98884" y="0">𝑘</tspan>
				<tspan style="font-size: 11.21px; " x="24.063576" y="0">)</tspan>
				<tspan style="font-size: 11.21px; " x="30.900455" y="0">=</tspan>
				<tspan style="font-size: 11.21px; " x="41.69376" y="0">𝑥</tspan>
				<tspan style="font-size: 11.21px; " x="47.858158" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="51.590424" y="0">𝑘</tspan>
				<tspan style="font-size: 11.21px; " x="60.153336" y="0">+</tspan>
				<tspan style="font-size: 11.21px; " x="70.318993" y="0">1</tspan>
				<tspan style="font-size: 11.21px; " x="75.922989" y="0">)</tspan>
				<tspan style="font-size: 11.21px; " x="82.15464" y="0">−</tspan>
				<tspan style="font-size: 11.21px; " x="92.320297" y="0">𝑥</tspan>
				<tspan style="font-size: 11.21px; " x="98.484695" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="102.21696" y="0">𝑘</tspan>
				<tspan style="font-size: 11.21px; " x="108.29169" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>, and <svg style="vertical-align:-2.10374pt;width:124.4625px;" id="M7" height="12.2125" version="1.1" viewBox="0 0 124.4625 12.2125" width="124.4625"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,12.2125)">
		<g transform="translate(72,-62.23)">
			<text transform="matrix(1,0,0,-1,-71.95,64.37)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑓</tspan>
				<tspan style="font-size: 11.21px; " x="9.3250561" y="0">∶</tspan>
				<tspan style="font-size: 11.21px; " x="18.168167" y="0">ℤ</tspan>
				<tspan style="font-size: 11.21px; " x="26.316383" y="0">(</tspan>
				<tspan style="font-size: 11.21px; " x="30.048649" y="0">1</tspan>
				<tspan style="font-size: 11.21px; " x="35.652649" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="40.326385" y="0">𝑇</tspan>
				<tspan style="font-size: 11.21px; " x="46.490784" y="0">)</tspan>
				<tspan style="font-size: 11.21px; " x="52.711224" y="0">×</tspan>
				<tspan style="font-size: 11.21px; " x="62.37252" y="0">ℝ</tspan>
				<tspan style="font-size: 11.21px; " x="75.732452" y="0">→</tspan>
				<tspan style="font-size: 11.21px; " x="91.087418" y="0">ℝ</tspan>
			</text>

		</g>
	</g>
</svg> is continuous. In argument, Morse inequalities play an important role.

Jianmin Guo

Caixia Guo

Existence and Multiplicity of Solutions for Discrete Nonlinear Two-Point Boundary Value Problems

A class of discrete-time Cohen-Grossberg neural networks with delays is investigated in this paper. By analyzing the corresponding characteristic equations, the asymptotical stability of the null solution and the existence of Neimark-Sacker bifurcations are discussed. By applying the normal form theory and the center manifold theorem, the direction of the Neimark-Sacker bifurcation and the stability of bifurcating periodic solutions are obtained. Numerical simulations are given to illustrate the obtained results.

Qiming Liu

Rui Xu

Zhiping Wang

Stability and Bifurcation of a Class of Discrete-Time Cohen-Grossberg Neural Networks with Delays

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<p><em>Discrete Dynamics in Nature and Society</em> publishes research that links basic and applied research relating to discrete dynamics of complex systems encountered in the natural and social sciences.</p>


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<p>Chief Editor, Dr&nbsp;Renna, is an associate professor at the University of Basilicata, Italy. His research interests include manufacturing systems, production planning and enterprise networks.&nbsp;</p>


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Discrete Dynamics in Nature and Society - Table of contents 2011

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Discrete Dynamics in Nature and Society publishes research that links basic and applied research relating to discrete dynamics of complex systems encountered in the natural and social sciences.

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Discrete Dynamics in Nature and Society