Analysis of intellectual property cooperation behavior based on stochastic catastrophe theory and the QSIM algorithm

Figures & data

Figure 1. The catastrophe model of intellectual property cooperation behavior.

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Figure 2. The qualitative variables evaluated based on fuzzy arithmetic.

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Table 1. The fuzzy value of $QS(f,t).$

$QS(f,t)$	The fuzzy value	$QS(f,t)$	The fuzzy value
<-1,dec>	[-1, 0.8,0.2,0]	<1,dec>	[0.2,0.4,0.4,0.8]
<-1,std>	[-0.6, 0.4,0.4,0.2]	<1,std>	[0.4,0.6,0.2,0.6]
<-1,inc>	[-0.4, 0.2,0.8,0.4]	<1,inc>	[0.8,1,0,0.2]

Source: Authors' creations.

Table 2. The fuzzy value of $QS(u,t)$ and $QS(v,t).$

$QS(u,t)QS(v,t)$	The fuzzy value	$QS(u,t)QS(v,t)$	The fuzzy value	$QS(u,t)QS(v,t)$	The fuzzy value
<-2,dec>	[-1, 1,0,0]	<-1,inc>	[-0.6, 0.6,0,0.2]	< +1,std>	[0.2,0.4,0,0.2]
<-2,std>	[-1, 0.8,0.2,0]	<0,dec>	[-0.4, 0.4,0.4,0]	< +1,inc>	[0.6,0.4,0,0.2]
<-2,inc>	[-0.8, 0.6,0,0.2]	<0,std>	[-0.4,0.2,0.2,0.4]	< +2,dec>	[0.8,0.6,0.2,0]
<-1,dec>	[-0.6, 0.4,0.2,0]	<0,inc>	[0.4,0.4,0,0.4]	< +2,std>	[0.8,1,0.2,0]
<-1,std>	[-0.4, 0.2,0.2,0]	< +1,dec>	[0.6,0.6,0.2,0]	< +2,inc>	[1,1,0,0]

Source: Authors' creations.

Table 3. Reliability test of the antecedent variables.

Antecedent variable	Dimension	The number of items	Cronbach’s α of each item	Cronbach’s α of each variable
Knowledge-sharing willingness	Knowledge-contribute willingness	6	0.747	0.836
	Knowledge-acquire willingness	4	0.748
Enterprise reputation	Emotional reputation	4	0.701	0.762
	Cognitive reputation	4	0.734
	Fair-trade reputation	4	0.714

Source: Authors' creations.

Table 4. Extraction of empirical sample.

Time	Catastrophe behavior sample
$t$	$\left\{QS\prime (f,t),QS\prime (u,t),QS\prime (v,t)\right\}=\left\{(1,\mathit{dec}),(-1,\mathit{dec}),(-1,\mathit{dec})\right\}$
	$\left\{QS\prime (f,t),QS\prime (u,t),QS\prime (v,t)\right\}=\left\{(1,\mathit{dec}),(-1,\mathit{dec}),(-2,\mathit{dec})\right\}$
	$\left\{QS\prime (f,t),QS\prime (u,t),QS\prime (v,t)\right\}=\left\{(1,\mathit{dec}),(-2,\mathit{dec}),(-1,\mathit{dec})\right\}$
	$\left\{QS\prime (f,t),QS\prime (u,t),QS\prime (v,t)\right\}=\left\{(1,\mathit{dec}),(-2,\mathit{dec}),(-2,\mathit{dec})\right\}$
$t+1$	$\left\{QS\prime (f,t+1),QS\prime (u,t+1),QS\prime (v,t+1)\right\}=\left\{(1,\mathit{std}),(-1,\mathit{std}),(-1,\mathit{std})\right\}$
	$\left\{QS\prime (f,t+1),QS\prime (u,t+1),QS\prime (v,t+1)\right\}=\left\{(1,\mathit{std}),(-1,\mathit{std}),(-2,\mathit{std})\right\}$
	$\left\{QS\prime (f,t+1),QS\prime (u,t+1),QS\prime (v,t+1)\right\}=\left\{(1,\mathit{std}),(-2,\mathit{std}),(-1,\mathit{std})\right\}$
	$\left\{QS\prime (f,t+1),QS\prime (u,t+1),QS\prime (v,t+1)\right\}=\left\{(1,\mathit{std}),(-2,\mathit{std}),(-2,\mathit{std})\right\}$

Note: at time $t,$ $u$ and $v$ have two states:$(-1,\mathit{dec})$ and $(-2,\mathit{dec}),$ at time $t+1,$ after the catastrophe phenomenon occurs in enterprises' intellectual property cooperation behavior, the direction of $f$ $u$ and $v$ is a stable state.

Source: Authors' creations.

Table 5. General function state transition.

$QS(X,t)$	$QS(X,t+1)$
$(1,\mathit{inc})$	$(1,\mathit{inc}),(+1,\mathit{std})$
$(1,\mathit{std})$	$(1,\mathit{inc}),(1,\mathit{std}),(1,\mathit{dec})$
$(1,\mathit{dec})$	$(-1,\mathit{std}),(1,\mathit{dec})$

Source: Authors' creations.

Table 6. Fuzzy arithmetic algorithm.

Calculation	Result	Condition
$p+q$	$(e1+e3,e2+e4,\tau +\gamma ,\beta +\delta )$	Random value of$p$and$q$
$p*q$	$(e1e3,e2e4,e1\gamma +e3\tau -\tau \gamma ,e2\delta +e4\beta +\beta \delta )$	$p>0,q>0$
	$(e1e4,e2e3,e4\tau -e1\delta +\tau \delta ,-e2\gamma +e3\beta -\beta \gamma )$	$p<0,q>0$
	$(e2e3,e1e4,e2\gamma -e3\beta +\beta \gamma ,-e4\tau +e1\delta -\tau \delta )$	$p>0,q<0$
	$(e2e4,e1e3,-e2\delta -e4\beta -\beta \delta ,-e1\tau -e3\tau +\tau \gamma )$	$p<0,q<0$
$\mathit{power}(p,q)$	$\left[2(e2-e1)+\beta +\gamma \right]/2$	Random value of$p$and$q$
$\mathit{centre}(p,q)$	$(e1+e2)/2$	Random value of$p$and$q$

Note:$p=[e1,e2,\tau ,\beta ],q=[e3,e4,\gamma ,\delta ].$

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Figure 3. The catastrophe model of intellectual property cooperation behavior.

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Figure 4. $u=2,v=-1,f=1,t=10.$

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Figure 5. $u=-2,v=-1,f=1,t=10.$

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Figure 6. $u=2,v=-1,f=1,t=50.$

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Figure 7. $u=-2,v=-1,f=1,t=50.$

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Figure 8. The division of bifurcation points set.

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Figure 9. $u$ forces on $f.$

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Figure 10. $v$ forces on $f.$

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Figure 11. $u$ forces on $f.$

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Figure 12. $v$ forces on $f.$

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Data availability

Not applicable